Christian Bar¨ Nicolas Ginoux Frank Pfafﬂe¨ · Christian Bar¨ Nicolas Ginoux Frank Pfafﬂe¨...

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Christian BarNicolas GinouxFrank Pfaffle

Wave Equations on LorentzianManifolds and Quantization

http://arXiv.org/abs/0806.1036v1

Preface

In General Relativity spacetime is described mathematically by a Lorentzian manifold.Gravitation manifests itself as the curvature of this manifold. Physical fields, such as theelectromagnetic field, are defined on this manifold and have to satisfy a wave equation.This book provides an introduction to the theory of linear wave equations on Lorentzianmanifolds. In contrast to other texts on this topic [Friedlander1975, Gunther1988] wedevelop the global theory. This means, we ask for existence and uniqueness of solutionswhich are defined on all of the underlying manifold. Such results are of great importanceand are already used much in the literature despite the fact that published proofs aremissing. Tracing back the references one typically ends at Leray’s unpublished lecturenotes [Leray1953] or their exposition [Choquet-Bruhat1968].

In this text we develop the global theory from scratch in a modern geometric language.In the first chapter we provide basic definitions and facts about distributions on manifolds,Lorentzian geometry, and normally hyperbolic operators. We study the building blocksfor local solutions, the Riesz distributions, in some detail. In the second chapter we showhow to solve wave equations locally. Using Riesz distributions and a formal recursive pro-cedure one first constructs formal fundamental solutions. These are formal series solvingthe equations formally but in general they do not converge. Using suitable cut-offs onegets “almost solutions” from these formal solutions. They are well-defined distributionsbut solve the equation only up to an error term. This is then corrected by some furtheranalysis which yields true local fundamental solutions.

This procedure is similar to the construction of the heat kernel for a Laplace typeoperator on a compact Riemannian manifold. The analogy goeseven further. Similar tothe short-time asymptotics for the heat kernel, the formal fundamental solution turns outto be an asymptotic expansion of the true fundamental solution. Along the diagonal thecoefficients of this asymptotic expansion are given by the same algebraic expression inthe curvature of the manifold, the coefficients of the operator, and their derivatives as theheat kernel coefficients.

In the third chapter we use the local theory to study global solutions. This means weconstruct global fundamental solutions, Green’s operators, and solutions to the Cauchyproblem. This requires assumptions on the geometry of the underlying manifold. InLorentzian geometry one has to deal with the problem that there is no good analog forthe notion of completeness of Riemannian manifolds. In our context globally hyperbolicmanifolds turn out to be the right class of manifolds to consider. Most basic models inGeneral Relativity turn out to be globally hyperbolic but there are exceptions such as

iv

anti-deSitter spacetime. This is why we also include a section in which we study caseswhere one can guarantee existence (but not uniqueness) of global solutions on certainnon-globally hyperbolic manifolds.

In the last chapter we apply the analytical results and describe the basic mathematicalconcepts behind field quantization. The aim of quantum field theory on curved spacetimesis to provide a partial unification of General Relativity with Quantum Physics where thegravitational field is left classical while the other fields are quantized. We develop thetheory ofC∗-algebras and CCR-representations in full detail to the extent that we need.Then we construct the quantization functors and check that the Haag-Kastler axioms of alocal quantum field theory are satisfied. We also construct the Fock space and the quantumfield.

From a physical perspective we just enter the door to quantumfield theory but donot go very far. We do not discussn-point functions, states, renormalization, nonlinearfields, nor physical applications such as Hawking radiation. For such topics we refer tothe corresponding literature. However, this book should provide the reader with a firmmathematical basis to enter this fascinating branch of physics.

In the appendix we collect background material on category theory, functional analy-sis, differential geometry, and differential operators that is used throughout the text. Thiscollection of material is included for the convenience of the reader but cannot replacea thorough introduction to these topics. The reader should have some experience withdifferential geometry. Despite the fact that normally hyperbolic operators on Lorentzianmanifolds look formally exactly like Laplace type operators on Riemannian manifoldstheir analysis is completely different. The elliptic theory of Laplace type operators is notneeded anywhere in this text. All results on hyperbolic equations which are relevant tothe subject are developed in full detail. Therefore no priorknowledge on the theory ofpartial differential equations is needed.

Acknowledgements

This book would not have been possible without the help of andthe inspired discus-sions with many colleagues. Special thanks go to Helga Baum,Klaus Fredenhagen, OlafMuller, Miguel Sanchez, Alexander Strohmaier, Rainer Verch, and all the other partici-pants of the workshop on “Hyperbolic operators on Lorentzian manifolds and quantiza-tion” in Vienna, 2005.

We would also like to thank the Schwerpunktprogramm (1154) “Globale Differential-geometrie” funded by the Deutsche Forschungsgemeinschaftfor financial support.

CHRISTIAN BAR, NICOLAS GINOUX , AND FRANK PFAFFLE,Potsdam, October 2006

Contents

Preface iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Preliminaries 11.1 Distributions on manifolds . . . . . . . . . . . . . . . . . . . . . . . .. 11.2 Riesz distributions on Minkowski space . . . . . . . . . . . . . .. . . . 101.3 Lorentzian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Riesz distributions on a domain . . . . . . . . . . . . . . . . . . . . .. . 291.5 Normally hyperbolic operators . . . . . . . . . . . . . . . . . . . . .. . 33

2 The local theory 372.1 The formal fundamental solution . . . . . . . . . . . . . . . . . . . .. . 382.2 Uniqueness of the Hadamard coefficients . . . . . . . . . . . . . .. . . 392.3 Existence of the Hadamard coefficients . . . . . . . . . . . . . . .. . . . 412.4 True fundamental solutions on small domains . . . . . . . . . .. . . . . 442.5 The formal fundamental solution is asymptotic . . . . . . . .. . . . . . 562.6 Solving the inhomogeneous equation on small domains . . .. . . . . . . 62

3 The global theory 673.1 Uniqueness of the fundamental solution . . . . . . . . . . . . . .. . . . 683.2 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3 Fundamental solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.4 Green’s operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.5 Non-globally hyperbolic manifolds . . . . . . . . . . . . . . . . .. . . . 92

4 Quantization 1034.1 C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2 The canonical commutator relations . . . . . . . . . . . . . . . . .. . . 1164.3 Quantization functors . . . . . . . . . . . . . . . . . . . . . . . . . . . .1234.4 Quasi-localC∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.5 Haag-Kastler axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.6 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.7 The quantum field defined by a Cauchy hypersurface . . . . . . .. . . . 147

vi CONTENTS

A Background material 155A.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.2 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157A.3 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .160A.4 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170A.5 More on Lorentzian geometry . . . . . . . . . . . . . . . . . . . . . . . 172

Bibliography 181

Figures 185

Symbols 186

Index 190

Chapter 1

Preliminaries

We want to study solutions to wave equations on Lorentzian manifolds. In this first chap-ter we develop the basic concepts needed for this task. In theappendix the reader will findthe background material on differential geometry, functional analysis and other fields ofmathematics that will be used throughout this text without further comment.

A wave equation is given by a certain differential operator of second order called a “nor-mally hyperbolic operator”. In general, these operators act on sections in vector bundleswhich is the geometric way of saying that we are dealing with systems of equations andnot just with scalar equations. It is important to allow thatthe sections may have certainsingularities. This is why we work with distributional sections rather than with smooth orcontinuous sections only.

The concept of distributions on manifolds is explained in the first section. One nice featureof distributions is the fact that one can apply differentialoperators to them and againobtain a distribution without any further regularity assumption.

The simplest example of a normally hyperbolic operator on a Lorentzian manifold is givenby the d’Alembert operator on Minkowski space. Its fundamental solution, a concept tobe explained later, can be described explicitly. This givesrise to a family of distributionson Minkowski space, the Riesz distributions, which will provide the building blocks forsolutions in the general case later.

After explaining the relevant notions from Lorentzian geometry we will show how to“transplant” Riesz distributions from the tangent space into the Lorentzian manifold. Wewill also derive the most important properties of the Riesz distributions.

1.1 Distributions on manifolds

Let us start by giving some definitions and by fixing the terminology for distributions onmanifolds. We will confine ourselves to those facts that we will actually need later on. Asystematic and much more complete introduction may be founde. g. in [Friedlander1998].

2 Chapter 1. Preliminaries

1.1.1 Preliminaries on distributions

Let M be a manifold equipped with a smooth volume density dV. Lateron we will usethe volume density induced by a Lorentzian metric but this isirrelevant for now. Weconsider a real or complex vector bundleE → M. We will always writeK = R or K = C

depending on whetherE is a real or complex. The space of compactly supported smoothsections inE will be denoted byD(M,E). We equipE andT∗M with connections, bothdenoted by∇. They induce connections on the tensor bundlesT∗M ⊗ ·· · ⊗T∗M ⊗E,again denoted by∇. For a continuously differentiable sectionϕ ∈C1(M,E) the covariantderivative is a continuous section inT∗M⊗E, ∇ϕ ∈ C0(M,T∗M ⊗E). More generally,for ϕ ∈Ck(M,E) we get∇kϕ ∈C0(M,T∗M⊗·· ·⊗T∗M︸︷︷︸

k factors

⊗E).

We choose a Riemannian metric onT∗M and a Riemannian or Hermitian metric onEdepending on whetherE is real or complex. This induces metrics on all bundlesT∗M⊗·· ·⊗T∗M⊗E. Hence the norm of∇kϕ is defined at all points ofM.For a subsetA⊂ M andϕ ∈Ck(M,E) we define theCk-normby

‖ϕ‖Ck(A) := maxj=0,...,k

supx∈A

|∇ jϕ(x)|. (1.1)

If A is compact, then different choices of the metrics and the connections yield equivalentnorms‖ · ‖Ck(A). For this reason there will usually be no need to explicitly specify themetrics and the connections.The elements ofD(M,E) are referred to astest sectionsin E. We define a notion ofconvergence of test sections.

Definition 1.1.1. Let ϕ ,ϕn ∈ D(M,E). We say that the sequence(ϕn)n converges toϕin D(M,E) if the following two conditions hold:

(1) There is a compact setK ⊂ M such that the supports of allϕn are contained inK,i. e., supp(ϕn) ⊂ K for all n.

(2) The sequence(ϕn)n converges toϕ in all Ck-norms overK, i. e., for eachk∈ N

‖ϕ −ϕn‖Ck(K) −→n→∞0.

We fix a finite-dimensionalK-vector spaceW. Recall thatK = R or K = C depending onwhetherE is real or complex.

Definition 1.1.2. A K-linear mapF : D(M,E∗) → W is called adistribution in E withvalues in Wif it is continuous in the sense that for all convergent sequencesϕn → ϕ inD(M,E∗) one hasF [ϕn] → F[ϕ ]. We writeD ′(M,E,W) for the space of allW-valueddistributions inE.

Note that sinceW is finite-dimensional all norms| · | onW yield the same topology onW.Hence there is no need to specify a norm onW for Definition 1.1.2 to make sense. Notemoreover, that distributions inE act on test sections inE∗.

1.1. Distributions on manifolds 3

Lemma 1.1.3. Let F be a W-valued distribution in E and let K⊂ M be compact. Thenthere is a nonnegative integer k and a constant C> 0 such that for allϕ ∈ D(M,E∗) withsupp(ϕ) ⊂ K we have

|F [ϕ ]| ≤C · ‖ϕ‖Ck(K) . (1.2)

The smallestk for which inequality (1.2) holds is called theorderof F overK.

Proof. Assume (1.2) does not hold for any pair ofC and k. Then for every positiveintegerk we can find a nontrivial sectionϕk ∈D(M,E∗) with supp(ϕk)⊂K and|F[ϕk]| ≥k · ‖ϕk‖Ck. We define sectionsψk := 1

|F [ϕk]|ϕk. Obviously, theseψk satisfy supp(ψk) ⊂ Kand

‖ψk‖Ck(K) = 1|F[ϕk]|‖ϕk‖Ck(K) ≤ 1

k .

Hence fork≥ j‖ψk‖C j (K) ≤ ‖ψk‖Ck(K) ≤ 1

k .

Therefore the sequence(ψk)k converges to 0 inD(M,E∗). SinceF is a distribution we

getF [ψk] → F [0] = 0 for k → ∞. On the other hand,|F [ψk]| =∣∣∣ 1|F [ϕk]|F [ϕk]

∣∣∣= 1 for all

k, which yields a contradiction.

Lemma 1.1.3 states that the restriction of any distributionto a (relatively) compact set isof finite order. We say that a distributionF is of orderm if m is the smallest integer suchthat for each compact subsetK ⊂ M there exists a constantC so that

|F [ϕ ]| ≤C · ‖ϕ‖Cm(K)

for all ϕ ∈ D(M,E∗) with supp(ϕ) ⊂ K. Such a distribution extends uniquely to a con-tinuous linear map onDm(M,E∗), the space ofCm-sections inE∗ with compact support.Convergence inDm(M,E∗) is defined similarly to that of test sections. We say thatϕn

converge toϕ in Dm(M,E∗) if the supports of theϕn andϕ are contained in a commoncompact subsetK ⊂ M and‖ϕ −ϕn‖Cm(K) → 0 asn→ ∞.Next we give two important examples of distributions.

Example 1.1.4.Pick a bundleE → M and a pointx∈ M. Thedelta-distributionδx is anE∗

x -valued distribution inE. Forϕ ∈ D(M,E∗) it is defined by

δx[ϕ ] = ϕ(x).

Clearly,δx is a distribution of order 0.

Example 1.1.5. Every locally integrable sectionf ∈ L1loc(M,E) can be interpreted as a

K-valued distribution inE by setting for anyϕ ∈ D(M,E∗)

f [ϕ ] :=∫

Mϕ( f ) dV.

As a distributionf is of order 0.


Lemma 1.1.6. Let M and N be differentiable manifolds equipped with smoothvolumedensities. Let E→ M and F → N be vector bundles. Let K⊂ N be compact and letϕ ∈Ck(M×N,E⊠F∗) be such that supp(ϕ)⊂M×K. Let m≤ k and let T∈D ′(N,F,K)be a distribution of order m. Then the map

f : M → E,

x 7→ T[ϕ(x, ·)],

defines a Ck−m-section in E with support contained in the projection ofsupp(ϕ) to thefirst factor, i. e.,supp( f ) ⊂ x∈ M |∃y ∈ K such that(x,y) ∈ supp(ϕ). In particular, ifϕ is smooth with compact support, and T is any distribution in F, then f is a smoothsection in E with compact support.Moreover, x-derivatives up to order k−m may be interchanged with T . More precisely, ifP is a linear differential operator of order≤ k−m acting on sections in E, then

P f = T[Pxϕ(x, ·)].

HereE ⊠ F∗ denotes the vector bundle overM ×N whose fiber over(x,y) ∈ M ×N isgiven byEx⊗F∗

y .

Proof. There is a canonical isomorphism

Ex⊗Dk(N,F∗) → Dk(N,Ex⊗F∗),

v⊗s 7→ (y 7→ v⊗s(y)).

Thus we can apply idEx ⊗T to ϕ(x, ·) ∈ Dk(N,Ex⊗F∗) ∼= Ex⊗Dk(N,F∗) and we obtain(idEx ⊗T)[ϕ(x, ·)] ∈ Ex. We briefly writeT[ϕ(x, ·)] instead of(idEx ⊗T)[ϕ(x, ·)].To see that the sectionx 7→ T[ϕ(x, ·)] in E is of regularityCk−m we may assume thatM isan open ball inRp and that the vector bundleE → M is trivialized overM, E = M×Kn,because differentiability and continuity are local properties.For fixedy ∈ N the mapx 7→ ϕ(x,y) is aCk-mapU → Kn⊗F∗

y . We perform a Taylorexpansion atx0 ∈U , see [Friedlander1998, p. 38f]. Forx∈U we get

ϕ(x,y)

= ∑|α |≤k−m−1

1α ! D

αx ϕ(x0,y)(x−x0)

α

+ ∑|α |=k−m

k−mα!

∫ 1

0(1− t)k−m−1Dα

x ϕ((1− t)x0+ tx,y)(x−x0)α dt

= ∑|α |≤k−m

1α ! D

αx ϕ(x0,y)(x−x0)

α +

∑|α |=k−m

k−mα !

∫ 1

0(1− t)k−m−1(Dα

x ϕ((1− t)x0+ tx,y)−Dαx ϕ(x0,y)) dt · (x−x0)

α .

Here we used the usual multi-index notation,α = (α1, . . . ,αp) ∈ Np, |α| = α1+ · · ·+αp,

Dαx = ∂ |α|

(∂x1)α1 ···(∂xp)αp , andxα = xα11 · · ·xαp

p . For|α| ≤ k−mwe certainly haveDαx ϕ(·, ·) ∈


Cm(U ×N,Kn⊗F∗) and, in particular,Dαx ϕ(x0, ·) ∈ Dm(N,Kn⊗F∗). We applyT to get

T[ϕ(x, ·)] (1.3)

= ∑|α |≤k−m

1α ! T[Dα

x ϕ(x0, ·)](x−x0)α +

∑|α |=k−m

k−mα ! T

[∫ 1

0(1− t)k−m−1(Dα

x ϕ((1− t)x0+ tx, ·)−Dαx ϕ(x0, ·)) dt

](x−x0)

α .

Restricting thex to a compact convex neighborhoodU ′ ⊂ U of x0 theDαx ϕ(·, ·) and all

their y-derivatives up to orderm areuniformlycontinuous onU ′×K. Givenε > 0 thereexistsδ > 0 so that|∇ j

yDαx ϕ(x,y)−∇ j

yDαx ϕ(x0,y)| ≤ ε

m+1 whenever|x− x0| ≤ δ , j =0, . . . ,m. Thus forx with |x−x0| ≤ δ

∥∥∥∥∫ 1

0(1− t)k−m−1(Dα

x ϕ((1− t)x0+ tx, ·)−Dαx ϕ(x0, ·)) dt

∥∥∥∥Cm(M)

=

∥∥∥∥∫ 1

0(1− t)k−m−1(Dα

x ϕ((1− t)x0+ tx, ·)−Dαx ϕ(x0, ·)) dt

∥∥∥∥Cm(K)

≤∫ 1

0(1− t)k−m−1‖Dα

x ϕ((1− t)x0+ tx, ·)−Dαx ϕ(x0, ·)‖Cm(K) dt

≤∫ 1

0(1− t)k−m−1ε dt

=ε

k−m.

SinceT is of orderm this implies in (1.3) thatT[∫ 1

0 · · · dt] → 0 asx → x0. Thereforethe mapx 7→ T[ϕ(x, ·)] is k−m times differentiable with derivativesDα

x |x=x0T[ϕ(x, ·)] =T[Dα

x ϕ(x0, ·)]. The same argument also shows that these derivatives are continuous inx.

1.1.2 Differential operators acting on distributions

Let E andF be twoK-vector bundles over the manifoldM, K = R or K = C. Consider alinear differential operatorP :C∞(M,E)→C∞(M,F). There is a unique linear differentialoperatorP∗ : C∞(M,F∗) → C∞(M,E∗) called theformal adjointof P such that for anyϕ ∈ D(M,E) andψ ∈ D(M,F∗)

∫

Mψ(Pϕ) dV =

∫

M(P∗ψ)(ϕ) dV. (1.4)

If P is of orderk, then so isP∗ and (1.4) holds for allϕ ∈Ck(M,E) andψ ∈ Ck(M,F∗)such that supp(ϕ)∩ supp(ψ) is compact. With respect to the canonical identificationE = (E∗)∗ we have(P∗)∗ = P.Any linear differential operatorP : C∞(M,E) →C∞(M,F) extends canonically to a linearoperatorP : D ′(M,E,W) → D ′(M,F,W) by

(PT)[ϕ ] := T[P∗ϕ ]


whereϕ ∈ D(M,F∗). If a sequence(ϕn)n converges inD(M,F∗) to 0, then the sequence(P∗ϕn)n converges to 0 as well becauseP∗ is a differential operator. Hence(PT)[ϕn] =T[P∗ϕn] → 0. ThereforePT is again a distribution.The mapP : D ′(M,E,W) → D ′(M,F,W) is K-linear. If P is of orderk andϕ is aCk-section inE, seen as aK-valued distribution inE, then the distributionPϕ coincides withthe continuous section obtained by applyingP to ϕ classically.An important special case occurs whenP is of order 0, i. e.,P ∈ C∞(M,Hom(E,F)).ThenP∗ ∈ C∞(M,Hom(F∗,E∗)) is the pointwise adjoint. In particular, for a functionf ∈C∞(M) we have

( f T)[ϕ ] = T[ f ϕ ].

1.1.3 Supports

Definition 1.1.7. Thesupportof a distributionT ∈ D ′(M,E,W) is defined as the set

supp(T)

:= x∈ M |∀ neighborhoodU of x ∃ϕ ∈ D(M,E) with supp(ϕ) ⊂U andT[ϕ ] 6= 0.

It follows from the definition that the support ofT is a closed subset ofM. In caseT is aL1

loc-section this notion of support coincides with the usual onefor sections.If for ϕ ∈ D(M,E∗) the supports ofϕ andT are disjoint, thenT[ϕ ] = 0. Namely, foreachx∈ supp(ϕ) there is a neighborhoodU of x such thatT[ψ ] = 0 whenever supp(ψ)⊂U . Cover the compact set supp(ϕ) by finitely many such open setsU1, . . . ,Uk. Using apartition of unity one can writeϕ = ψ1+ · · ·+ψk with ψ j ∈D(M,E∗) and supp(ψ j)⊂U j .Hence

T[ϕ ] = T[ψ1 + · · ·+ ψk] = T[ψ1]+ · · ·+T[ψk] = 0.

Be aware that it is not sufficient to assume thatϕ vanishes on supp(T) in order to ensureT[ϕ ] = 0. For example, ifM = R andE is the trivial K-line bundle letT ∈ D ′(R,K)be given byT[ϕ ] = ϕ ′(0). Then supp(T) = 0 but T[ϕ ] = ϕ ′(0) may well be nonzerowhile ϕ(0) = 0.If T ∈ D ′(M,E,W) and ϕ ∈ C∞(M,E∗), then the evaluationT[ϕ ] can be defined ifsupp(T)∩ supp(ϕ) is compact even if the support ofϕ itself is noncompact. To do thispick a functionσ ∈ D(M,R) that is constant 1 on a neighborhood of supp(T)∩supp(ϕ)and put

T[ϕ ] := T[σϕ ].

This definition is independent of the choice ofσ since for another choiceσ ′ we have

T[σϕ ]−T[σ ′ϕ ] = T[(σ −σ ′)ϕ ] = 0

because supp((σ −σ ′)ϕ) and supp(T) are disjoint.Let T ∈ D ′(M,E,W) and letΩ ⊂ M be an open subset. Each test sectionϕ ∈ D(Ω,E∗)can be extended by 0 and yields a test sectionϕ ∈ D(M,E∗). This defines an embed-ding D(Ω,E∗) ⊂ D(M,E∗). By the restriction ofT to Ω we mean its restriction fromD(M,E∗) to D(Ω,E∗).


Definition 1.1.8. Thesingular supportsingsupp(T) of a distributionT ∈ D ′(M,E,W) isthe set of points which do not have a neighborhood restrictedto whichT coincides witha smooth section.

The singular support is also closed and we always have singsupp(T) ⊂ supp(T).

Example 1.1.9.For the delta-distributionδx we have supp(δx) = singsupp(δx) = x.

1.1.4 Convergence of distributions

The spaceD ′(M,E) of distributions inE will always be given theweak topology. Thismeans thatTn → T in D ′(M,E,W) if and only if Tn[ϕ ] → T[ϕ ] for all ϕ ∈ D(M,E∗).Linear differential operatorsP are always continuous with respect to the weak topology.Namely, ifTn → T, then we have for everyϕ ∈ D(M,E∗)

PTn[ϕ ] = Tn[P∗ϕ ] → T[P∗ϕ ] = PT[ϕ ].

HencePTn → PT.

Lemma 1.1.10.Let Tn,T ∈C0(M,E) and suppose‖Tn−T‖C0(M) → 0. Consider Tn andT as distributions.Then Tn → T in D ′(M,E). In particular, for every linear differential operator P wehavePTn → PT.

Proof. Let ϕ ∈ D(M,E). Since‖Tn−T‖C0(M) → 0 andϕ ∈ L1(M,E), it follows fromLebesgue’s dominated convergence theorem:

limn→∞

Tn[ϕ ] = limn→∞

∫

MTn(x) ·ϕ(x) dV(x)

=∫

Mlimn→∞

(Tn(x) ·ϕ(x)) dV(x)

=

∫

M( limn→∞

Tn(x)) ·ϕ(x) dV(x)

=

∫

MT(x) ·ϕ(x) dV(x)

= T[ϕ ].

1.1.5 Two auxiliary lemmas

The following situation will arise frequently. LetE, F , andG beK-vector bundles overM equipped with metrics and with connections which we all denote by∇. We giveE⊗FandF∗ ⊗G the induced metrics and connections. Here and henceforthF∗ will denotethe dual bundle toF . The natural pairingF ⊗F∗ → K given by evaluation of the secondfactor on the first yields a vector bundle homomorphismE⊗F ⊗F∗⊗G→ E⊗G whichwe write asϕ ⊗ψ 7→ ϕ ·ψ . 1

1If one identifiesE⊗F with Hom(E∗,F) andF∗⊗G with Hom(F,G), thenϕ ·ψ corresponds toψ ϕ .


Lemma 1.1.11.For all Ck-sectionsϕ in E⊗F andψ in F∗⊗G and all A⊂ M we have

‖ϕ ·ψ‖Ck(A) ≤ 2k · ‖ϕ‖Ck(A) · ‖ψ‖Ck(A).

Proof. The casek = 0 follows from the Cauchy-Schwarz inequality. Namely, for fixedx∈M we choose an orthonormal basisfi , i = 1, . . . , r, for Fx. Let f ∗i be the basis ofF∗

x dualto fi . We writeϕ(x) = ∑r

i=1ei ⊗ fi for suitableei ∈ Ex and similarlyψ(x) = ∑ri=1 f ∗i ⊗gi,

gi ∈ Gx. Thenϕ(x) ·ψ(x) = ∑ri=1ei ⊗gi and we see

|ϕ(x) ·ψ(x)|2 = |r

∑i=1

ei ⊗gi|2

=r

∑i, j=1

〈ei ⊗gi,ej ⊗g j〉

=r

∑i, j=1

〈ei ,ej〉〈gi ,g j〉

≤√

r

∑i, j=1

〈ei ,ej〉2 ·√

r

∑i, j=1

〈gi,g j〉2

≤√

r

∑i, j=1

|ei |2|ej |2 ·√

r

∑i, j=1

|gi |2|g j |2

=

√r

∑i=1

|ei |2r

∑j=1

|ej |2 ·√

r

∑i=1

|gi |2r

∑j=1

|g j |2

=r

∑i=1

|ei |2 ·r

∑i=1

|gi|2

= |ϕ(x)|2 · |ψ(x)|2.

Now we proceed by induction onk.

‖∇k+1(ϕ ·ψ)‖C0(A) ≤ ‖∇(ϕ ·ψ)‖Ck(A)

= ‖(∇ϕ) ·ψ + ϕ ·∇ψ‖Ck(A)

≤ ‖(∇ϕ) ·ψ‖Ck(A) +‖ϕ ·∇ψ‖Ck(A)

≤ 2k · ‖∇ϕ‖Ck(A) · ‖ψ‖Ck(A) +2k · ‖ϕ‖Ck(A) · ‖∇ψ‖Ck(A)

≤ 2k · ‖ϕ‖Ck+1(A) · ‖ψ‖Ck+1(A) +2k · ‖ϕ‖Ck+1(A) · ‖ψ‖Ck+1(A)

= 2k+1 · ‖ϕ‖Ck+1(A) · ‖ψ‖Ck+1(A).

Thus

‖ϕ ·ψ‖Ck+1(A) = max‖ϕ ·ψ‖Ck(A),‖∇k+1(ϕ ·ψ)‖C0(A)≤ max2k · ‖ϕ‖Ck(A) · ‖ψ‖Ck(A),2

k+1 · ‖ϕ‖Ck+1(A) · ‖ψ‖Ck+1(A)= 2k+1 · ‖ϕ‖Ck+1(A) · ‖ψ‖Ck+1(A).


This lemma allows us to estimate theCk-norm of products of sections in terms of theCk-norms of the factors. The next lemma allows us to deal with compositions of functions.We recursively define the following universal constants:

α(k,0) := 1,

α(k, j) := 0

for j > k and for j < 0 and

α(k+1, j) := maxα(k, j), 2k ·α(k, j −1) (1.5)

if 1 ≤ j ≤ k. The precise values of theα(k, j) are not important. The definition was madein such a way that the following lemma holds.

Lemma 1.1.12.Let Γ be a real valued Ck-function on a Lorentzian manifold M and letσ : R → R be a Ck-function. Then for all A⊂ M and I⊂ R such thatΓ(A) ⊂ I we have

‖σ Γ‖Ck(A) ≤ ‖σ‖Ck(I) · maxj=0,...,k

α(k, j)‖Γ‖ jCk(A)

.

Proof. We again perform an induction onk. The casek = 0 is obvious. By Lemma 1.1.11

‖∇k+1(σ Γ)‖C0(A) = ‖∇k[(σ ′ Γ) ·∇Γ]‖C0(A)

≤ ‖(σ ′ Γ) ·∇Γ‖Ck(A)

≤ 2k · ‖σ ′ Γ‖Ck(A) · ‖∇Γ‖Ck(A)

≤ 2k · ‖σ ′ Γ‖Ck(A) · ‖Γ‖Ck+1(A)

≤ 2k · ‖σ ′‖Ck(I) · maxj=0,...,k

α(k, j)‖Γ‖ jCk+1(A)

· ‖Γ‖Ck+1(A)

≤ 2k · ‖σ‖Ck+1(I) · maxj=0,...,k

α(k, j)‖Γ‖ j+1Ck+1(A)

= 2k · ‖σ‖Ck+1(I) · maxj=1,...,k+1

α(k, j −1)‖Γ‖ jCk+1(A)

.

Hence

‖σ Γ‖Ck+1(A) = max‖σ Γ‖Ck(A),‖∇k+1(σ Γ)‖C0(A)≤ max‖σ‖Ck(I) · max

j=0,...,kα(k, j)‖Γ‖ j

Ck(A),

2k · ‖σ‖Ck+1(I) · maxj=1,...,k+1

α(k, j −1)‖Γ‖ jCk+1(A)

≤ ‖σ‖Ck+1(I) · maxj=0,...,k+1

maxα(k, j),2kα(k, j −1)‖Γ‖ jCk+1(A)

= ‖σ‖Ck+1(I) · maxj=0,...,k+1

α(k+1, j)‖Γ‖ jCk+1(A)

.


1.2 Riesz distributions on Minkowski space

The distributionsR+(α) andR−(α) to be defined below were introduced by M. Riesz inthe first half of the 20th century in order to find solutions to certain differential equations.He collected his results in [Riesz1949]. We will derive all relevant facts in full detail.LetV be ann-dimensional real vector space, let〈·, ·〉 be a nondegenerate symmetric bilin-ear form of index 1 onV. Hence(V,〈·, ·〉) is isometric ton-dimensional Minkowski space(Rn,〈·, ·〉0) where〈x,y〉0 = −x1y1 +x2y2 + · · ·+xnyn. Set

γ : V → R, γ(X) := −〈X,X〉. (1.6)

A nonzero vectorX ∈ V \ 0 is calledtimelike(or lightlike or spacelike) if and only ifγ(X) > 0 (or γ(X) = 0 or γ(X) < 0 respectively). The zero vectorX = 0 is consideredas spacelike. The setI(0) of timelike vectors consists of two connected components.We choose atimeorientationon V by picking one of these two connected components.Denote this component byI+(0) and call its elementsfuture directed. PutJ+(0) := I+(0),C+(0) := ∂ I+(0), I−(0) := −I+(0), J−(0) := −J+(0), andC−(0) := −C+(0).

b0

C+(0)

I+(0)

C−(0)

I−(0)

Fig. 1: Light cone in Minkowski space

Definition 1.2.1. For any complex numberα with Re(α) > n let R+(α) andR−(α) bethe complex-valued continuous functions onV defined by

R±(α)(X) :=

C(α,n)γ(X)

α−n2 , if X ∈ J±(0),

0, otherwise,

whereC(α,n) := 21−α π2−n

2

( α2 −1)!( α−n

2 )!andz 7→ (z−1)! is the Gamma function.

1.2. Riesz distributions on Minkowski space 11

For α ∈ C with Re(α) ≤ n this definition no longer yields continuous functions dueto the singularities alongC±(0). This requires a more careful definition ofR±(α) as adistribution which we will give below. Even forRe(α) > n we will from now on considerthe continuous functionsR±(α) as distributions as explained in Example 1.1.5.Since the Gamma function has no zeros the mapα 7→C(α,n) is holomorphic onC. Hencefor each fixed testfunctionϕ ∈ D(V,C) the mapα 7→ R±(α)[ϕ ] yields a holomorphicfunction onRe(α) > n.There is a natural differential operator acting on functions onV, f := ∂e1∂e1 f −∂e2∂e2 f −·· ·− ∂en∂en f wheree1, . . . ,en is any basis ofV such that−〈e1,e1〉 = 〈e2,e2〉 =· · · = 〈en,en〉 = 1 and〈ei ,ej〉= 0 for i 6= j. Such a basise1, . . . ,en is calledLorentzian or-thonormal. The operator is called thed’Alembert operator. The formula in Minkowskispace with respect to the standard basis may look more familiar to the reader,

=∂ 2

(∂x1)2 −∂ 2

(∂x2)2 −·· ·− ∂ 2

(∂xn)2 .

The definition of the d’Alembertian on general Lorentzian manifolds can be found in thenext section. In the following lemma the application of differential operators such as totheR±(α) is to be taken in the distributional sense.

Lemma 1.2.2. For all α ∈ C with Re(α) > n we have

(1) γ ·R±(α) = α(α −n+2)R±(α +2),

(2) (gradγ) ·R±(α) = 2α gradR±(α +2),

(3) R±(α +2) = R±(α).

(4) The mapα 7→ R±(α) extends uniquely toC as a holomorphic family of distribu-tions. In other words, for eachα ∈ C there exists a unique distribution R±(α) onV such that for each testfunctionϕ the mapα 7→ R±(α)[ϕ ] is holomorphic.

Proof. Identity (1) follows from

C(α,n)

C(α +2,n)=

2(1−α) (α+22 −1)! (α+2−n

2 )!

2(1−α−2) (α2 −1)! (α−n

2 )!= α (α −n+2).

To show (2) we choose a Lorentzian orthonormal basise1, . . . ,en of V and we denotedifferentiation in directionei by ∂i . We fix a testfunctionϕ and integrate by parts:

∂iγ ·R±(α)[ϕ ] = C(α,n)∫

J±(0)γ(X)

α−n2 ∂iγ(X)ϕ(X) dX

=2C(α,n)

α +2−n

∫

J±(0)∂i(γ(X)

α−n+22 )ϕ(X) dX

= −2αC(α +2,n)∫

J±(0)γ(X)

α−n+22 ∂iϕ(X) dX

= −2αR±(α +2)[∂iϕ ]

= 2α∂iR±(α +2)[ϕ ],


which proves (2). Furthermore, it follows from (2) that

∂ 2i R±(α +2) = ∂i

(1

2α∂iγ ·R±(α)

)

=1

2α

(∂ 2

i γ ·R±(α)+ ∂iγ ·(

12(α −2)

∂iγ ·R±(α −2)

))

=1

2α∂ 2

i γ ·R±(α)+1

4α(α −2)(∂iγ)2 (α −2)(α −n)

γ·R±(α)

=

(1

2α∂ 2

i γ +α −n4α

· (∂iγ)2

γ

)·R±(α),

so that

R±(α +2) =

(nα

+α −n4α

· 4γγ

)R±(α)

= R±(α).

To show (4) we first note that for fixedϕ ∈ D(V,C) the mapRe(α) > n → C, α 7→R±(α)[ϕ ], is holomorphic. ForRe(α) > n−2 we set

R±(α) := R±(α +2). (1.7)

This defines a distribution onV. The mapα 7→ R±(α) is then holomorphic onRe(α) >n− 2. By (3) we haveR±(α) = R±(α) for Re(α) > n, so thatα 7→ R±(α) extendsα 7→R±(α) holomorphically toRe(α) > n−2. We proceed inductively and construct aholomorphic extension ofα 7→R±(α) onRe(α) > n−2k (wherek∈N\0) from thaton Re(α) > n−2k+ 2 just as above. Note that these extensions necessarily coincideon their common domain since they are holomorphic and they coincide on an open subsetof C. We therefore obtain a holomorphic extension ofα 7→ R±(α) to the whole ofC,which is necessarily unique.

Lemma 1.2.2 (4) definesR±(α) for all α ∈ C, not as functions but as distributions.

Definition 1.2.3. We callR+(α) theadvanced Riesz distributionandR−(α) theretardedRiesz distributiononV for α ∈ C.

The following illustration shows the graphs of Riesz distributionsR+(α) for n = 2 andvarious values ofα. In particular, one sees the singularities alongC+(0) for Re(α) ≤ 2.

α = 0.1 α = 1


α = 2 α = 3

α = 4 α = 5

Fig. 2: Graphs of Riesz distributionsR+(α) in two dimensions

We now collect the important facts on Riesz distributions.

Proposition 1.2.4. The following holds for allα ∈ C:

(1) γ ·R±(α) = α(α −n+2)R±(α +2),

(2) (gradγ)R±(α) = 2α grad(R±(α +2)),

(3) R±(α +2) = R±(α),

(4) For everyα ∈ C\ (0,−2,−4, . . .∪n−2,n−4, . . .), we havesupp(R±(α)) = J±(0) andsingsupp(R±(α)) ⊂C±(0).

(5) For everyα ∈ 0,−2,−4, . . .∪n−2,n−4, . . ., we havesupp(R±(α)) = singsupp(R±(α)) ⊂C±(0).

(6) For n≥ 3 andα = n−2,n−4, . . .,1 or 2 respectively, we havesupp(R±(α)) = singsupp(R±(α)) = C±(0).

(7) R±(0) = δ0.

(8) For Re(α) > 0 the order of R±(α) is bounded from above by n+1.

(9) If α ∈ R, then R±(α) is real, i. e., R±(α)[ϕ ] ∈ R for all ϕ ∈ D(V,R).

Proof. Assertions (1), (2), and (3) hold forRe(α) > n by Lemma 1.2.2. Since, afterinsertion of a fixedϕ ∈ D(V,C), all expressions in these equations are holomorphic inαthey hold for allα.


Proof of (4). Letϕ ∈ D(V,C) with supp(ϕ)∩ J±(0) = /0. Since supp(R±(α)) ⊂ J±(0)for Re(α) > n, it follows for thoseα that

R±(α)[ϕ ] = 0,

and then for allα by Lemma 1.2.2 (4). Therefore supp(R±(α)) ⊂ J±(0) for all α.

On the other hand, ifX ∈ I±(0), thenγ(X) > 0 and the mapα 7→ C(α,n)γ(X)α−n

2 iswell-defined and holomorphic on all ofC. By Lemma 1.2.2 (4) we have forϕ ∈ D(V,C)with supp(ϕ) ⊂ I±(0)

R±(α)[ϕ ] =

∫

supp(ϕ)C(α,n)γ(X)

α−n2 ϕ(X)dX

for every α ∈ C. Thus R±(α) coincides on I±(0) with the smooth functionC(α,n)γ(·) α−n

2 and therefore singsupp(R±(α)) ⊂ C±(0). Since furthermore the func-tion α 7→ C(α,n) vanishes only on0,−2,−4, . . . ∪ n− 2,n− 4, . . . (caused bythe poles of the Gamma function), we haveI±(0) ⊂ supp(R±(α)) for every α ∈ C \(0,−2,−4, . . .∪n−2,n−4, . . .). Thus supp(R±(α)) = J±(0). This proves (4).Proof of (5). Forα ∈ 0,−2,−4, . . . ∪ n− 2,n− 4, . . . we haveC(α,n) = 0 andthereforeI±(0)∩ supp(R±(α)) = /0. Hence singsupp(R±(α)) ⊂ supp(R±(α)) ⊂ C±(0).It remains to show supp(R±(α)) ⊂ singsupp(R±(α)). Let X 6∈ singsupp(R±(α)).Then R±(α) coincides with a smooth functionf on a neighborhood ofX. Sincesupp(R±(α)) ⊂ C±(0) and sinceC±(0) has a dense complement inV, we havef ≡ 0.ThusX 6∈ supp(R±(α)). This proves (5).Before we proceed to the next point we derive a more explicit formula for the Rieszdistributions evaluated on testfunctions of a particular form. Introduce linear coordinatesx1, . . . ,xn onV such thatγ(x) = −(x1)2 +(x2)2 + · · ·+(xn)2 and such that thex1-axis isfuture directed. Letf ∈ D(R,C) andψ ∈ D(Rn−1,C) and putϕ(x) := f (x1)ψ(x) wherex = (x2, . . . ,xn). Choose the functionψ such that onJ+(0) we haveϕ(x) = f (x1).

Rn−1

x1

supp( f )

| |

ψ ≡ 1

Fig. 3: Support ofϕ


Claim: If Re(α) > 1, then

R+(α)[ϕ ] =1

(α −1)!

∫ ∞

0rα−1 f (r)dr.

Proof of the Claim. Since both sides of the equation are holomorphic inα for Re(α) > 1it suffices to show it forRe(α) > n. In that case we have by the definition ofR+(α)

R+(α)[ϕ ] = C(α,n)

∫

J+(0)ϕ(X)γ(X)

α−n2 dX

= C(α,n)

∫ ∞

0

∫

|x|<x1ϕ(x1, x)((x1)2−|x|2) α−n

2 dx dx1

= C(α,n)∫ ∞

0f (x1)

∫

|x|<x1((x1)2−|x|2) α−n

2 dx dx1

= C(α,n)

∫ ∞

0f (x1)

∫ x1

0

∫

Sn−2((x1)2− t2)

α−n2 tn−2dω dt dx1,

whereSn−2 is the(n−2)-dimensional round sphere anddω its standard volume element.Renamingx1 we get

R+(α)[ϕ ] = vol(Sn−2)C(α,n)

∫ ∞

0f (r)

∫ r

0(r2− t2)

α−n2 tn−2dt dr.

Using∫ r

0 (r2− t2)α−n

2 tn−2dt = 12rα−1 ( α−n

2 )!( n−32 )!

( α−12 )!

we obtain

R+(α)[ϕ ] =vol(Sn−2)

2C(α,n)

∫ ∞

0f (r)rα−1 (α−n

2 )!(n−32 )!

(α−12 )!

dr

=12

2π (n−1)/2

(n−12 −1)!

· 21−απ1−n/2

(α/2−1)!(α−n2 )!

· (α−n2 )!(n−3

2 )!

(α−12 )!

·∫ ∞

0f (r)rα−1dr

=

√π ·21−α

(α/2−1)!(α−12 )!

·∫ ∞

0f (r)rα−1dr.

Legendre’s duplication formula (see [Jeffrey1995, p. 218])

(α2−1)

!

(α +1

2−1

)! = 21−α√π (α −1)! (1.8)

yields the Claim.To show (6) recall first from (5) that we know already

singsupp(R±(α)) = supp(R±(α)) ⊂C±(0)

for α = n− 2,n− 4, . . . ,2 or 1 respectively. Note also that the distributionR±(α) isinvariant under timeorientation-preserving Lorentz transformations, that is, for any suchtransformationA of V we have

R±(α)[ϕ A] = R±(α)[ϕ ]


for every testfunctionϕ . Hence supp(R±(α)) as well as singsupp(R±(α)) are also in-variant under the group of those transformations. Under theaction of this group the orbitdecomposition ofC±(0) is given by

C±(0) = 0∪ (C±(0)\ 0).

Thus supp(R±(α)) = singsupp(R±(α)) coincides either with0 or with C±(0).

The Claim shows for the testfunctionsϕ considered there

R+(2)[ϕ ] =

∫ ∞

0r f (r)dr.

Hence the support ofR+(2) cannot be contained in0. If n is even, we concludesupp(R+(2)) = C+(0) and then also supp(R+(α)) = C+(0) for α = 2,4, . . . ,n−2.

Taking the limitα ց 1 in the Claim yields

R+(1)[ϕ ] =

∫ ∞

0f (r)dr.

Now the same argument shows for oddn that supp(R+(1)) = C+(0) and then alsosupp(R+(α)) = C+(0) for α = 1,3, . . . ,n−2. This concludes the proof of (6).

Proof of (7). Fix a compact subsetK ⊂ V. Let σK ∈ D(V,R) be a function such thatσ|K ≡ 1. For anyϕ ∈ D(V,C) with supp(ϕ) ⊂ K write

ϕ(x) = ϕ(0)+n

∑j=1

x jϕ j(x)

with suitable smooth functionsϕ j . Then

R±(0)[ϕ ] = R±(0)[σKϕ ]

= R±(0)[ϕ(0)σK +n

∑j=1

x jσKϕ j ]

= ϕ(0)R±(0)[σK ]︸︷︷︸=:cK

+n

∑j=1

(x jR±(0))︸︷︷︸=0 by (2)

[σKϕ j ]

= cKϕ(0).

The constantcK actually does not depend onK since forK′ ⊃ K and supp(ϕ)⊂ K(⊂ K′),

cK′ϕ(0) = R+(0)[ϕ ] = cKϕ(0),

so thatcK = cK′ =: c. It remains to showc = 1.

1.3. Lorentzian geometry 17

We again look at testfunctionsϕ as in the Claim and compute using (3)

c ·ϕ(0) = R+(0)[ϕ ]

= R+(2)[ϕ ]

=

∫ ∞

0r f ′′(r)dr

= −∫ ∞

0f ′(r)dr

= f (0)

= ϕ(0).

This concludes the proof of (7).Proof of (8). By its definition, the distributionR±(α) is a continuous function ifRe(α) >n, therefore it is of order 0. Since is a differential operator of order 2, the order ofR±(α) is at most that ofR±(α) plus 2. It then follows from (3) that:• If n is even: for everyα with Re(α) > 0 we haveRe(α)+ n = Re(α)+ 2 · n

2 > n, sothat the order ofR±(α) is not greater thann (and son+1).• If n is odd: for everyα with Re(α) > 0 we haveRe(α)+n+1= Re(α)+2· n+1

2 > n,so that the order ofR±(α) is not greater thann+1.This concludes the proof of (8).Assertion (9) is clear by definition wheneverα > n. For generalα ∈ R choosek ∈ N solarge thatα +2k > n. Using (3) we get for anyϕ ∈ D(V,R)

R±(α)[ϕ ] = kR±(α +2k)[ϕ ] = R±(α +2k)[kϕ ] ∈ R

becausekϕ ∈ D(V,R) as well.

In the following we will need a slight generalization of Lemma 1.2.2 (4):

Corollary 1.2.5. For ϕ ∈ Dk(V,C) the mapα 7→ R±(α)[ϕ ] defines a holomorphic func-tion onα ∈ C | Re(α) > n−2[ k

2].

Proof. Let ϕ ∈ Dk(V,C). By the definition ofR±(α) the mapα 7→ R±(α)[ϕ ] is clearlyholomorphic onRe(α) > n. Using (3) of Proposition 1.2.4 we get the holomorphicextension to the setRe(α) > n−2[ k

2].

1.3 Lorentzian geometry

We now summarize basic concepts of Lorentzian geometry. We will assume famil-iarity with semi-Riemannian manifolds, geodesics, the Riemannian exponential mapetc. A summary of basic notions in differential geometry canbe found in Ap-pendix A.3. A thorough introduction to Lorentzian geometrycan e. g. be found in[Beem-Ehrlich-Easley1996] or in [O’Neill1983]. Further results of more technical na-ture which could distract the reader at a first reading but which will be needed later arecollected in Appendix A.5.


Let M be a timeoriented Lorentzian manifold. A piecewiseC1-curve inM is calledtime-like, lightlike, causal, spacelike, future directed, or past directedif its tangent vectorsare timelike, lightlike, causal, spacelike, future directed, or past directed respectively. ApiecewiseC1-curve inM is calledinextendible, if no piecewiseC1-reparametrization ofthe curve can be continuously extended to any of the end points of the parameter interval.

Thechronological futureIM+ (x) of a pointx ∈ M is the set of points that can be reached

from x by future directed timelike curves. Similarly, thecausal futureJM+ (x) of a point

x ∈ M consists of those points that can be reached fromx by causal curves and ofxitself. In the following, the notationx < y (or x ≤ y) will meany∈ IM

+ (x) (or y ∈ JM+ (x)

respectively). Thechronological futureof a subsetA ⊂ M is defined to beIM+ (A) := ∪

x∈A

IM+ (x). Similarly, thecausal futureof A is JM

+ (A) := ∪x∈A

JM+ (x). Thechronological past

IM− (A) and thecausal pastJM

− (A) are defined by replacing future directed curves by pastdirected curves. One has in general thatIM

± (A) is the interior ofJM± (A) and thatJM

± (A) iscontained in the closure ofIM

± (A). The chronological future and past are open subsets butthe causal future and past are not always closed even ifA is closed (see also Section A.5in Appendix).

A

JM+ (A)

IM+ (A)

bc

JM− (A)

IM− (A)

Fig. 4: Causal and chronological future and past of subsetA of Minkowski space with one point removed

We will also use the notationJM(A) := JM− (A)∪ JM

+ (A). A subsetA ⊂ M is calledpastcompactif A∩ JM

− (p) is compact for allp ∈ M. Similarly, one definesfuture compactsubsets.


A

bp

JM− (p)

Fig. 5: The subsetA is past compact

Definition 1.3.1. A subsetΩ⊂M in a timeoriented Lorentzian manifold is calledcausallycompatibleif for all pointsx∈ Ω

JΩ±(x) = JM

± (x)∩Ω

holds.

Note that the inclusion “⊂” always holds. The condition of being causally compatiblemeans that whenever two points inΩ can be joined by a causal curve inM this can alsobe done insideΩ.

p

JM+ (p)∩Ω = JΩ

+(p)

ΩFig. 6: Causally compatible subset of Minkowski space

+p

JM+ (p)∩Ω

+p

JΩ+(p)

Fig. 7: Domain which is not causally compatible in Minkowskispace


If Ω ⊂ M is a causally compatible domain in a timeoriented Lorentzian manifold, then weimmediately see that for each subsetA⊂ Ω we have

JΩ±(A) = JM

± (A)∩Ω.

Note also that being causally compatible is transitive: IfΩ ⊂ Ω′ ⊂ Ω′′, if Ω is causallycompatible inΩ′, and ifΩ′ is causally compatible inΩ′′, then so isΩ in Ω′′.

Definition 1.3.2. A domainΩ ⊂ M in a Lorentzian manifold is called

• geodesically starshapedwith respect to a fixed pointx∈ Ω if there exists an opensubsetΩ′ ⊂ TxM, starshaped with respect to 0, such that the Riemannian exponen-tial map expx mapsΩ′ diffeomorphically ontoΩ.

• geodesically convex(or simplyconvex) if it is geodesically starshaped with respectto all of its points.

Ω′

Ω

M

TxM

x

0

expx

b

b

Fig. 8: Ω is geodesically starshaped w. r. t.x

If Ω is geodesically starshaped with respect tox, then expx(I±(0)∩ Ω′) = IΩ± (x) and

expx(J±(0)∩Ω′) = JΩ±(x). We putCΩ

±(x) := expx(C±(0)∩Ω′).On a geodesically starshaped domainΩ we define the smooth positive functionµx : Ω →R by

dV = µx · (exp−1x )∗ (dz) , (1.9)

where dV is the Lorentzian volume density anddz is the standard volume densityon TxΩ. In other words,µx = det(dexpx) exp−1

x . In normal coordinates aboutx,µx =

√|det(gi j )|.

For each open covering of a Lorentzian manifold there existsa refinement consisting ofconvex open subsets, see [O’Neill1983, Chap. 5, Lemma 10].

Definition 1.3.3. A domainΩ is calledcausalif Ω is contained in a convex domainΩ′

and if for anyp,q∈ Ω the intersectionJΩ′+ (p)∩JΩ′

− (q) is compact and contained inΩ.


Ωb

q

b

p

Ω′b

b

convex, but not causal

Ω′

r r

Ωbq

b

p

b

b

causal

Fig. 9: Convexity versus causality

Definition 1.3.4. A subsetS of a connected timeoriented Lorentzian manifold is calledachronal(or acausal) if and only if each timelike (respectively causal) curve meetsS atmost once.A subsetSof a connected timeoriented Lorentzian manifold is aCauchy hypersurfaceifeach inextendible timelike curve inM meetsSat exactly one point.

M

Sb

Fig. 10: Cauchy hypersurfaceSmet by a timelike curve

Obviously every acausal subset is achronal, but the reverseis wrong. However, ev-ery achronal spacelike hypersurface is acausal (see Lemma 42 from Chap. 14 in


[O’Neill1983]).Any Cauchy hypersurface is achronal. Moreover, it is a closed topological hypersurfaceand it is hit by each inextendible causal curve in at least onepoint. Any two Cauchy hy-persurfaces inM are homeomorphic. Furthermore, the causal future and past of a Cauchyhypersurface is past- and future-compact respectively. This is a consequence of e. g.[O’Neill1983, Ch. 14, Lemma 40].

Definition 1.3.5. The Cauchy developmentof a subsetS of a timeoriented LorentzianmanifoldM is the setD(S) of points ofM through which every inextendible causal curvein M meetsS.

S

Cauchy develop-ment ofS

Fig. 11: Cauchy development

Remark 1.3.6. It follows from the definition thatD(D(S))= D(S) for every subsetS⊂M.Hence ifT ⊂ D(S), thenD(T) ⊂ D(D(S)) = D(S).Of course, ifS is achronal, then every inextendible causal curve inM meetsS at mostonce. The Cauchy developmentD(S) of every acausalhypersurfaceS is open, see[O’Neill1983, Chap. 14, Lemma 43].

Definition 1.3.7. A Lorentzian manifold is said to satisfy thecausality conditionif it doesnot contain any closed causal curve.A Lorentzian manifold is said to satisfy thestrong causality conditionif there are noalmost closed causal curves. More precisely, for each pointp ∈ M and for each openneighborhoodU of p there exists an open neighborhoodV ⊂U of p such that each causalcurve inM starting and ending inV is entirely contained inU .

b

b

b

p

VU

forbidden!

Fig. 12: Strong causality condition


Obviously, the strong causality condition implies the causality condition. Convex opensubsets of a Lorentzian manifold satisfy the strong causality condition.

Definition 1.3.8. A connected timeoriented Lorentzian manifold is calledglobally hyper-bolic if it satisfies the strong causality condition and if for allp,q ∈ M the intersectionJM+ (p)∩JM

− (q) is compact.

Remark 1.3.9. If M is a globally hyperbolic Lorentzian manifold, then a nonempty opensubsetΩ ⊂ M is itself globally hyperbolic if and only if for anyp,q∈ Ω the intersectionJΩ+(p)∩ JΩ

− (q) ⊂ Ω is compact. Indeed non-existence of almost closed causal curves inM directly implies non-existence of such curves inΩ.

We now state a very useful characterization of globally hyperbolic manifolds.

Theorem 1.3.10. Let M be a connected timeoriented Lorentzian manifold. Thenthefollowing are equivalent:

(1) M is globally hyperbolic.

(2) There exists a Cauchy hypersurface in M.

(3) M is isometric toR×S with metric−βdt2 +gt whereβ is a smooth positive func-tion, gt is a Riemannian metric on S depending smoothly on t∈ R and eacht×Sis a smooth spacelike Cauchy hypersurface in M.

Proof. That (1) implies (3) has been shown by Bernal and Sanchez in[Bernal-Sanchez2005, Thm. 1.1] using work of Geroch [Geroch1970, Thm. 11].See also [Ellis-Hawking1973, Prop. 6.6.8] and [Wald1984, p. 209] for earlier mention-ings of this fact. That (3) implies (2) is trivial and that (2)implies (1) is well-known, seee. g. [O’Neill1983, Cor. 39, p. 422].

Examples 1.3.11.Minkowski space is globally hyperbolic. Every spacelike hyperplaneis a Cauchy hypersurface. One can write Minkowski space asR×Rn−1 with the metric− dt2 +gt wheregt is the Euclidean metric onRn−1 and does not depend ont.Let (S,g0) be a connected Riemannian manifold andI ⊂ R an interval. The manifoldM = I ×S with the metricg = − dt2 + g0 is globally hyperbolic if and only if(S,g0) iscomplete. This applies in particular ifS is compact.More generally, iff : I → R is a smooth positive function we may equipM = I ×Swiththe metricg = − dt2 + f (t)2 · g0. Again, (M,g) is globally hyperbolic if and only if(S,g0) is complete, see Lemma A.5.14.Robertson-Walker spacetimesand, in particular,Friedmann cosmological models, are of this type. They are used to discuss big bang,expansion of the universe, and cosmological redshift, compare [Wald1984, Ch. 5 and 6] or[O’Neill1983, Ch. 12]. Another example of this type isdeSitter spacetime, whereI = R,S= Sn−1, g0 is the canonical metric ofSn−1 of constant sectional curvature 1, andf (t) =cosh(t). Anti-deSitter spacetimewhich we will discuss in more detail in Section 3.5 is notglobally hyperbolic.The interior and exteriorSchwarzschild spacetimesare globally hyperbolic. They modelthe universe in the neighborhood of a massive static rotationally symmetric body such


as a black hole. They are used to investigate perihelion advance of Mercury, the bend-ing of light near the sun and other astronomical phenomena, see [Wald1984, Ch. 6] and[O’Neill1983, Ch. 13].

Corollary 1.3.12. On every globally hyperbolic Lorentzian manifold M there exists asmooth function h: M → R whose gradient is past directed timelike at every point and allof whose level-sets are spacelike Cauchy hypersurfaces.

Proof. Defineh to be the compositiont Φ whereΦ : M → R×S is the isometry givenin Theorem 1.3.10 andt : R×S→ R is the projection onto the first factor.

Such a functionh on a globally hyperbolic Lorentzian manifold will be referred to as aCauchy time-function. Note that a Cauchy time-function is strictly monotonically increas-ing along any future directed causal curve.We quote an enhanced form of Theorem 1.3.10, due to A. Bernal and M. Sanchez (see[Bernal-Sanchez2006, Theorem 1.2]), which will be neededin Chapter 3.

Theorem 1.3.13.Let M be a globally hyperbolic manifold and S be a spacelike smoothCauchy hypersurface in M. Then there exists a Cauchy time-function h: M →R such thatS= h−1(0).

Any given smooth spacelike Cauchy hypersurface in a (necessarily globally hyperbolic)Lorentzian manifold is therefore the leaf of a foliation by smooth spacelike Cauchy hy-persurfaces.Recall that thelengthL[c] of a piecewiseC1-curvec : [a,b]→M on a Lorentzian manifold(M,g) is defined by

L[c] :=∫ b

a

√|g(c(t), c(t))|dt.

Definition 1.3.14. The time-separationon a Lorentzian manifold(M,g) is the functionτ : M×M → R∪∞ defined by

τ(p,q) :=

supL[c] |c future directed causal curve fromp to q, if p < q

0, otherwise,

for all p, q in M.

The properties ofτ which will be needed later are the following:

Proposition 1.3.15.Let M be a timeoriented Lorentzian manifold. Let p, q, and r∈ M.Then

(1) τ(p,q) > 0 if and only if q∈ IM+ (p).

(2) The functionτ is lower semi-continuous on M×M. If M is convex or globallyhyperbolic, thenτ is finite and continuous.

(3) The functionτ satisfies theinverse triangle inequality: If p ≤ q≤ r, then

τ(p, r) ≥ τ(p,q)+ τ(q, r). (1.10)


See e. g. Lemmas 16, 17, and 21 from Chapter 14 in [O’Neill1983] for a proof.

Now letM be a Lorentzian manifold. For a differentiable functionf : M →R, thegradientof f is the vector field

gradf := (d f)♯. (1.11)

Hereω 7→ ω♯ denotes the canonical isomorphismT∗M → TM induced by the Lorentzianmetric, i. e., forω ∈ T∗

x M the vectorω♯ ∈ TxM is characterized by the fact thatω(X) =〈ω♯,X〉 for all X ∈ TxM. The inverse isomorphismTM → T∗M is denoted byX 7→ X.One easily checks that for differentiable functionsf ,g : M → R

grad( f g) = ggradf + f gradg. (1.12)

Locally, the gradient off can be written as

gradf =n

∑j=1

ε j d f(ej )ej

wheree1, . . . ,en is a local Lorentz orthonormal frame ofTM, ε j = 〈ej ,ej〉 = ±1. For adifferentiable vector fieldX onM thedivergenceis the function

divX := tr(∇X) =n

∑j=1

ε j〈ej ,∇ej X〉

If X is a differentiable vector field andf a differentiable function onM, then one imme-diately sees that

div( f X) = f divX + 〈gradf ,X〉. (1.13)

There is another way to characterize the divergence. Let dV be the volume form in-duced by the Lorentzian metric. Inserting the vector fieldX yields an(n− 1)-formdV(X, ·, . . . , ·). Henced( dV(X, ·, . . . , ·)) is an n-form and can therefore be written asa function times dV, namely

d( dV(X, ·, . . . , ·) = divX · dV. (1.14)

This shows that the divergence operator depends only mildlyon the Lorentzian metric.If two Lorentzian (or more generally, semi-Riemannian) metrics have the same volumeform, then they also have the same divergence operator. Thisis certainly not true for thegradient.The divergence is important because of Gauss’ divergence theorem:

Theorem 1.3.16. Let M be a Lorentzian manifold and let D⊂ M be a domain withpiecewise smooth boundary. We assume that the induced metric on the smooth partof the boundary is non-degenerate, i. e., it is either Riemannian or Lorentzian on eachconnected component. Letn denote the exterior normal field along∂D, normalized to〈n,n〉 =: εn = ±1.Then for every smooth vector field X on M such thatsupp(X)∩D is compact we have

∫

Ddiv(X) dV =

∫

∂Dεn〈X,n〉 dA .


Let e1, . . . ,en be a Lorentz orthonormal basis ofTxM. Then(ξ 1, . . . ,ξ n) 7→ expx(∑ j ξ jej)is a local diffeomorphism of a neighborhood of 0 inRn onto a neighborhood ofx in M.This defines coordinatesξ 1, . . . ,ξ n on any open neighborhood ofx which is geodesicallystarshaped with respect tox. Such coordinates are callednormal coordinatesabout thepointx.We express the vectorX in normal coordinates aboutx and writeX = ∑ j η j ∂

∂ξ j . From

(1.14) we conclude, using dV= µx ·dξ 1∧ . . .∧dξ n

div(µ−1x X) · dV = d( dV(µ−1

x X, ·, . . . , ·)

= d

(∑

j(−1) j−1η j dξ 1∧ . . .∧ dξ j ∧ . . .∧dξ n

)

= ∑j

(−1) j−1dη j ∧dξ 1∧ . . .∧ dξ j ∧ . . .∧dξ n

= ∑j

∂η j

∂ξ j dξ 1∧ . . .∧dξ n

= ∑j

∂η j

∂ξ j µ−1x dV.

Thus

µx div(µ−1x X) = ∑

j

∂η j

∂ξ j . (1.15)

For aC2-function f theHessianat x is the symmetric bilinear form

Hess( f )|x : TxM×TxM → R, Hess( f )|x(X,Y) := 〈∇X gradf ,Y〉.

Thed’Alembert operatoris defined by

f := − tr(Hess( f )) = −divgradf .

If f : M → R andF : R → R areC2 a straightforward computation yields

(F f ) = −(F ′′ f )〈d f,d f〉+(F ′ f ) f . (1.16)

Lemma 1.3.17.Let Ω be a domain in M, geodesically starshaped with respect to x∈ Ω.Then the functionµx defined in (1.9) satisfies

µx(x) = 1, dµx|x = 0, Hess(µx)|x = −13

ricx, (µx)(x) =13

scal(x),

wherericx denotes the Ricci curvature considered as a bilinear form onTxΩ andscalisthe scalar curvature.

Proof. LetX ∈TxΩ be fixed. Lete1, . . . ,en be a Lorentz orthonormal basis ofTxΩ. Denote

by J1, . . . ,Jn the Jacobi fields alongc(t) = expx(tX) satisfyingJj(0) = 0 and∇Jjdt (0) = ej


for every 1≤ j ≤ n. The differential of expx at tX is, for everyt for which it is defined,given by

dtX expx(ej) =1tJj(t),

j = 1, . . . ,n. From the definition ofµx we have

µx(expx(tX))e1∧ . . .∧en = det(dtX expx)e1∧ . . .∧en

= (dtX expx(e1))∧ . . .∧ (dtX expx(en))

=1t

J1(t)∧ . . .∧ 1tJn(t).

Jacobi fieldsJ along the geodesicc(t) = expx(tX) satisfy the Jacobi field equation∇2

dt2J(t) = −R(J(t), c(t))c(t), whereR denotes the curvature tensor of the Levi-Civita

connection∇. Differentiating this once more yields∇3

dt3J(t) = −∇R

dt (J(t), c(t))c(t)−R( ∇

dt J(t), c(t))c(t). For J = Jj and t = 0 we haveJj(0) = 0,∇Jjdt (0) = ej ,

∇2Jj

dt2(0) =

−R(0, c(0))c(0) = 0, and∇3Jj

dt3(0) = −R(ej ,X)X whereX = c(0). IdentifyingJj(t) with

its parallel translate toTxΩ alongc the Taylor expansion ofJj up to order 3 reads as

Jj(t) = tej −t3

6R(ej ,X)X +O(t4).

This implies

1t

J1(t)∧ . . .∧ 1tJn(t) = e1∧ . . .∧en

− t2

6

n

∑j=1

e1∧ . . .∧R(ej ,X)X∧ . . .∧en +O(t3)

= e1∧ . . .∧en

− t2

6

n

∑j=1

ε j〈R(ej ,X)X,ej〉e1∧ . . .∧en +O(t3)

=(

1− t2

6ric(X,X)+O(t3)

)e1∧ . . .∧en.

Thus

µx(expx(tX)) = 1− t2

6ric(X,X)+O(t3)

and therefore

µx(x) = 1, dµx(X) = 0, Hess(µx)(X,X) = −13

ric(X,X).

Taking a trace yields the result for the d’Alembertian.

Lemma 1.3.17 and (1.16) withf = µx andF(t) = t−1/2 yield:


Corollary 1.3.18. Under the assumptions of Lemma 1.3.17 one has

(µ−1/2x )(x) = −1

6scal(x).

Let Ω be a domain in a Lorentzian manifoldM, geodesically starshaped with respect tox∈ Ω. Set

Γx := γ exp−1x : Ω → R (1.17)

whereγ is defined as in (1.6) withV = TxΩ.

Lemma 1.3.19.Let M be a timeoriented Lorentzian manifold. Let the domainΩ ⊂ M begeodesically starshaped with respect to x∈ Ω. Then the following holds onΩ:

(1) 〈gradΓx,gradΓx〉 = −4Γx.

(2) On IΩ+(x) (or on IΩ−(x)) the gradientgradΓx is a past directed (or future directedrespectivel) timelike vector field.

(3) Γx−2n = −〈gradΓx,grad(log(µx))〉.

Proof. Proof of (1). Lety∈ Ω andZ ∈ TyΩ. The differential ofγ at a pointp is given bydpγ = −2〈p, ·〉. Hence

dyΓx(Z) = dexp−1x (y)γ dyexp−1

x (Z)

= −2〈exp−1x (y),dyexp−1

x (Z)〉.

Applying the Gauss Lemma [O’Neill1983, p. 127], we obtain

dyΓx(Z) = −2〈dexp−1x (y) expx(exp−1

x (y)),Z〉.

Thusgrady Γx = −2dexp−1

x (y) expx(exp−1x (y)). (1.18)

It follows again from the Gauss Lemma that

〈grady Γx,grady Γx〉 = 4〈dexp−1x (y) expx(exp−1

x (y)),dexp−1x (y) expx(exp−1

x (y))〉= 4〈exp−1

x (y),exp−1x (y)〉

= −4Γx(y).

Proof of (2). OnIΩ+(x) the functionΓx is positive, hence〈gradΓx,gradΓx〉 = −4Γx < 0.

Thus gradΓx is timelike. For a future directed timelike tangent vectorZ ∈ TxΩ the curvec(t) := expx(tZ) is future directed timelike andΓx increases alongc. Hence 0≤ d

dt (Γx c) = 〈gradΓx, c〉. Thus gradΓx is past directed alongc. Since every point inIΩ

+(x) can bewritten in the form expx(Z) for a future directed timelike tangent vectorZ this proves theassertion forIΩ

+ (x). The argument forIΩ−(x) is analogous.

1.4. Riesz distributions on a domain 29

Proof of (3). Using (1.13) withf = µ−1x andX = gradΓx we get

div(µ−1x gradΓx) = µ−1

x divgradΓx + 〈grad(µ−1x ),gradΓx〉

and therefore

Γx = 〈grad(log(µ−1x )),gradΓx〉− µx div(µ−1

x gradΓx)

= −〈grad(log(µx)),gradΓx〉− µx div(µ−1x gradΓx).

It remains to showµx div(µ−1x gradΓx) = −2n. We check this in normal coordinates

ξ 1, . . . ,ξ n aboutx. By (1.18) we have gradΓx = −2∑ j ξ j ∂∂ξ j so that (1.15) implies

µx div(µ−1x gradΓx) = −2∑

j

∂ξ j

∂ξ j = −2n.

Remark 1.3.20. If Ω is convex andτ is the time-separation function ofΩ, then one cancheck that

τ(p,q) =

√Γ(p,q), if p < q

0, otherwise.

1.4 Riesz distributions on a domain

Riesz distributions have been defined on all spaces isometric to Minkowski space. Theyare therefore defined on the tangent spaces at all points of a Lorentzian manifold. We nowshow how to construct Riesz distributions defined in small open subsets of the Lorentzianmanifold itself. The passage from the tangent space to the manifold will be provided bythe Riemannian exponential map.Let Ω be a domain in a timeorientedn-dimensional Lorentzian manifold,n≥ 2. SupposeΩ is geodesically starshaped with respect to some pointx∈ Ω. In particular, the Rieman-nian exponential function expx is a diffeomorphism fromΩ′ := exp−1(Ω) ⊂ TxΩ to Ω.Let µx : Ω → R be defined as in (1.9). Put

RΩ±(α,x) := µx exp∗x R±(α),

that is, for every testfunctionϕ ∈ D(Ω,C),

RΩ±(α,x)[ϕ ] := R±(α)[(µxϕ)expx].

Note that supp((µxϕ)expx) is contained inΩ′. Extending the function(µxϕ)expx byzero we can regard it as a testfunction onTxΩ and thus applyR±(α) to it.

Definition 1.4.1. We callRΩ+(α,x) theadvanced Riesz distributionandRΩ

−(α,x) the re-tarded Riesz distributionon Ω at x for α ∈ C.

The relevant properties of the Riesz distributions are collected in the following proposi-tion.


Proposition 1.4.2. The following holds for allα ∈ C and all x∈ Ω:

(1) If Re(α) > n, then RΩ±(α,x) is the continuous function

RΩ±(α,x) =

C(α,n)Γ

α−n2

x on JΩ±(x),

0 elsewhere.

(2) For every fixed testfunctionϕ the mapα 7→ RΩ±(α,x)[ϕ ] is holomorphic onC.

(3) Γx ·RΩ±(α,x) = α(α −n+2)RΩ

±(α +2,x)

(4) grad(Γx) ·RΩ±(α,x) = 2α gradRΩ

±(α +2,x)

(5) If α 6= 0, thenRΩ±(α +2,x) =

(Γx−2n

2α +1)

RΩ±(α,x)

(6) RΩ±(0,x) = δx

(7) For everyα ∈C\(0,−2,−4, . . .∪n−2,n−4, . . .) we havesupp(RΩ±(α,x)

)=

JΩ±(x) andsingsupp

(RΩ±(α,x)

)⊂CΩ

±(x).

(8) For everyα ∈ 0,−2,−4, . . . ∪ n− 2,n− 4, . . . we havesupp(RΩ±(α,x)

)=

singsupp(RΩ±(α,x)

)⊂CΩ

±(x).

(9) For n≥ 3 andα = n−2,n−4, . . . ,1 or 2 respectively we havesupp(RΩ±(α,x)

)=

singsupp(RΩ±(α,x)

)= CΩ

±(x).

(10) ForRe(α) > 0 we haveord(RΩ±(α,x)) ≤ n+1. Moreover, there exists a neighbor-

hood U of x and a constant C> 0 such that

|RΩ±(α,x′)[ϕ ]| ≤C · ‖ϕ‖Cn+1(Ω)

for all ϕ ∈ D(Ω,C) and all x′ ∈U.

(11) If U ⊂ Ω is an open neighborhood of x such thatΩ is geodesically starshapedwith respect to all x′ ∈U and if V∈ D(U ×Ω,C), then the function U→ C, x′ 7→RΩ±(α,x′)[y 7→V(x′,y)], is smooth.

(12) If U ⊂ Ω is an open neighborhood of x such thatΩ is geodesically starshaped withrespect to all x′ ∈U, if Re(α) > 0, and if V∈Dn+1+k(U ×Ω,C), then the functionU → C, x′ 7→ RΩ

±(α,x′)[y 7→V(x′,y)], is Ck.

(13) For everyϕ ∈ Dk(Ω,C) the mapα 7→ RΩ±(α,x)[ϕ ] is a holomorphic function on

α ∈ C | Re(α) > n−2[ k2].

(14) If α ∈ R, then RΩ±(α,x) is real, i. e., RΩ

±(α,x)[ϕ ] ∈ R for all ϕ ∈ D(Ω,R).

1.4. Riesz distributions on a domain 31

Proof. It suffices to prove the statements for the advanced Riesz distributions.

Proof of (1). LetRe(α) > n andϕ ∈ D(Ω,C). Then

RΩ+(α,x)[ϕ ] = RΩ

+(α,x)[(µxexpx) · (ϕ expx)]

= C(α,n)

∫

J+(0)γ

α−n2 · (ϕ expx) ·µxdz

= C(α,n)∫

JΩ+ (x)

Γα−n

2x ·ϕ dV.

Proof of (2). This follows directly from the definition ofRΩ+(α,x) and from

Lemma 1.2.2 (4).

Proof of (3). By (1) this obviously holds forRe(α) > n sinceC(α,n) = α(α − n+2)C(α +2,n). By analyticity ofα 7→ RΩ

+(α,x) it must hold for allα.

Proof of (4). Considerα with Re(α) > n. By (1) the functionRΩ+(α + 2,x) is thenC1.

OnJΩ+(x) we compute

2α gradRΩ+(α +2,x) = 2αC(α +2,n)grad

(Γ

α+2−n2

x

)

= 2αC(α +2,n)α +2−n

2︸︷︷︸C(α ,n)

Γα−n

2x gradΓx

= RΩ+(α,x)gradΓx.

For arbitraryα ∈ C assertion (4) follows from analyticity ofα 7→ RΩ+(α,x).

Proof of (5). Letα ∈ C with Re(α) > n+ 2. SinceRΩ+(α + 2,x) is thenC2, we can

computeRΩ+(α +2,x) classically. This will show that (5) holds for allα with Re(α) >

n+2. Analyticity then implies (5) for allα.

RΩ+(α +2,x) = −div

(gradRΩ

+(α +2,x))

(4)= − 1

2αdiv(

RΩ+(α,x) ·grad(Γx)

)

(1.13)=

12α

Γx ·RΩ+(α,x)− 1

2α〈gradΓx,gradRΩ

+(α,x)〉(4)=

12α

Γx ·RΩ+(α,x)− 1

2α ·2(α −2)〈gradΓx,gradΓx ·RΩ

+(α −2,x)〉

Lemma 1.3.19(1)=

12α

Γx ·RΩ+(α,x)+

1α(α −2)

Γx ·RΩ+(α −2,x)

(3)=

12α

Γx ·RΩ+(α,x)+

(α −2)(α −n)

α(α −2)RΩ

+(α,x)

=

(Γx−2n

2α+1

)RΩ

+(α,x).


Proof of (6). Letϕ be a testfunction onΩ. Then by Proposition 1.2.4 (7)

RΩ+(0,x)[ϕ ] = R+(0)[(µxϕ)expx]

= δ0[(µxϕ)expx]

= ((µxϕ)expx)(0)

= µx(x)ϕ(x)

= ϕ(x)

= δx[ϕ ].

Proof of (11). LetA(x,x′) : TxΩ → Tx′Ω be a timeorientation preserving linear isometry.Then

RΩ+(α,x′)[V(x′, ·)] = R+(α)[(µx′ ·V(x′, ·))expx′ A(x,x′)]

whereR+(α) is, as before, the Riesz distribution onTxΩ. Hence if we chooseA(x,x′) todepend smoothly onx′, then(µx′ ·V(x′,y)) expx′ A(x,x′) is smooth inx′ andy and theassertion follows from Lemma 1.1.6.Proof of (10). Since ord(R±(α)) ≤ n + 1 by Proposition 1.2.4 (8) we haveord(RΩ

±(α,x)) ≤ n+ 1 as well. From the definitionRΩ±(α,x) = µx exp∗x R±(α) it is clear

that the constantC may be chosen locally uniformly inx.Proof of (12). By (10) we can applyRΩ

±(α,x′) to V(x′, ·). Now the same argument as for(11) shows that the assertion follows from Lemma 1.1.6.The remaining assertions follow directly from the corresponding properties of the Rieszdistributions on Minkowski space. For example (13) is a consequence of Corollary 1.2.5.

Advanced and retarded Riesz distributions are related as follows.

Lemma 1.4.3. LetΩ be a convex timeoriented Lorentzian manifold. Letα ∈ C. Then forall u ∈ D(Ω×Ω,C) we have

∫

ΩRΩ

+(α,x) [y 7→ u(x,y)] dV(x) =∫

ΩRΩ−(α,y) [x 7→ u(x,y)] dV(y).

Proof. The convexity condition forΩ ensures that the Riesz distributionsRΩ±(α,x) are

defined for allx ∈ Ω. By Proposition 1.4.2 (11) the integrands are smooth. Sinceu hascompact support contained inΩ×Ω the integrandRΩ

+(α,x) [y 7→ u(x,y)] (as a function inx) has compact support contained inΩ. A similar statement holds for the integrand of theright hand side. Hence the integrals exist. By Proposition 1.4.2 (13) they are holomorphicin α. Thus it suffices to show the equation forα with Re(α) > n.For such anα ∈C the Riesz distributionsR+(α,x) andR−(α,y) are continuous functions.From the explicit formula (1) in Proposition 1.4.2 we see

R+(α,x)(y) = R−(α,y)(x)

1.5. Normally hyperbolic operators 33

for all x,y∈ Ω. By Fubini’s theorem we get

∫

ΩRΩ

+(α,x) [y 7→ u(x,y)] dV(x) =

∫

Ω

(∫

ΩRΩ

+(α,x)(y) u(x,y) dV(y)

)dV(x)

=

∫

Ω

(∫

ΩRΩ−(α,y)(x) u(x,y) dV(x)

)dV(y)

=

∫


As a technical tool we will also need a version of Lemma 1.4.3 for certain nonsmoothsections.

Lemma 1.4.4. Let Ω be a causal domain in a timeoriented Lorentzian manifold of di-mension n. LetRe(α) > 0 and let k≥ n+1. Let K1, K2 be compact subsets ofΩ and letu∈Ck(Ω×Ω) so thatsupp(u) ⊂ JΩ

+(K1)×JΩ−(K2). Then

∫

ΩRΩ

+(α,x) [y 7→ u(x,y)] dV(x) =∫


Proof. For fixed x, the support of the functiony 7→ u(x,y) is contained inJΩ−(K2).

SinceΩ is causal, it follows from Lemma A.5.3 that the subsetJΩ−(K2)∩ JΩ

+ (x) is rel-atively compact inΩ. Therefore the intersection of the supports ofy 7→ u(x,y) andRΩ

+(α,x) is compact and contained inΩ. By Proposition 1.4.2 (10) one can then ap-ply RΩ

+(α,x) to theCk-functiony 7→ u(x,y). Furthermore, the support of the continuousfunction x 7→ RΩ

+(α,x) [y 7→ u(x,y)] is contained inJΩ+(K1)∩ JΩ

−(supp(y 7→ u(x,y))) ⊂JΩ+(K1)∩ JΩ

− (JΩ−(K2)) = JΩ

+(K1)∩ JΩ+(K2), which is relatively compact inΩ, again by

Lemma A.5.3. Hence the functionx 7→ RΩ+(α,x) [y 7→ u(x,y)] has compact support in

Ω, so that the left-hand-side makes sense. Analogously the right-hand-side is well-defined. Our considerations also show that the integrals depend only on the values ofu on

(JΩ+(K1)∩JΩ

−(K2))×(JΩ+(K1)∩JΩ

−(K2))

which is a relatively compact set. Ap-plying a cut-off function argument we may assume without loss of generality thatu hascompact support. Proposition 1.4.2 (13) says that the integrals depend holomorphicallyon α on the domainRe(α) > 0. Therefore it suffices to show the equality forα withsufficiently large real part, which can be done exactly as in the proof of Lemma 1.4.3.

1.5 Normally hyperbolic operators

Let M be a Lorentzian manifold and letE → M be a real or complex vector bundle. Fora summary on basics concerning linear differential operators see Appendix A.4. A lineardifferential operatorP : C∞(M,E) → C∞(M,E) of second order will be callednormallyhyperbolicif its principal symbol is given by the metric,

σP(ξ ) = −〈ξ ,ξ 〉 · idEx


for all x∈ M and allξ ∈ T∗x M. In other words, if we choose local coordinatesx1, . . . ,xn

onM and a local trivialization ofE, then

P = −n

∑i, j=1

gi j (x)∂ 2

∂xi∂x j +n

∑j=1

A j(x)∂

∂x j +B1(x)

whereA j andB1 are matrix-valued coefficients depending smoothly onx and(gi j )i j is theinverse matrix of(gi j )i j with gi j = 〈 ∂

∂xi ,∂

∂xj 〉.Example 1.5.1. Let E be the trivial line bundle so that sections inE are just functions.The d’Alembert operatorP = is normally hyperbolic because

σgrad(ξ ) f = f ξ ♯, σdiv(ξ )X = ξ (X)

and soσ(ξ ) f = −σdiv(ξ )σgrad(ξ ) f = −ξ ( f ξ ♯) = −〈ξ ,ξ 〉 f .

Recall thatξ 7→ ξ ♯ denotes the isomorphismT∗x M → TxM induced by the Lorentzian

metric, compare (1.11).

Example 1.5.2.Let E be a vector bundle and let∇ be a connection onE. This connectiontogether with the Levi-Civita connection onT∗M induces a connection onT∗M ⊗E,again denoted∇. We define theconnection-d’Alembert operator∇ to be minus thecomposition of the following three maps

C∞(M,E)∇−→C∞(M,T∗M⊗E)

∇−→C∞(M,T∗M⊗T∗M⊗E)tr⊗idE−−−−→C∞(M,E)

where tr :T∗M⊗T∗M → R denotes the metric trace, tr(ξ ⊗η) = 〈ξ ,η〉. We compute theprincipal symbol,

σ∇(ξ )ϕ = −(tr⊗idE)σ∇(ξ )σ∇(ξ )(ϕ) = −(tr⊗idE)(ξ ⊗ ξ ⊗ϕ) = −〈ξ ,ξ 〉ϕ .

Hence∇ is normally hyperbolic.

Example 1.5.3. Let E = ΛkT∗M be the bundle ofk-forms. Exterior differentiationd :C∞(M,ΛkT∗M)→C∞(M,Λk+1T∗M) increases the degree by one while the codifferentialδ : C∞(M,ΛkT∗M) → C∞(M,Λk−1T∗M) decreases the degree by one, see [Besse1987,p. 34] for details. Whiled is independent of the metric, the codifferentialδ does dependon the Lorentzian metric. The operatorP = dδ + δd is normally hyperbolic.

Example 1.5.4.If M carries a Lorentzian metric and a spin structure, then one can definethe spinor bundleΣM and the Dirac operator

D : C∞(M,ΣM) →C∞(M,ΣM),

see [Bar-Gauduchon-Moroianu2005] or [Baum1981] for the definitions. The principalsymbol ofD is given by Clifford multiplication,

σD(ξ )ψ = ξ ♯ ·ψ .

HenceσD2(ξ )ψ = σD(ξ )σD(ξ )ψ = ξ ♯ ·ξ ♯ ·ψ = −〈ξ ,ξ 〉ψ .

ThusP = D2 is normally hyperbolic.

1.5. Normally hyperbolic operators 35

The following lemma is well-known, see e. g. [Baum-Kath1996, Prop. 3.1]. It says thateach normally hyperbolic operator is a connection-d’Alembert operator up to a term oforder zero.

Lemma 1.5.5. Let P : C∞(M,E) → C∞(M,E) be a normally hyperbolic operator on aLorentzian manifold M. Then there exists a unique connection ∇ on E and a uniqueendomorphism field B∈C∞(M,Hom(E,E)) such that

P = ∇ +B.

Proof. First we prove uniqueness of such a connection. Let∇′ be an arbitrary connectiononE. For any sections∈C∞(M,E) and any functionf ∈C∞(M) we get

∇′

( f ·s) = f · (∇′s)−2∇′

gradf s+( f ) ·s. (1.19)

Now suppose that∇ satisfies the condition in Lemma 1.5.5. ThenB = P−∇ is an

endomorphism field and we obtain

f ·(

P(s)−∇s)

= P( f ·s)−∇( f ·s).

By (1.19) this yields

∇gradf s= 12 f ·P(s)−P( f ·s)+ ( f ) ·s . (1.20)

At a given pointx ∈ M every tangent vectorX ∈ TxM can be written in the formX =gradx f for some suitably chosen functionf . Thus (1.20) shows that∇ is determined byP and (which is determined by the Lorentzian metric).

To show existence one could use (1.20) to define a connection∇ as in the statement.We follow an alternative path. Let∇′ be some connection onE. SinceP and

∇′are

both normally hyperbolic operators acting on sections inE, the differenceP−∇′

is adifferential operator of first order and can therefore be written in the form

P−∇′

= A′ ∇′ +B′,

for someA′ ∈ C∞(M,Hom(T∗M ⊗E,E)) and B′ ∈ C∞(M,Hom(E,E)). Set for everyvector fieldX onM and sections in E

∇Xs := ∇′Xs− 1

2A′(X⊗s).

This defines a new connection∇ onE. Lete1, . . . ,en be a local Lorentz orthonormal basisof TM. Write as beforeε j = 〈ej ,ej〉 = ±1. We may assume that at a given pointp∈ M


we have∇ej ej(p) = 0. Then we compute atp

∇′

s+A′ ∇′s =n

∑j=1

ε j

−∇′

ej∇′

ejs+A′(e

j ⊗∇′ej

s)

=n

∑j=1

ε j

− (∇ej +

12

A′(ej ⊗·))(∇ej s+

12

A′(ej ⊗s))

+A′(ej ⊗∇ej s)+

12

A′(ej ⊗A′(e

j ⊗s))

=n

∑j=1

ε j

−∇ej ∇ej s−

12

∇ej (A′(e

j ⊗s))

+12

A′(ej ⊗∇ej s)+

14

A′(ej ⊗A′(e

j ⊗s))

= ∇s+

14

n

∑j=1

ε j

A′(e

j ⊗A′(ej ⊗s))−2(∇ej A

′)(ej ⊗s)

,

where∇ in ∇ej A′ stands for the induced connection on Hom(T∗M⊗E,E). We observe

thatQ(s) := ∇′

s+ A′ ∇′s−∇s= 1

4 ∑nj=1 ε j

A′(e

j ⊗A′(ej ⊗ s))−2(∇ej A

′)(ej ⊗ s)

is of order zero. Hence

P = ∇′

+A′ ∇′+B′ = ∇s+Q(s)+B′(s)

is the desired expression withB = Q+B′.

The connection in Lemma 1.5.5 will be called theP-compatibleconnection. We shallhenceforth always work with theP-compatible connection. We restate (1.20) as a lemma.

Lemma 1.5.6.Let P= ∇ +B be normally hyperbolic. For f∈C∞(M) and s∈C∞(M,E)

one getsP( f ·s) = f ·P(s)−2∇gradf s+ f ·s.

Chapter 2

The local theory

Now we start with our detailed study of wave equations. By a wave equation we mean anequation of the formPu= f whereP is a normally hyperbolic operator acting on sectionsin a vector bundle. The right-hand-sidef is given and the sectionu is to be found. In thischapter we deal with local problems, i. e., we try to find solutions defined on sufficientlysmall domains. This can be understood as a preparation for the global theory which wepostpone to the third chapter. Solving wave equations on allof the Lorentzian manifold is,in general, possible only under the geometric assumption ofthe manifold being globallyhyperbolic.There are various techniques available in the theory of partial differential equations thatcan be used to settle the local theory. We follow an approach based on Riesz distributionsand Hadamard coefficients as in [Gunther1988]. The centraltask is to construct funda-mental solutions. This means that one solves the wave equation where the right-hand-sidef is a delta-distribution.The construction consists of three steps. First one writes down a formal series in Rieszdistributions with unknown coefficients. The wave equationyields recursive relations forthese Hadamard coefficients known as transport equations. Since the transport equationsare ordinary differential equations along geodesics they can be solved uniquely. There isno reason why the formal solution constructed in this way should be convergent.In the second step one makes the series convergent by introducing certain cut-off func-tions. This is similar to the standard proof showing that each formal power series is theTaylor series of some smooth function. Since there are errorterms produced by the cut-offfunctions the result is convergent but no longer solves the wave equation. We call it anapproximate fundamental solution.Thirdly, we turn the approximate fundamental solution intoa true one using certain in-tegral operators. Once the existence of fundamental solutions is established one can findsolutions to the wave equation for an arbitrary smoothf with compact support. The sup-port of these solutions is contained in the future or in the past of the support off .Finally, we show that the formal fundamental solution constructed in the first step isasymptotic to the true fundamental solution. This implies that the singularity structure ofthe fundamental solution is completely determined by the Hadamard coefficients which

38 Chapter 2. The local theory

are in turn determined by the geometry of the manifold and thecoefficients of the operator.

2.1 The formal fundamental solution

In this chapter the underlying Lorentzian manifold will typically be denoted byΩ. Later,in Chapter 3, when we apply the local resultsΩ will play the role of a small neighborhoodof a given point.

Definition 2.1.1. Let Ω be a timeoriented Lorentzian manifold, letE → Ω be a vectorbundle and letP : C∞(Ω,E) → C∞(Ω,E) be normally hyperbolic. Letx ∈ Ω. A funda-mental solutionof P at x is a distributionF ∈ D ′(Ω,E,E∗

x ) such that

PF = δx.

In other words, for allϕ ∈ D(Ω,E∗) we have

F [P∗ϕ ] = ϕ(x).

If supp(F(x))⊂ JΩ+ (x), then we callF anadvanced fundamental solution, if supp(F(x))⊂

JΩ−(x), then we callF a retarded fundamental solution.

For flat Minkowski space withP = acting on functions Proposition 1.2.4 (3) and (7)show that the Riesz distributionsR±(2) are fundamental solutions atx = 0. More pre-cisely, R+(2) is an advanced fundamental solution because its support is contained inJ+(0) andR−(2) is a retarded fundamental solution.On a general timeoriented Lorentzian manifoldΩ the situation is more complicated evenif P = . The reason is the factorΓx−2n

2α + 1 in Proposition 1.4.2 (5) which cannot beevaluated forα = 0 unlessΓx−2n vanished identically. It will turn out thatRΩ

±(2,x)does not suffice to construct fundamental solutions. We willalso need Riesz distributionsRΩ±(2+2k,x) for k≥ 1.

Let Ω be geodesically starshaped with respect to some fixedx ∈ Ω so that the RieszdistributionsRΩ

±(α,x) = RΩ±(α,x) are defined. LetE → Ω be a real or complex vector

bundle and letP be a normally hyperbolic operatorP acting onC∞(Ω,E). In this sectionwe start constructing fundamental solutions. We make the following formal ansatz:

R±(x) :=∞

∑k=0

Vkx RΩ

±(2+2k,x)

whereVkx ∈ C∞(Ω,E ⊗ E∗

x) are smooth sections yet to be found. Forϕ ∈ D(Ω,E∗)the functionVk

x ·ϕ is anE∗x -valued testfunction and we have(Vk

x ·RΩ±(2+ 2k,x))[ϕ ] =

RΩ±(2+ 2k,x)[Vk

x ·ϕ ] ∈ E∗x . Hence each summandVk

x ·RΩ±(2+ 2k,x) is a distribution in

D ′(Ω,E,E∗x ).

By formal termwise differentiation using Lemma 1.5.6 and Proposition 1.4.2 we trans-late the condition ofR±(x) being a fundamental solution atx into conditions on theVk

x . To do this let∇ be theP-compatible connection onE, that is,P = ∇ + B where

2.2. Uniqueness of the Hadamard coefficients 39

B∈C∞(Ω,End(E)), compare Lemma 1.5.5. We compute

RΩ±(0,x) = δx = PR±(x) =

∞

∑k=0

P(Vkx RΩ

±(2+2k,x))

=∞

∑k=0

Vkx ·RΩ

±(2+2k,x)−2∇gradRΩ±(2+2k,x)V

kx +PVk

x ·RΩ±(2+2k,x)

= V0x ·RΩ

±(2,x)−2∇gradRΩ±(2,x)V

0x

+∞

∑k=1

Vk

x ·( 1

2Γx−n2k +1

)RΩ±(2k,x)− 2

4k ∇gradΓx RΩ±(2k,x)V

kx

+PVk−1x ·RΩ

±(2k,x)

= V0x ·RΩ

±(2,x)−2∇gradRΩ±(2,x)V

0x

+∞

∑k=1

12k

(12Γx−n+2k

)Vk

x −∇gradΓxVkx +2kPVk−1

x

RΩ±(2k,x).(2.1)

Comparing the coefficients ofRΩ±(2k,x) we get the conditions

2∇gradRΩ±(2,x)V

0x −RΩ

±(2,x) ·V0x +RΩ

±(0,x) = 0 and (2.2)

∇gradΓxVkx −

(12Γx−n+2k

)Vk

x = 2kPVk−1x for k≥ 1. (2.3)

We take a look at what condition (2.3) would mean fork = 0. We multiply this equationby RΩ

±(α,x):∇gradΓx RΩ

±(α ,x)V0x −

(12Γx−n

)V0

x ·RΩ±(α,x) = 0.

By Proposition 1.4.2 (4) and (5) we obtain

∇2α gradRΩ±(α+2,x)V

0x −

(αRΩ

±(α +2,x)−αRΩ±(α,x)

)V0

x = 0.

Division byα and the limitα → 0 yield

2∇gradRΩ±(2,x)V

0x −

(RΩ

±(2,x)−RΩ±(0,x)

)V0

x = 0.

Therefore we recover condition (2.2) if and only ifV0x (x) = idEx.

To get formal fundamental solutionsR±(x) for P we hence needVkx ∈ C∞(Ω,E ⊗E∗

x )satisfying

∇gradΓxVkx −

(12Γx−n+2k

)Vk

x = 2kPVk−1x (2.4)

for all k≥ 0 with “initial condition” V0x (x) = idEx. In particular, we have the same condi-

tions onVkx for R+(x) and forR−(x). Equations (2.4) are known astransport equations.

2.2 Uniqueness of the Hadamard coefficients

This and the next section are devoted to uniqueness and existence of solutions to thetransport equations.


Definition 2.2.1. Let Ω be timeoriented and geodesically starshaped with respect to x∈Ω. SectionsVk

x ∈C∞(Ω,E⊗E∗x) are calledHadamard coefficientsfor P atx if they satisfy

the transport equations (2.4) for allk≥ 0 andV0x (x) = idEx. Given Hadamard coefficients

Vkx for P at x we call the formal series

R+(x) =∞

∑k=0

Vkx ·RΩ

+(2+2k,x)

a formal advanced fundamental solution for P at xand

R−(x) =∞

∑k=0

Vkx ·RΩ

−(2+2k,x)

a formal retarded fundamental solution for P at x.

In this section we show uniqueness of the Hadamard coefficients (and hence of the formalfundamental solutionsR±(x)) by deriving explicit formulas for them. These formulaswill also be used in the next section to prove existence.Fory∈ Ω we denote the∇-parallel translation along the (unique) geodesic fromx to y by

Πxy : Ex → Ey.

We haveΠxx = idEx and(Πx

y)−1 = Πy

x. Note that the mapΦ : Ω× [0,1] → Ω, Φ(y,s) =

expx(s· exp−1x (y)), is well-defined and smooth sinceΩ is geodesically starshaped with

respect tox.

Lemma 2.2.2. Let Vkx be Hadamard coefficients for P at x. Then they are given by

V0x (y) = µ−1/2

x (y)Πxy (2.5)

and for k≥ 1

Vkx (y) = −kµ−1/2

x (y)Πxy

∫ 1

0µ1/2

x (Φ(y,s))sk−1 ΠΦ(y,s)x (PVk−1

x (Φ(y,s)))ds. (2.6)

Proof. We putρ :=√|Γx|. On Ω\C(x) whereC(x) = expx(C(0)) is the light cone ofx

we haveΓx(y) = −ερ2(y) whereε = 1 if exp−1x (y) is spacelike andε = −1 if exp−1

x (y)

is timelike. Using the identities12Γx−n = − 12∂gradΓx logµx = −∂gradΓx log(µ1/2

x ) from

Lemma 1.3.19 (3) and∂gradΓx(logρk) = k∂−2ερ gradρ logρ = −2εkρ ∂gradρ ρρ = −2k we

reformulate (2.4):

∇gradΓxVkx + ∂gradΓx log

(µ1/2

x ·ρk)

Vkx = 2kPVk−1

x .

This is equivalent to

∇gradΓx

(µ1/2

x ·ρk ·Vkx

)= µ1/2

x ·ρk∇gradΓxVkx + ∂gradΓx(µ1/2

x ·ρk)Vkx

= µ1/2x ·ρk ·2k ·PVk−1

x . (2.7)

2.3. Existence of the Hadamard coefficients 41

For k = 0 one has∇gradΓx(µ1/2x V0

x ) = 0. Henceµ1/2x V0

x is ∇-parallel along the timelikeand spacelike geodesics starting inx. By continuity it is∇-parallel along any geodesic

starting atx. Sinceµ1/2x (x)V0

x (x) = 1 · idEx = Πxx we concludeµ1/2

x (y)V0x (y) = Πx

y for ally∈ Ω. This shows (2.5).Next we determineVk

x for k ≥ 1. We consider some pointy∈ Ω \C(x) outside the lightcone ofx. We put η := exp−1

x (y). Then c(t) := expx(e2t · η) gives a reparametriza-

tion of the geodesicβ (t) = expx(tη) from x to y such that ˙c(t) = 2e2t β (e2t). ByLemma 1.3.19 (1)

〈c(t), c(t)〉 = 4e4t〈β (e2t), β (e2t)〉= 4e4t〈η ,η〉 = −4γ(e2tη)

= −4Γx(c(t)) = 〈gradΓx,gradΓx〉.Thusc is an integral curve of the vector field−gradΓx. Equation (2.7) can be rewrittenas

− ∇dt

(µ1/2

x ·ρk ·Vkx

)(c(t)) =

(µ1/2

x ·ρk ·2k ·PVk−1x

)(c(t)),

which we can solve explicitly:(

µ1/2x ·ρk ·Vk

x

)(c(t))

= −Πxc(t)

(∫ t

−∞Πc(τ)

x

(µ1/2

x ·ρk ·2k ·PVk−1x

)(c(τ))dτ

)

= −2kΠxc(t)

(∫ t

−∞µ1/2

x (c(τ))ρ(c(τ))kΠc(τ)x

(PVk−1

x (c(τ)))

dτ)

.

We haveρ(c(τ))k = ρ(expx(e2τ η))k = |γ(e2τ η)|k/2 = |e4τ γ(η)|k/2 = e2kτ |γ(η)|k/2.

Sincey 6∈C(x) we can divide by|γ(η)| 6= 0:

e2kt(

µ1/2x Vk

x

)(c(t))

= −2kΠxc(t)

(∫ t

−∞µ1/2

x (c(τ))e2kτ Πc(τ)x

(PVk−1

x (c(τ)))

dτ)

= −2kΠxc(t)

∫ e2t

0µ1/2

x (expx(s·η))sk Πexpx(s·η)x (PVk−1

x (expx(s·η)))ds2s

where we used the substitutions= e2τ . Fort = 0 this yields (2.6).

Corollary 2.2.3. Let Ω be timeoriented and geodesically starshaped with respect to x∈Ω. Let P be a normally hyperbolic operator acting on sections in a vector bundle E overΩ.Then the Hadamard coefficients Vk

x for P at x are unique for all k≥ 0.

2.3 Existence of the Hadamard coefficients

Let Ω be timeoriented and geodesically starshaped with respect to x ∈ Ω. Let P be anormally hyperbolic operator acting on sections in a real orcomplex vector bundleE over


Ω. To construct Hadamard coefficients forP atx we use formulas (2.5) and (2.6) obtainedin the previous section as definitions:

V0x (y) := µ−1/2

x (y) ·Πxy and

Vkx (y) := −kµ−1/2

x (y)Πxy

∫ 1

0µ1/2

x (Φ(y,s))sk−1 ΠΦ(y,s)x (PVk−1

x (Φ(y,s)))ds.

We observe that this defines smooth sectionsVkx ∈C∞(Ω,E⊗E∗

x). We have to check forall k≥ 0

∇gradΓx

(µ1/2

x ρkVkx

)= µ1/2

x ρk ·2k ·PVk−1x , (2.8)

from which Equation (2.4) follows as we have already seen. For k = 0 Equation (2.8)obviously holds:

∇gradΓx

(µ1/2

x V0x

)= ∇gradΓxΠ = 0.

Fork≥ 1 we check:

∇gradΓx

(µ1/2

x ρkVkx

)(y)

= −k∇gradΓx Πxy

∫ 1

0µ1/2

x (Φ(y,s)) ρ(y)k ·sk

︸︷︷︸=ρ(Φ(y,s))k

ΠΦ(y,s)x PVk−1

x (Φ(y,s))dss

= −k∇gradΓx Πxy

∫ 1

0

(µ1/2

x ρk ΠxΦ(y,s)(PVk−1

x ))

(Φ(y,s))dss

s=e2τ= −2k∇gradΓx Πx

y

∫ 0

−∞

(µ1/2


x ))

(Φ(y,e2τ )︸︷︷︸integral curvefor −gradΓx

)dτ

= 2k Πxy

ddt

∣∣∣t=0

∫ 0

−∞

(µ1/2


x ))(

Φ(Φ(y,e2t ),e2τ)) dτ

= 2k Πxy

ddt

∣∣∣t=0

∫ 0

−∞

(µ1/2


x ))(

Φ(

y,e2(τ+t)))

dτ

τ ′=τ+t= 2k Πx

yddt

∣∣∣t=0

∫ t

−∞

(µ1/2


x ))(

Φ(

y,e2τ ′))

dτ ′

= 2k Πxy

(µ1/2


x ))

(Φ(y,e0)︸︷︷︸=y

)

= 2kµ1/2x (y)ρk(y)

(PVk−1

x

)(y)

which is (2.8). This shows the existence of the Hadamard coefficients and, therefore, wehave found formal fundamental solutionsR±(x) for P at fixedx∈ Ω.Now we letx vary. We assume there exists an open subsetU ⊂ Ω such thatΩ is geodesi-cally starshaped with respect to allx ∈ U . This ensures that the Riesz distributionsRΩ±(α,x) are defined for allx ∈ U . We writeVk(x,y) := Vk

x (y) for the Hadamard coef-ficients atx. ThusVk(x,y) ∈ Hom(Ex,Ey) = E∗

x ⊗Ey. The explicit formulas (2.5) and

2.3. Existence of the Hadamard coefficients 43

(2.6) show that the Hadamard coefficientsVk also depend smoothly onx, i. e.,

Vk ∈C∞(U ×Ω,E∗⊠E).

Recall thatE∗⊠ E is the bundle with fiber(E∗

⊠ E)(x,y) = E∗x ⊗Ey. We have formal

fundamental solutions forP at all x∈U :

R±(x) =∞

∑k=0

Vk(x, ·)RΩ±(2+2k,x).

We summarize our results about Hadamard coefficients obtained so far.

Proposition 2.3.1.LetΩ be a Lorentzian manifold, let U⊂ Ω be a nonempty open subsetsuch thatΩ is geodesically starshaped with respect to all points x∈U. Let P=

∇ +Bbe a normally hyperbolic operator acting on sections in a real or complex vector bundleoverΩ. Denote the∇-parallel transport byΠ.Then at each x∈U there are unique Hadamard coefficients Vk(x, ·) for P, k≥ 0. They aresmooth, Vk ∈C∞(U ×Ω,E∗

⊠E), and are given by

V0(x,y) = µ−1/2x (y) ·Πx

y

and for k≥ 1

Vk(x,y) = −kµ−1/2x (y)Πx

y

∫ 1

0µ1/2

x (Φ(y,s))sk−1 ΠΦ(y,s)x (P(2)Vk−1)(x,Φ(y,s))ds

where P(2) denotes the action of P on the second variable of Vk−1.

These formulas become particularly simple along the diagonal, i. e., forx = y. We havefor any normally hyperbolic operatorP

V0(x,x) = µx(x)−1/2Πx

x = idEx.

Fork≥ 1 we get

Vk(x,x) = −kµ−1/2x (x)︸︷︷︸

=1

·1 · Πxx︸︷︷︸

=id

∫ 1

0sk−1 Πx

x︸︷︷︸=id

(P(2)Vk−1)(x,x)µ−1/2x (x)ds

= −(P(2)Vk−1)(x,x).

We computeV1(x,x) for P = ∇ +B. By (2.6) and Lemma 1.5.6 we have

V1(x,x) = −(P(2)V0)(x,x)

= −P(µ−1/2x Πx

•)(x)

= −µ−1/2x (x) ·P(Πx

•)(x)+2∇gradµx(x)︸︷︷︸=0

Πx•(x)− (µ−1/2

x )(x) · idEx

= −(∇ +B)(Πx•)(x)− (µ−1/2

x )(x) · idEx

= −B|x− (µ−1/2x )(x) · idEx.


From Corollary 1.3.18 we conclude

V1(x,x) =scal(x)

6idEx −B|x.

Remark 2.3.2. We compare our definition of Hadamard coefficients with the definitionused in [Gunther1988] and in [Baum-Kath1996]. In [Gunther1988, Chap. 3, Prop. 1.3]Hadamard coefficientsUk are solutions of the differential equations

L[Γx,Uk(x, ·)]+ (M(x, ·)+2k)Uk(x, ·) = −PUk−1(x, ·) (2.9)

with initial conditionsU0(x) = idEx, where, in our terminology, L[ f , ·] denotes−∇gradf (·)for theP-compatible connection∇, andM(x, ·) = 1

2Γx−n. Hence (2.9) reads as

−∇gradΓxUk(x, ·)+(

12Γx−n+2k

)Uk(x, ·) = −PUk−1(x, ·).

We recover our defining equations (2.3) after the substitution

Uk = 12k·k!

Vk.

2.4 True fundamental solutions on small domains

In this section we show existence of “true” fundamental solutions in the sense of Defini-tion 2.1.1 on sufficiently small causal domains in a timeoriented Lorentzian manifoldM.Assume thatΩ′ ⊂ M is a geodesically convex open subset. We then have the HadamardcoefficientsVj ∈C∞(Ω′×Ω′,E∗

⊠E) and for allx∈ Ω′ the formal fundamental solutions

R±(x) =∞

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x).

Fix an integerN ≥ n2 wheren is the dimension of the manifoldM. Then for all j ≥ N the

distributionRΩ′± (2+2 j,x) is a continuous function onΩ′. Hence we can split the formal

fundamental solutions

R±(x) =N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)+

∞

∑j=N

Vj(x, ·)RΩ′± (2+2 j,x)

where∑N−1j=0 Vj(x, ·)RΩ′

± (2+ 2 j,x) is a well-definedE∗x -valued distribution inE over Ω′

and∑∞j=NVj(x, ·)RΩ′

± (2+ 2 j,x) is a formal sum of continuous sections,Vj(x, ·)RΩ′± (2+

2 j,x) ∈C0(Ω′,E∗x ⊗E) for j ≥ N.

Using suitable cut-offs we will now replace the infinite formal part of the series by aconvergent series. Letσ : R → R be a smooth function vanishing outside[−1,1], suchthat σ ≡ 1 on [− 1

2, 12] and 0≤ σ ≤ 1 everywhere. We need the following elementary

lemma.

2.4. True fundamental solutions on small domains 45

Lemma 2.4.1. For every l∈ N and everyβ ≥ l + 1 there exists a constant c(l ,β ) suchthat for all 0 < ε ≤ 1 we have

∥∥∥∥dl

dtl(σ(t/ε)tβ )

∥∥∥∥C0(R)

≤ ε ·c(l ,β ) · ‖σ‖Cl (R).

Proof.

∥∥∥∥dl

dtl(σ(t/ε)tβ )

∥∥∥∥C0(R)

≤l

∑m=0

(lm

)∥∥∥∥1

εmσ (m)(t/ε) ·β (β −1) · · ·(β − l +m+1)tβ−l+m

∥∥∥∥C0(R)

=l

∑m=0

(lm

)·β (β −1) · · ·(β − l +m+1)εβ−l

∥∥∥(t/ε)β−l+mσ (m)(t/ε)∥∥∥

C0(R).

Now σ (m)(t/ε) vanishes for |t|/ε ≥ 1 and thus ‖(t/ε)β−m+lσ (m)(t/ε)‖C0(R) ≤‖σ (m)‖C0(R). Moreover,β − l ≥ 1, henceεβ−l ≤ ε. Therefore

∥∥∥∥dl

dtl(σ(t/ε)tβ )

∥∥∥∥C0(R)

≤ εl

∑m=0

(lm

)·β (β −1) · · ·(β − l +m+1)

∥∥∥σ (m)∥∥∥

C0(R)

≤ ε c(l ,β )‖σ‖Cl (R).

We defineΓ ∈ C∞(Ω′ ×Ω′,R) by Γ(x,y) := Γx(y) whereΓx is as in (1.17). Note thatΓ(x,y) = 0 if and only if the geodesic joiningx andy in Ω′ is lightlike. In other words,Γ−1(0) =

⋃x∈Ω′(CΩ′

+ (x)∪CΩ′− (x)).

Lemma 2.4.2. Let Ω ⊂⊂ Ω′ be a relatively compact open subset. Then there exists asequence ofε j ∈ (0,1], j ≥ N, such that for each k≥ 0 the series

(x,y) 7→∞

∑j=N+k

σ(Γ(x,y)/ε j )Vj(x,y)RΩ′± (2+2 j,x)(y)

=

∑∞

j=N+kC(2+2 j,n)σ(Γ(x,y)/ε j)Vj(x,y)Γ(x,y) j+1−n/2 if y ∈ JΩ′± (x)

0 otherwise

converges in Ck(Ω×Ω,E∗⊠E). In particular, the series

(x,y) 7→∞

∑j=N

σ(Γ(x,y)/ε j)Vj(x,y)RΩ′± (2+2 j,x)(y)

defines a continuous section overΩ×Ω and a smooth section over(Ω×Ω)\Γ−1(0).


Proof. For j ≥ N ≥ n2 the exponent inΓ(x,y) j+1−n/2 is positive. Therefore the piecewise

definition of thej-th summand yields a continuous section overΩ′.The factorσ(Γ(x,y)/ε j ) vanishes wheneverΓ(x,y) ≥ ε j . Hence forj ≥ N ≥ n

2 and 0<ε j ≤ 1∥∥∥(x,y) 7→ σ(Γ(x,y)/ε j )Vj(x,y)RΩ′

± (2+2 j,x)(y)∥∥∥

C0(Ω×Ω)

≤ C(2+2 j,n) ‖Vj‖C0(Ω×Ω) ε j+1−n/2j

≤ C(2+2 j,n) ‖Vj‖C0(Ω×Ω) ε j .

Hence if we chooseε j ∈ (0,1] such that

C(2+2 j,n) ‖Vj‖C0(Ω×Ω) ε j < 2− j ,

then the series converges in theC0-norm and therefore defines a continuous section.

For k ≥ 0 and j ≥ N + k ≥ n2 + k the functionΓ j+1−n

2 vanishes to(k + 1)-st orderalong Γ−1(0). Thus the j-th summand in the series is of regularityCk. Writingσ j(t) := σ(t/ε j)t j+1−n/2 we know from Lemma 2.4.1 that

‖σ j‖Ck(R) ≤ ε j ·c1(k, j,n) · ‖σ‖Ck(R)

where here and henceforthc1,c2, . . . denote certain universal positive constants whoseprecise values are of no importance. Using Lemmas 1.1.11 and1.1.12 we obtain

∥∥∥(x,y) 7→ σ(Γ(x,y)/ε j )Vj(x,y)RΩ′± (2+2 j,x)(y)

∥∥∥Ck(Ω×Ω)

≤ C(2+2 j,n)‖(σ j Γ) ·Vj‖Ck(Ω×Ω)

≤ c2(k, j,n) · ‖σ j Γ‖Ck(Ω×Ω) · ‖Vj‖Ck(Ω×Ω)

≤ c3(k, j,n) · ‖σ j‖Ck(R) · maxℓ=0,...,k

‖Γ‖ℓCk(Ω×Ω)

· ‖Vj‖Ck(Ω×Ω)

≤ c4(k, j,n) · ε j · ‖σ‖Ck(R) · maxℓ=0,...,k


· ‖Vj‖Ck(Ω×Ω).

Hence if we add the (finitely many) conditions onε j that

c4(k, j,n) · ε j · ‖Vj‖Ck(Ω×Ω) ≤ 2− j

for all k≤ j −N, then we have for fixedk∥∥∥(x,y) 7→ σ(Γ(x,y)/ε j )Vj(x,y)RΩ′

± (2+2 j,x)(y)∥∥∥

Ck(Ω×Ω)

≤ 2− j · ‖σ‖Ck(R) · maxℓ=0,...,k


for all j ≥ N+k. Thus the series

(x,y) 7→∞

∑j=N+k



converges inCk(Ω×Ω,E∗⊠ E). All summandsσ(Γ(x,y)/ε j )Vj(x,y)RΩ′

± (2+ 2 j,x)(y)are smooth onΩ×Ω\Γ−1(0), thus

(x,y) 7→∞

∑j=N


=N+k−1

∑j=N


+∞

∑j=N+k

σ(Γ(x,y)/ε j)Vj(x,y)RΩ′± (2+2 j,x)(y)

is Ck for all k, hence smooth on(Ω×Ω)\Γ−1(0).

Define distributionsR+(x) andR−(x) by

R±(x) :=N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)+

∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x).

By Lemma 2.4.2 and the properties of Riesz distributions we know that

supp(R±(x)) ⊂ JΩ′± (x), (2.10)

singsupp(R±(x)) ⊂CΩ′± (x), (2.11)

and that ord(R±(x)) ≤ n+1.

Lemma 2.4.3.Theε j in Lemma 2.4.2 can be chosen such that in addition to the assertionin Lemma 2.4.2 we have onΩ

P(2)R±(x) = δx +K±(x, ·) (2.12)

with smooth K± ∈C∞(Ω×Ω,E∗⊠E).

Proof. From properties (2.2) and (2.3) of the Hadamard coefficientswe know

P(2)

(N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)

)= δx +(P(2)VN−1(x, ·))RΩ′

± (2N,x). (2.13)

Moreover, by Lemma 1.1.10 we may interchangeP with the infinite sum and we get

P(2)

(∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)

)

=∞

∑j=N

P(2)

(σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′

± (2+2 j,x))

=∞

∑j=N

((2)(σ(Γ(x, ·)/ε j ))Vj(x, ·)RΩ′

± (2+2 j,x)−2∇(2)grad(2)σ(Γ(x,·)/ε j )

(Vj(x, ·)RΩ′± (2+2 j,x))

+σ(Γ(x, ·)/ε j)P(2)(Vj(x, ·)RΩ′± (2+2 j,x))

)


Here and in the following(2), grad(2), and∇(2) indicate that the operators are appliedwith respect to they-variable just as forP(2).

Abbreviating Σ1 := ∑∞j=N (2)(σ(Γ(x, ·)/ε j ))Vj(x, ·)RΩ′

± (2 + 2 j,x) and Σ2 :=

−2∑∞j=N ∇(2)

grad(2)σ(Γ(x,·)/ε j )(Vj(x, ·)RΩ′

± (2+2 j,x)) we have

P(2)

(∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)

)

= Σ1 + Σ2 +∞

∑j=N

σ(Γ(x, ·)/ε j )P(2)(Vj(x, ·)RΩ′± (2+2 j,x))

= Σ1 + Σ2 +∞

∑j=N

σ(Γ(x, ·)/ε j )((P(2)Vj(x, ·))RΩ′

± (2+2 j,x)−2∇(2)

grad(2)RΩ′± (2+2 j ,x)

Vj(x, ·)

+Vj(x, ·)(2)RΩ′± (2+2 j,x)

).

Properties (2.2) and (2.3) of the Hadamard coefficients tellus

Vj(x, ·)(2)RΩ′± (2+2 j,x)−2∇(2)

grad(2)RΩ′± (2+2 j ,x)

Vj(x, ·) = −P(2)(Vj−1(x, ·)RΩ′± (2+2 j,x))

and hence

P(2)

(∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)

)

= Σ1 + Σ2+∞

∑j=N

σ(Γ(x, ·)/ε j )((P(2)Vj(x, ·))RΩ′

± (2+2 j,x)−P(2)Vj−1RΩ′± (2 j,x)

)

= Σ1 + Σ2−σ(Γ(x, ·)/εN)P(2)VN−1RΩ′± (2N,x)

+∞

∑j=N

(σ(Γ(x, ·)/ε j )−σ(Γ(x, ·)/ε j+1)

)(P(2)Vj(x, ·))RΩ′

± (2+2 j,x).

Putting Σ3 := ∑∞j=N

(σ(Γ(x, ·)/ε j )−σ(Γ(x, ·)/ε j+1)

)(P(2)Vj(x, ·))RΩ′

± (2 + 2 j,x) andcombining with (2.13) yields

P(2)R±(x)−δx = (1−σ(Γ(x, ·)/εN−1))P(2)VN−1(x, ·)RΩ′± (2N,x)+Σ1+Σ2+Σ3. (2.14)

We have to show that the right hand side is actually smooth in both variables. Since

P(2)VN−1(x,y)RΩ′± (2N,x)(y) =

C(2N,n)P(2)VN−1(x,y)Γ(x,y)N−n/2, if y∈ JΩ′

± (x)0, otherwise

is smooth on(Ω′×Ω′)\Γ−1(0) and since 1−σ(Γ(x, ·)/ε j) vanishes on a neighborhoodof Γ−1(0) we have that

(x,y) 7→ (1−σ(Γ(x,y)/ε j)) ·P(2)VN−1(x,y)RΩ′± (2N,x)(y)


is smooth. Similarly, the individual terms in the three infinite sums are smooth sectionsbecauseσ(Γ/ε j)−σ(Γ/ε j+1), grad(2)(σ Γ

ε j), and(2)(σ Γ

ε j) all vanish on a neigh-

borhood ofΓ−1(0). It remains to be shown that the three series in (2.14) converge in allCk-norms.We start withΣ2. Let Sj := (x,y) ∈ Ω′×Ω′ | ε j

2 ≤ Γ(x,y) ≤ ε j.

time direction

JΩ′+ (x)

b

x

y∈ Ω′ | ε j

2 ≤ Γ(x,y) ≤ ε j

Fig. 13: Section ofSj for fixed x

Since grad(2)(σ Γε j

) vanishes outside the “strip”Sj , there exist constantsc1(k,n), c2(k,n)

andc3(k,n, j) such that

∥∥∥∥∥∇(2)

grad(2)(σ Γε j

)

(Vj(·, ·)RΩ′

± (2+2 j, ·))∥∥∥∥∥

Ck(Ω×Ω)

=

∥∥∥∥∥∇(2)

grad(2)(σ Γε j

)

(Vj(·, ·)RΩ′

± (2+2 j, ·))∥∥∥∥∥

Ck(Ω×Ω∩Sj )

≤ c1(k,n) ·∥∥∥σ Γ

ε j

∥∥∥Ck+1(Ω×Ω∩Sj )

·∥∥∥Vj(·, ·)RΩ′

± (2+2 j, ·)∥∥∥

Ck+1(Ω×Ω∩Sj )

≤ c2(k,n) · ‖σ‖Ck+1(R) · maxℓ=0,...,k+1

‖ Γε j‖ℓ

Ck+1(Ω×Ω∩Sj )

·‖Vj‖Ck+1(Ω×Ω∩Sj )· ‖RΩ′

± (2+2 j, ·)‖Ck+1(Ω×Ω∩Sj )

≤ c2(k,n) · 1

εk+1j

· ‖σ‖Ck+1(R) · maxℓ=0,...,k+1

‖Γ‖ℓCk+1(Ω×Ω)

·‖Vj‖Ck+1(Ω×Ω) · ‖RΩ′± (2+2 j, ·)‖Ck+1(Ω×Ω∩Sj )

≤ c3(k,n, j) · 1

εk+1j

· ‖σ‖Ck+1(R) · maxℓ=0,...,k+1


·‖Vj‖Ck+1(Ω×Ω) · ‖Γ1+ j−n/2‖Ck+1(Ω×Ω∩Sj ).


By Lemma 1.1.12 we have

‖Γ1+ j−n/2‖Ck+1(Ω×Ω∩Sj )

≤ c4(k) · ‖t 7→ t1+ j−n/2‖Ck+1([ε j /2,ε j ])· maxℓ=0,...,k+1

‖Γ‖ℓCk+1(Ω×Ω∩Sj )

≤ c5(k, j,n) · ε j−n/2−kj · max

ℓ=0,...,k+1‖Γ‖ℓ

Ck+1(Ω×Ω∩Sj ).

Thus∥∥∥∥∥∇(2)

grad(2)(σ Γε j

)

(Vj(·, ·)RΩ′

± (2+2 j, ·))∥∥∥∥∥

Ck(Ω×Ω)

≤ c6(k, j,n) · ‖σ‖Ck+1(R) ·(

maxℓ=0,...,k+1


)2

· ‖Vj‖Ck+1(Ω×Ω) · εj−2k−n/2−1j

≤ c6(k, j,n) · ‖σ‖Ck+1(R) · maxℓ=0,...,k+1

‖Γ‖2ℓCk+1(Ω×Ω)

· ‖Vj‖Ck+1(Ω×Ω) · ε j

if j ≥ 2k+n/2+2. Hence if we require the (finitely many) conditions

c6(k, j,n) · ‖Vj‖Ck+1(Ω×Ω) · ε j ≤ 2− j

on ε j for all k≤ j/2−n/4−1, then almost allj-th terms of the seriesΣ2 are bounded intheCk-norm by 2− j · ‖σ‖Ck+1(R) ·maxℓ=0,...,k+1‖Γ‖2ℓ

Ck+1(Ω×Ω). ThusΣ2 converges in the

Ck-norm for anyk and defines a smooth section inE∗⊠E overΩ×Ω.

The seriesΣ1 is treated similarly. To examineΣ3 we observe that forj ≥ k+ n2

∥∥∥((σ Γ

ε j)− (σ Γ

ε j+1))· (P(2)Vj) ·RΩ′

± (2+2 j, ·)∥∥∥

Ck(Ω×Ω)

≤ c7( j,n) ·∥∥∥((σ Γ

ε j)− (σ Γ

ε j+1))· (P(2)Vj) ·Γ1+ j−n/2

∥∥∥Ck(Ω×Ω)

≤ c8(k, j,n) ·∥∥∥((σ Γ

ε j)− (σ Γ

ε j+1))·Γk+1

∥∥∥Ck(Ω×Ω)

·∥∥P(2)Vj

∥∥Ck(Ω×Ω)

·∥∥∥Γ j−k−n/2

∥∥∥Ck(Ω×Ω)

≤ c8(k, j,n) ·(∥∥∥(σ Γ

ε j) ·Γk+1

∥∥∥Ck(Ω×Ω)

+∥∥∥(σ Γ

ε j+1) ·Γk+1

∥∥∥Ck(Ω×Ω)

)

·∥∥P(2)Vj

∥∥Ck(Ω×Ω)

·∥∥∥Γ j−k−n/2

∥∥∥Ck(Ω×Ω)

. (2.15)

Puttingσ j(t) := σ(t/ε j) · tk+1 we have(σ Γε j

) ·Γk+1 = σ j Γ. Hence by Lemmas 1.1.12

and 2.4.1∥∥∥(σ Γ

ε j) ·Γk+1

∥∥∥Ck(Ω×Ω)

=∥∥σ j Γ

∥∥Ck(Ω×Ω)

≤ c9(k,n) ·∥∥σ j∥∥

Ck(R)· maxℓ=0,...,k


≤ c10(k,n) · ε j · ‖σ‖Ck(R) · maxℓ=0,...,k

‖Γ‖ℓCk(Ω×Ω) .


Plugging this into (2.15) yields∥∥∥((σ Γ

ε j)− (σ Γ

ε j+1))· (P(2)Vj) ·RΩ′

± (2+2 j, ·)∥∥∥

Ck(Ω×Ω)

≤ c11(k, j,n) · (ε j + ε j+1) · ‖σ‖Ck(R) · maxℓ=0,...,k


·∥∥P(2)Vj

∥∥Ck(Ω×Ω)

·∥∥∥Γ j−k−n/2

∥∥∥Ck(Ω×Ω)

.

Hence if we add the conditions onε j that

c11(k, j,n) · ε j ·∥∥P(2)Vj

∥∥Ck(Ω×Ω)

·∥∥∥Γ j−k−n/2

∥∥∥Ck(Ω×Ω)

≤ 2− j−1

for all k≤ j − n2 and

c11(k, j −1,n) · ε j ·∥∥P(2)Vj−1

∥∥Ck(Ω×Ω)

·∥∥∥Γ j−1−k−n/2

∥∥∥Ck(Ω×Ω)

≤ 2− j−2

for all k ≤ j − 1− n2, then we have that almost allj-th terms inΣ3 are bounded in the

Ck-norm by 2− j · ‖σ‖Ck(R) ·maxℓ=0,...,k‖Γ‖ℓCk(Ω×Ω). ThusΣ3 defines a smooth section as

well.

Lemma 2.4.4. Theε j in Lemmas 2.4.2 and 2.4.3 can be chosen such that in additionthere is a constant C> 0 so that

|R±(x)[ϕ ]| ≤C · ‖ϕ‖Cn+1(Ω)

for all x ∈ Ω and all ϕ ∈ D(Ω,E∗). In particular, R(x) is of order at most n+1. More-over, the map x7→ R±(x)[ϕ ] is for every fixedϕ ∈ D(Ω,E∗) a smooth section in E∗,

R±(·)[ϕ ] ∈C∞(Ω,E∗).

We know already that for eachx∈ Ω the distributionR(x) is of order at mostn+1. Thepoint of the lemma is that the constantC in the estimate|R±(x)[ϕ ]| ≤C · ‖ϕ‖Cn+1(Ω) canbe chosen independently ofx.

Proof. Recall the definition ofR±(x),

R±(x) =N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)+

∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x).

By Proposition 1.4.2 (10) there are constantsCj > 0 such that|RΩ′± (2+ 2 j,x)[ϕ ]| ≤ Cj ·

‖ϕ‖Cn+1(Ω) for all ϕ and allx∈ Ω. Thus there is a constantC′ > 0 such that

∣∣∣∣∣N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)[ϕ ]

∣∣∣∣∣≤C′ · ‖ϕ‖Cn+1(Ω)


for all ϕ and all x ∈ Ω. The remainder term∑∞j=N σ(Γ(x,y)/ε j )Vj(x,y)RΩ′

± (2 +2 j,x)(y) =: f (x,y) is a continuous section, hence∣∣∣∣∣

∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)[ϕ ]

∣∣∣∣∣ ≤ ‖ f‖C0(Ω×Ω) ·vol(Ω) · ‖ϕ‖C0(Ω)

≤ ‖ f‖C0(Ω×Ω) ·vol(Ω) · ‖ϕ‖Cn+1(Ω)

for all ϕ and allx∈ Ω. ThereforeC := C′ +‖ f‖C0(Ω×Ω) ·vol(Ω) does the job.To see smoothness inx we fix k≥ 0 and we write

R±(x)[ϕ ] =N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)[ϕ ]+

N+k−1

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)[ϕ ]

+∞

∑j=N+k

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)[ϕ ].

By Proposition 1.4.2 (11) the summandsVj(x, ·)RΩ′± (2 + 2 j,x)[ϕ ] and

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2 + 2 j,x)[ϕ ] depend smoothly onx. By Lemma 2.4.2 the

remainder∑∞j=N+k σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′

± (2+2 j,x)[ϕ ] is Ck. Thusx 7→ R±(x)[ϕ ] is Ck

for everyk, hence smooth.

Definition 2.4.5. If M is a timeoriented Lorentzian manifold, then we call a subsetS⊂M ×M future-stretchedwith respect toM if y ∈ JM

+ (x) whenever(x,y) ∈ S. We call itstrictly future-stretchedwith respect toM if y∈ IM

+ (x) whenever(x,y) ∈ S. Analogously,we definepast-stretchedandstrictly past-stretchedsubsets.

We summarize the results obtained so far.

Proposition 2.4.6. Let M be an n-dimensional timeoriented Lorentzian manifoldand letP be a normally hyperbolic operator acting on sections in a vector bundle E over M.Let Ω′ ⊂ M be a convex open subset. Fix an integer N≥ n

2 and fix a smooth functionσ : R→ R satisfyingσ ≡ 1 outside[−1,1], σ ≡ 0 on [− 1

2, 12], and0≤ σ ≤ 1 everywhere.

Then for every relatively compact open subsetΩ ⊂⊂ Ω′ there exists a sequenceε j > 0,j ≥ N, such that for every x∈ Ω

R±(x) =N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)+

∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)

defines a distribution onΩ satisfying

(1) supp(R±(x)) ⊂ JΩ′± (x),

(2) singsupp(R±(x)) ⊂CΩ′± (x),

(3) P(2)R±(x) = δx +K±(x, ·) with smooth K± ∈C∞(Ω×Ω,E∗⊠E),

(4) supp(K+) is future-stretched andsupp(K−) is past-stretched with respect toΩ′,


(5) R±(x)[ϕ ] depends smoothly on x for every fixedϕ ∈ D(Ω,E∗),

(6) there is a constant C> 0 such that|R±(x)[ϕ ]| ≤ C · ‖ϕ‖Cn+1(Ω) for all x ∈ Ω andall ϕ ∈ D(Ω,E∗).

Proof. The only thing that remains to be shown is the statement (4). Recall from (2.14)that in the notation of the proof of Lemma 2.4.3

K±(x,y) = (1−σ(Γ(x,y)/εN−1)) ·P(2)VN−1(x,y) ·RΩ′± (2N,x)(y)+ Σ1 + Σ2+ Σ3.

The first term as well as all summands in the three infinite seriesΣ1, Σ2, andΣ3 contain afactorRΩ′

± (2 j,x)(y) for some j ≥ N. Hence ifK+(x,y) 6= 0, theny∈ supp(RΩ′± (2 j,x)) ⊂

JΩ′+ (x). In other words,(x,y) ∈ Ω×Ω |K+(x,y) 6= 0 is future-stretched with respect

to Ω′. SinceΩ′ is geodesically convex causal futures are closed. Hence supp(K+) =(x,y) ∈ Ω×Ω |K+(x,y) 6= 0 is future-stretched with respect toΩ′ as well. In the sameway one sees that supp(K−) is past-stretched.

Definition 2.4.7. If the ε j are chosen as in Proposition 2.4.6, then we call

R±(x) =N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)+

∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x)

anapproximate advancedor retarded fundamental solutionrespectively.

From now on we assume thatΩ ⊂⊂ Ω′ is a relatively compactcausalsubset. Then foreveryx ∈ Ω we haveJΩ

±(x) = JΩ′± (x)∩ Ω. We fix approximate fundamental solutions

R±(x).We use the correspondingK± as an integral kernel to define an integral operator. Set foru∈C0(Ω,E∗) andx∈ Ω

(K±u)(x) :=∫

ΩK±(x,y)u(y) dV(y). (2.16)

SinceK± is C∞ so isK±u, i. e.,K±u∈ C∞(Ω,E∗). By the properties of the support ofK± the integrandK±(x,y)u(y) vanishes unlessy∈ JΩ

±(x)∩supp(u). Hence(K±u)(x) = 0

if JΩ± (x)∩supp(u) = /0. In other words,

supp(K±u) ⊂ JΩ∓(supp(u)). (2.17)

If we putCk :=∫

Ω ‖K±(·,y)‖Ck(Ω) dV(y), then

‖K±u‖Ck(Ω) ≤Ck · ‖u‖C0(Ω).

Hence (2.16) defines a bounded linear map

K± : C0(Ω,E∗) →Ck(Ω,E∗)

for all k≥ 0.


Lemma 2.4.8. Let Ω ⊂⊂ Ω′ be causal. SupposeΩ is so small that

vol(Ω) · ‖K±‖C0(Ω×Ω) < 1. (2.18)

Thenid+K± : Ck(Ω,E∗) →Ck(Ω,E∗)

is an isomorphism with bounded inverse for all k= 0,1,2, . . .. The inverse is given by theseries

(id+K±)−1 =∞

∑j=0

(−K±) j

which converges in all Ck-operator norms. The operator(id+K+)−1K+ has a smoothintegral kernel with future-stretched support (with respect to Ω). The operator(id +K−)−1 K− has a smooth integral kernel with past-stretched support (with respect toΩ).

Proof. The operatorK± is bounded as an operatorC0(Ω,E∗) → Ck(Ω,E∗). Thus id+K± defines a bounded operatorCk(Ω,E∗) →Ck(Ω,E∗) for all k. Now

‖K±u‖C0(Ω) ≤ vol(Ω) · ‖K±‖C0(Ω×Ω) · ‖u‖C0(Ω)

= (1−η) · ‖u‖C0(Ω)

whereη := 1− vol(Ω) · ‖K±‖C0(Ω×Ω) > 0. Hence theC0-operator norm ofK± is less

than 1 so that the Neumann series∑∞j=0(−K±) j converges in theC0-operator norm and

gives the inverse of id+K± onC0(Ω,E∗).Next we replace theCk-norm‖ ·‖Ck(Ω) onCk(Ω,E∗) as defined in (1.1) by the equivalentnorm

9u9Ck(Ω) := ‖u‖C0(Ω) +η

2vol(Ω)‖K±‖Ck(Ω×Ω) +1‖u‖Ck(Ω).

Then

9K±u9Ck(Ω)

= ‖K±u‖C0(Ω) +η

2vol(Ω)‖K±‖Ck(Ω×Ω) +1‖K±u‖Ck(Ω)

≤ (1−η) · ‖u‖C0(Ω) +η

2vol(Ω)‖K±‖Ck(Ω×Ω) +1vol(Ω)‖K±‖Ck(Ω×Ω)‖u‖C0(Ω)

≤ (1− η2

)‖u‖C0(Ω)

≤ (1− η2

)9u9Ck(Ω) .

This shows that with respect to9 ·9Ck(Ω) theCk-operator norm ofK± is less than 1. Thus

the Neumann series∑∞j=0(−K±) j converges in allCk-operator norms and id+K± is an

isomorphism with bounded inverse on allCk(Ω,E∗).


For j ≥ 1 the integral kernel of(K±) j is given by

K( j)± (x,y) :=

∫

Ω· · ·∫

ΩK±(x,z1)K±(z1,z2) · · ·K±(zj−1,y) dV(z1) · · · dV(zj−1).

Thus supp(K( j)± ) ⊂

(x,y) ∈ Ω×Ω |y∈ JΩ

±(x)

and

‖K( j)± ‖Ck(Ω×Ω) ≤ ‖K±‖2

Ck(Ω×Ω)·vol(Ω) j−1 · ‖K±‖ j−2

C0(Ω×Ω)≤ δ j−2 ·vol(Ω) · ‖K±‖2

Ck(Ω×Ω)

whereδ := vol(Ω) · ‖K±‖C0(Ω×Ω) < 1. Hence the series

∞

∑j=1

(−1) j−1K( j)±

converges in allCk(Ω×Ω,E∗⊠ E). Since this series yields the integral kernel of(id +

K±)−1K± it is smooth and its support is contained in(x,y) ∈ Ω×Ω |y∈ JΩ

± (x)

.

Corollary 2.4.9. Let Ω ⊂⊂ Ω′ be as in Lemma 2.4.8. Then for each u∈C0(Ω,E)

supp((id+K±)−1u) ⊂ JΩ∓(supp(u)).

Proof. We observe that

(id+K±)−1u = u− (id+K±)−1K±u.

Now supp(u) ⊂ JΩ∓(supp(u)) and supp((id+K±)−1K±u) ⊂ JΩ

∓(supp(u)) by the proper-ties of the integral kernel of(id+K±)−1K±.

Fix ϕ ∈ D(Ω,E∗). Thenx 7→ R±(x)[ϕ ] defines a smooth section inE∗ over Ω withsupport contained inJΩ′

∓ (supp(ϕ))∩Ω = JΩ∓(supp(ϕ)). Hence

FΩ± (·)[ϕ ] := (id+K±)−1(R±(·)[ϕ ]) (2.19)

defines a smooth section inE∗ with

supp(FΩ± (·)[ϕ ]) ⊂ JΩ

∓(supp(R±(·)[ϕ ])) ⊂ JΩ∓(JΩ

∓(supp(ϕ))) = JΩ∓(supp(ϕ)). (2.20)

Lemma 2.4.10.For each x∈ Ω the mapD(Ω,E∗) 7→ E∗x , ϕ 7→ FΩ

+ (x)[ϕ ], is an advancedfundamental solution at x onΩ andϕ 7→ FΩ

− (x)[ϕ ] is a retarded fundamental solution atx onΩ.

Proof. We first check thatϕ 7→ FΩ± (x)[ϕ ] defines a distribution for any fixedx ∈ Ω. Let

ϕm → ϕ in D(Ω,E∗). Thenϕm → ϕ in Cn+1(Ω,E∗) and by the last point of Proposi-tion 2.4.6R±(·)[ϕm] → R±(·)[ϕ ] in C0(Ω,E∗). Since(id+K±)−1 is bounded onC0 wehaveFΩ

± (·)[ϕm] → FΩ± (·)[ϕ ] in C0. In particular,FΩ

± (x)[ϕm] → FΩ± (x)[ϕ ].


Next we check thatFΩ± (x) are fundamental solutions. We compute

P(2)FΩ± (·)[ϕ ] = FΩ

± (·)[P∗ϕ ]

= (id+K±)−1(R±(·)[P∗ϕ ])

= (id+K±)−1(P(2)R±(·)[ϕ ])

(2.12)= (id+K±)−1(ϕ +K±ϕ)

= ϕ .

Thus for fixedx∈ Ω,PFΩ

± (x)[ϕ ] = ϕ(x) = δx[ϕ ].

Finally, to see that supp(FΩ± (x))⊂ JΩ

±(x) let ϕ ∈D(Ω,E∗) such that supp(ϕ)∩JΩ± (x) = /0.

Thenx 6∈ JΩ∓(supp(ϕ)) and thusFΩ

± (x)[ϕ ] = 0 by (2.20).

We summarize the results of this section.

Proposition 2.4.11. Let M be a timeoriented Lorentzian manifold. Let P be a normallyhyperbolic operator acting on sections in a vector bundle E over M. LetΩ ⊂⊂ M be arelatively compact causal domain. Suppose thatΩ is sufficiently small in the sense that(2.18) holds.Then for each x∈ Ω

(1) the distributions FΩ+ (x) and FΩ− (x) defined in (2.19) are fundamental solutions for

P at x overΩ,

(2) supp(FΩ± (x)) ⊂ JΩ

±(x),

(3) for eachϕ ∈ D(Ω,E∗) the maps x′ 7→ FΩ± (x′)[ϕ ] are smooth sections in E∗ over

Ω.

Corollary 2.4.12. Let M be a timeoriented Lorentzian manifold. Let P be a normallyhyperbolic operator acting on sections in a vector bundle E over M.Then each point in M possesses an arbitrarily small causal neighborhoodΩ such that foreach x∈ Ω there exist fundamental solutions FΩ

± (x) for P overΩ at x. They satisfy

(1) supp(FΩ± (x)) ⊂ JΩ

±(x),

(2) for eachϕ ∈ D(Ω,E∗) the maps x7→ FΩ± (x)[ϕ ] are smooth sections in E∗.

2.5 The formal fundamental solution is asymptotic

Let M be a timeoriented Lorentzian manifold. LetP be a normally hyperbolic operatoracting on sections in a vector bundleE over M. Let Ω′ ⊂ M be a convex domain andlet Ω ⊂ Ω′ be a relatively compact causal domain withΩ ⊂ Ω′. We assume thatΩ is so

2.5. The formal fundamental solution is asymptotic 57

small that Corollary 2.4.12 applies. Using Riesz distributions and Hadamard coefficientswe have constructed the formal fundamental solutions atx∈ Ω

R±(x) =∞

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x),

the approximate fundamental solutions

R±(x) =N−1

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x)+

∞

∑j=N

σ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′± (2+2 j,x),

whereN ≥ n2 is fixed, and the true fundamental solutionsFΩ

± (x),

FΩ± (·)[ϕ ] = (id+K±)−1(R±(·)[ϕ ]).

The purpose of this section is to show that, in a suitable sense, the formal fundamentalsolution is an asymptotic expansion of the true fundamentalsolution. Fork≥ 0 we definethetruncated formal fundamental solution

RN+k± (x) :=

N−1+k

∑j=0

Vj(x, ·)RΩ′± (2+2 j,x).

Hence we cut the formal fundamental solution at the(N+k)-th term. The truncated for-mal fundamental solution is a well-defined distribution onΩ′, RN+k

± (x) ∈ D ′(Ω′,E,E∗x ).

We will show that the true fundamental solution coincides with the truncated formal fun-damental solution up to an error term which is very regular along the light cone. Thelargerk is, the more regular is the error term.

Proposition 2.5.1. For every k∈ N and every x∈ Ω the difference of distributionsFΩ± (x)−RN+k

± (x) is a Ck-section in E. In fact,

(x,y) 7→(

FΩ± (x)−RN+k

± (x))

(y)

is of regularity Ck on Ω×Ω.

Proof. We write(

FΩ± (x)−RN+k

± (x))

(y) =(

FΩ± (x)− R±(x)

)(y)+

(R±(x)−RN+k

± (x))

(y)

and we show that(R±(x)−RN+k

± (x))

(y) and(

FΩ± (x)− R±(x)

)(y) are bothCk in

(x,y). Now

(R±(x)−RN+k

± (x))

(y) =N+k−1

∑j=N

(σ(Γ(x,y)/ε j)−1)Vj(x,y)RΩ′± (2+2 j,x)(y)

+∞

∑j=N+k

σ(Γ(x,y)/ε j )Vj(x,y)RΩ′± (2+2 j,x)(y).


From Lemma 2.4.2 we know that the infinite part(x,y) 7→∑∞

j=N+k σ(Γ(x,y)/ε j )Vj(x,y)RΩ′± (2 + 2 j,x)(y) is Ck. The finite part (x,y) 7→

∑N+k−1j=N (σ(Γ(x,y)/ε j )−1)Vj(x,y)RΩ′

± (2 + 2 j,x)(y) is actually smooth since

σ(Γ/ε j) − 1 vanishes on a neighborhood ofΓ−1(0) which is precisely the locuswhere(x,y) 7→ RΩ′

± (2+2 j,x)(y) is nonsmooth. Furthermore,

FΩ± (·)[ϕ ]− R±(·)[ϕ ] =

((id+K±)−1− id

)(R±(·)[ϕ ])

= −((id+K±)−1 K±

)(R±(·)[ϕ ]).

By Lemma 2.4.8 the operator−((id+K±)−1K±

)has a smooth integral kernel

L±(x,y) whose support is future or past-stretched respectively. Hence

FΩ± (x)[ϕ ]− R±(x)[ϕ ]

=

∫

ΩL±(x,y)R±(y)[ϕ ] dV(y)

=N−1

∑j=0

∫

ΩL±(x,y)Vj(y, ·)RΩ′

± (2+2 j,y)[ϕ ] dV(y)

+N+k−1

∑j=N

∫

ΩL±(x,y)σ(Γ(y, ·)/ε j )Vj(y, ·)RΩ′

± (2+2 j,y)[ϕ ] dV(y)

+

∫

Ω×ΩL±(x,y) f (y,z)ϕ(z) dV(z) dV(y)

where f (y,z) = ∑∞j=N+k σ(Γ(y,z)/ε j )Vj(y,z)RΩ′

± (2 + 2 j,y)(z) is Ck by Lemma 2.4.2.

Thus (x,z) 7→∫

Ω L±(x,y) f (y,z) dV(y) is a Ck-section. WriteVj(y,z) := Vj(y,z) if j ≤N−1 andVj(y,z) := σ(Γ(y,z)/ε j )Vj(y,z) if j ≥ N. It follows from Lemma 1.4.4

∫

ΩL±(x,y)Vj(y, ·)RΩ′

± (2+2 j,y)[ϕ ] dV(y)

=

∫

ΩRΩ′± (2+2 j,y)[z 7→ L±(x,y)Vj(y,z)ϕ(z)] dV(y)

=

∫

ΩRΩ′∓ (2+2 j,z)[y 7→ L±(x,y)Vj(y,z)ϕ(z)] dV(z)

=

∫

ΩRΩ′∓ (2+2 j,z)[y 7→ L±(x,y)Vj(y,z)]ϕ(z) dV(z)

=∫

ΩWj(x,z)ϕ(z) dV(z)

whereWj(x,z) = RΩ′∓ (2 + 2 j,z)[y 7→ L±(x,y)Vj(y,z)] is smooth in(x,z) by Proposi-

tion 1.4.2 (11). Hence

(FΩ± (x)− R±(x)

)(z) =

N+k−1

∑j=0

Wj(x,z)+

∫

ΩL±(x,y) f (y,z) dV(y)

is Ck in (x,z).


The following theorem tells us that the formal fundamental solutions are asymptotic ex-pansions of the true fundamental solutions near the light cone.

Theorem 2.5.2. Let M be a timeoriented Lorentzian manifold. Let P be a normallyhyperbolic operator acting on sections in a vector bundle E.Let Ω ⊂ M be a relativelycompact causal domain and let x∈ Ω. Let FΩ

± denote the fundamental solutions of P at xandRN+k

± (x) the truncated formal fundamental solutions.Then for each k∈ N there exists a constant Ck such that

∥∥∥(

FΩ± (x)−RN+k

± (x))

(y)∥∥∥≤Ck · |Γ(x,y)|k

for all (x,y) ∈ Ω×Ω.

Here‖ · ‖ denotes an auxiliary norm onE∗⊠E. The proof requires some preparation.

Lemma 2.5.3.Let M be a smooth manifold. Let H1,H2 ⊂M be two smooth hypersurfacesglobally defined by the equationsϕ1 = 0 andϕ2 = 0 respectively, whereϕ1,ϕ2 : M → R

are smooth functions on M satisfying dxϕi 6= 0 for every x∈ Hi , i = 1,2. We assume thatH1 and H2 intersect transversally.Let f : M → R be a Ck-function on M, k∈ N. Let k1,k2 ∈ N such that k1 + k2 ≤ k. We

assume that f vanishes to order ki along Hi , i. e., in local coordinates∂|α| f

∂xα (x) = 0 forevery x∈ Hi and every multi-indexα with |α| ≤ ki −1.Then there exists a continuous function F: M → R such that

f = ϕk11 ϕk2

2 F.

Proof of Lemma 2.5.3.We first prove the existence of aCk−k1-functionF1 : M → R suchthat

f = ϕk11 F1.

This is equivalent to saying that the functionf/ϕk11 being well-defined andCk on M \H1

extends to aCk−k1-functionF1 on M. Since it suffices to prove this locally, we introducelocal coordinatesx1, . . . ,xn so thatϕ1(x) = x1. Hence in this local chartH1 = x1 = 0.

Since f (0,x2, . . . ,xn) = ∂ j f∂ (x1) j (0,x2, . . . ,xn) = 0 for any (x2, . . . ,xn) and j ≤ k1 − 1 we

obtain from the Taylor expansion off in thex1-direction to the orderk1−1 with integralremainder term

f (x1,x2, . . . ,xn) =∫ x1

0

(x1− t)k1−1

(k1−1)!∂ k1 f

∂ (x1)k1(t,x2, . . . ,xn)dt.

In particular, forx1 6= 0

f (x1,x2, . . . ,xn) = (x1)k1−1∫ x1

0

1(k1−1)!

(x1− t

x1

)k1−1 ∂ k1 f∂ (x1)k1

(t,x2, . . . ,xn)dt

=(x1)k1−1

(k1−1)!

∫ 1

0(1−u)k1−1x1 ∂ k1 f

∂ (x1)k1(x1u,x2, . . . ,xn)du

=(x1)k1

(k1−1)!

∫ 1

0(1−u)k1−1 ∂ k1 f

∂ (x1)k1(x1u,x2, . . . ,xn)du.


Now F1(x1, . . . ,xn) := 1(k1−1)!

∫ 10 (1 − u)k1−1 ∂ k1 f

∂ (x1)k1(x1u,x2, . . . ,xn)du yields a Ck−k1-

function because∂ k1 f∂ (x1)k1

is Ck−k1. Moreover, we have

f = (x1)k1 ·F1 = ϕk1 ·F1.

On M \H1 we haveF1 = f/ϕk11 and soF1 vanishes to the orderk2 on H2 \H1 becausef

does. SinceH1 andH2 intersect transversally the subsetH2\H1 is dense inH2. Thereforethe functionF1 vanishes to the orderk2 on all of H2. Applying the considerations aboveto F1 yields aCk−k1−k2-functionF : M → R such thatF1 = ϕk2

2 ·F . This concludes theproof.

Lemma 2.5.4.Let f : Rn →R a C3k+1-function. We equipRn with its standard Minkowskiproduct〈·, ·〉 and we assume that f vanishes on all spacelike vectors.Then there exists a continuous function h: Rn → R such that

f = h · γk

whereγ(x) = −〈x,x〉.

Proof of Lemma 2.5.4.The problem here is that the hypersurfaceγ = 0 is the light conewhich contains 0 as a singular point so that Lemma 2.5.3 does not apply directly. We willget around this difficulty by resolving the singularity.Let π : M := R×Sn−1 → Rn be the map defined byπ(t,x) := tx. It is smooth onM =R×Sn−1 and outsideπ−1(0) = 0×Sn−1 it is a two-fold covering ofRn \ 0. Thefunction f := f π : M → R is C3k+1 since f is.Consider the functionsγ : M → R, γ(t,x) := γ(x), andπ1 : M → R, π1(t,x) := t. Thesefunctions are smooth and have only regular points onM. For γ this follows fromdxγ 6= 0for everyx∈ Sn−1. ThereforeC(0) := γ−1(0) and0×Sn−1 = π−1

1 (0) are smoothembedded hypersurfaces. Since the differentials ofγ and ofπ1 are linearly independentthe hypersurfaces intersect transversally. Furthermore,one obviously hasπ(C(0)) =C(0)andπ(0×Sn−1) = 0.Since f isC3k+1 and vanishes on all spacelike vectorsf vanishes to the order 3k+2 alongC(0) (and in particular at 0). Hencef vanishes to the order 3k+2 alongC(0) and along0×Sn−1. Applying Lemma 2.5.3 tof , ϕ1 := π1 andϕ2 := γ, with k1 := 2k+ 1 andk2 := k, yields a continuous functionF : R×Sn−1 → R such that

f = π2k+11 · γ k · F. (2.21)

Fory∈ Rn we set

h(y) :=

‖y‖ · F(‖y‖, y

‖y‖ ) if y 6= 00 if y = 0,

where‖ ·‖ is the standard Euclidean norm onRn. The functionh is obviously continuous


onRn. It remains to showf = γk ·h. Fory∈ Rn\ 0 we have

f (y) = f

(‖y‖ · y

‖y‖

)

= f

(‖y‖, y

‖y‖

)

(2.21)= ‖y‖2k+1 · γ

(y‖y‖

)k

· F(‖y‖, y

‖y‖

)

= ‖y‖2k · γ(

y‖y‖

)k

·h(y)

= γ(y)k ·h(y).

Fory = 0 the equationf (y) = γ(y)k ·h(y) holds trivially.

Proof of Theorem 2.5.2.Repeatedly using Proposition 1.4.2 (3) we find constantsC′j such

that(

FΩ± (x)−RN+k

± (x))

(y)

=(

FΩ± (x)−RN+3k+1

± (x))

(y)+N+3k

∑j=N+k

Vj(x,y) ·RΩ′± (2+2 j,x)(y)

=(

FΩ± (x)−RN+3k+1

± (x))

(y)+N+3k

∑j=N+k

Vj(x,y) ·C′j ·Γ(x,y)k ·RΩ′

± (2+2( j −k),x)(y).

Now h j(x,y) := C′j ·Vj(x,y) ·RΩ′

± (2+ 2( j − k),x)(y) is continuous since 2+ 2( j − k) ≥2+2N≥ 2+n> n. By Proposition 2.5.1 the section(x,y) 7→

(FΩ± (x)−RN+3k+1

± (x))(y)

is of regularityC3k+1. Moreover, we know supp(FΩ± (x)−RN+3k+1

± (x)) ⊂ JΩ±(x). Hence

we may apply Lemma 2.5.4 in normal coordinates and we obtain acontinuous sectionhsuch that (

FΩ± (x)−RN+3k+1

± (x))

(y) = Γ(x,y)k ·h(x,y).

This shows

(FΩ± (x)−RN+k

± (x))

(y) =

(h(x,y)+

N+3k

∑j=N+k

h j(x,y)

)Γ(x,y)k.

NowCk := ‖h+ ∑N+3kj=N+k h j‖C0(Ω×Ω) does the job.

Remark 2.5.5. It is interesting to compare Theorem 2.5.2 to a similar situation arisingin the world of Riemannian manifolds. IfM is ann-dimensional compact Riemannianmanifold, then the operators analogous to normally hyperbolic operators on Lorentzianmanifolds are theLaplace typeoperators. They are defined formally just like normallyhyperbolic operators, namely their principal symbol must be given by the metric. Analyt-ically however, they behave very differently because they are elliptic.


If L is a nonnegative formally selfadjoint Laplace type operator onM, then it is essentiallyselfadjoint and one can form the semi-groupt 7→ e−tL whereL is the selfadjoint extensionof L. Fort > 0 the operatore−tL has a smooth integral kernelKt(x,y). One can show thatthere is an asymptotic expansion of this “heat kernel”

Kt(x,x) ∼1

(4πt)n/2

∞

∑k=0

αk(x)tk

ast ց 0. The coefficientsαk(x) are given by a universal expression in the coefficients ofL and their covariant derivatives and the curvature ofM and its covariant derivatives.

Even though this asymptotic expansion is very different in nature from the one in The-orem 2.5.2, it turns out that the Hadamard coefficients on thediagonalVk(x,x) of a nor-mally hyperbolic operatorP on ann-dimensional Lorentzian manifold aregiven by thesame universal expressionin the coefficients ofP and their covariant derivatives and thecurvature ofM and its covariant derivatives asαk(x). This is due to the fact that the re-cursive relations definingαk are formally the same as the transport equations (2.4) forP.See e. g. [Berline-Getzler-Vergne1992] for details on Laplace type operators.

2.6 Solving the inhomogeneous equation on small do-mains

In the next chapter we will show uniqueness of the fundamental solutions. For this weneed to be able to solve the inhomogeneous equationPu = v for given v with smallsupport. LetΩ be a relatively compact causal subset ofM as in Corollary 2.4.12. LetFΩ± (x) be the corresponding fundamental solutions forP at x∈ Ω overΩ. Recall that for

ϕ ∈ D(Ω,E∗) the mapsx 7→ FΩ± (x)[ϕ ] are smooth sections inE∗. Using the natural pair-

ing E∗x ⊗Ex →K, ℓ⊗e 7→ ℓ ·e, we obtain a smoothK-valued functionx 7→ FΩ

± (x)[ϕ ] ·v(x)with compact support. We put

u±[ϕ ] :=∫

ΩFΩ± (x)[ϕ ] ·v(x) dV(x). (2.22)

This defines distributionsu± ∈ D ′(Ω,E) because ifϕm → ϕ in D(Ω,E∗), thenFΩ± (·)[ϕm]→ FΩ

± (·)[ϕ ] in C0(Ω,E∗) by Lemma 2.4.4 and (2.19). Henceu±[ϕm]→ u±[ϕ ].

Lemma 2.6.1. The distributions u± defined in (2.22) satisfy

Pu± = v

and

supp(u±) ⊂ JΩ±(supp(v)).

2.6. Solving the inhomogeneous equation on small domains 63

Proof. Let ϕ ∈ D(Ω,E∗). We compute

Pu±[ϕ ] = u±[P∗ϕ ]

=∫

ΩFΩ± (x)[P∗ϕ ] ·v(x) dV(x)

=

∫

ΩP(2)F

Ω± (x)[ϕ ] ·v(x) dV(x)

=

∫

Ωϕ(x) ·v(x) dV(x).

ThusPu± = v. Now assume supp(ϕ)∩JΩ±(supp(v)) = /0. Then supp(v)∩JΩ

∓(supp(ϕ)) =/0. Since JΩ

∓(supp(ϕ)) contains the support ofx 7→ FΩ± (x)[ϕ ] we have supp(v) ∩

supp(FΩ± (·)[ϕ ]) = /0. Hence the integrand in (2.22) vanishes identically and therefore

u±[ϕ ] = 0. This proves supp(u±) ⊂ JΩ±(supp(v)).

Lemma 2.6.2. Let Ω be causal and contained in a convex domainΩ′. Let S1,S2 ⊂ Ω becompact subsets. Let V∈C∞(Ω×Ω,E∗

⊠E). LetΦ ∈Cn+1(Ω,E∗) andΨ ∈Cn+1(Ω,E)be such thatsupp(Φ) ⊂ JΩ

∓(S1) andsupp(Ψ) ⊂ JΩ±(S2).

Then for all j≥ 0∫

Ω

(V(x, ·)RΩ′

± (2+2 j,x))

[Φ]·Ψ(x) dV(x)=

∫

ΩΦ(y)·

(V(·,y)RΩ′

∓ (2+2 j,y))

[Ψ] dV(y).

Proof. Since supp(RΩ′± (2 + 2 j,x)) ∩ supp(Φ) ⊂ JΩ

±(x) ∩ JΩ∓(S1) is compact

(Lemma A.5.7) and since the distributionRΩ′± (2 + 2 j,x) is of order ≤ n + 1

we may apply V(x, ·)RΩ′± (2 + 2 j,x) to Φ. By Proposition 1.4.2 (12) the

section x 7→ V(x, ·)RΩ′± (2 + 2 j,x)[Φ] is continuous. Moreover, supp(x 7→

V(x, ·)RΩ′± (2 + 2 j,x)[Φ]) ∩ supp(Ψ) ⊂ JΩ

∓(supp(Φ)) ∩ JΩ±(S2) ⊂ JΩ

∓(S1) ∩ JΩ±(S2) is

also compact and contained inΩ. Hence the integrand of the left hand side is a compactlysupported continuous function and the integral is well-defined. Similarly, the integral onthe right hand side is well-defined. By Lemma 1.4.3

∫

Ω

(V(x, ·)RΩ′

± (2+2 j,x))

[Φ] ·Ψ(x) dV(x)

=

∫

ΩRΩ′± (2+2 j,x)[y 7→V(x,y)∗Φ(y)] ·Ψ(x) dV(x)

=

∫

ΩRΩ′± (2+2 j,x)[y 7→ Φ(y)V(x,y)Ψ(x)] dV(x)

=

∫

ΩRΩ′∓ (2+2 j,y)[x 7→ Φ(y)V(x,y)Ψ(x)] dV(y)

=∫

ΩΦ(y) ·

(V(·,y)RΩ

∓(2+2 j,y)[Ψ])

dV(y).

Lemma 2.6.3. Let Ω ⊂ M be a relatively compact causal domain satisfying (2.18) inLemma 2.4.8.Then the distributions u± defined in (2.22) are smooth sections in E, i. e., u± ∈C∞(Ω,E).


Proof. Let ϕ ∈D(Ω,E∗). PutS:= supp(ϕ). Let L± ∈C∞(Ω×Ω,E∗⊠E) be the integral

kernel of(id+K±)−1 K±. We recall from (2.19)

FΩ± (·)[ϕ ] = (id+K±)−1(R±(·)[ϕ ]) = R±(·)[ϕ ]− (id+K±)−1K±(R±(·)[ϕ ]).

Therefore

u±[ϕ ] =∫

ΩFΩ± (x)[ϕ ] ·v(x) dV(x)

=

∫

ΩR±(x)[ϕ ] ·v(x) dV(x)−

∫

Ω

∫

ΩL±(y,x) · R±(x)[ϕ ] ·v(y) dV(x) dV(y)

=

∫

ΩR±(x)[ϕ ] ·w(x) dV(x)

where w(x) := v(x) − ∫Ω v(y) · L±(y,x) dV(y) ∈ Ex. Obviously, w ∈ C∞(Ω,E). By

Lemma 2.4.8 supp(L±) ⊂ (y,x) ∈ Ω×Ω |x ∈ JΩ±(y). Hence supp(w) ⊂ JΩ

±(supp(v)).We may therefore apply Lemma 2.6.2 withΦ = ϕ andΨ = w to obtain

∫

ΩVj(x, ·)RΩ′

± (2+2 j,x)[ϕ ] ·w(x) dV(x) =∫

Ωϕ(y)Vj(·,y)RΩ′

∓ (2+2 j,y)[w] dV(y)

for j = 0, . . . ,N−1 and∫

Ωσ(Γ(x, ·)/ε j )Vj(x, ·)RΩ′

± (2+2 j,x)[ϕ ] ·w(x) dV(x)

=

∫

Ωϕ(y)σ(Γ(·,y)/ε j )Vj(·,y)RΩ′

∓ (2+2 j,y)[w] dV(y)

for j ≥ N. Note that the contribution of the zero set∂Ω in the above integrals vanishes,hence we integrate overΩ instead ofΩ. Summation overj yields

u±[ϕ ] =

∫

ΩR±(x)[ϕ ] ·w(x) dV(x)

=N−1

∑j=0

∫

Ωϕ(y)Vj(·,y)RΩ′

∓ (2+2 j,y)[w] dV(y)

+∞

∑j=N

∫

Ωϕ(y)σ(Γ(·,y)/ε j )Vj(·,y)RΩ′

∓ (2+2 j,y)[w] dV(y).

Thus

u±(y) =N−1

∑j=0

(Vj(·,y)RΩ′

∓ (2+2 j,y))

[w]+∞

∑j=N

(σ(Γ(·,y)/ε j )Vj(·,y)RΩ′

∓ (2+2 j,y))

[w].

Proposition 1.4.2 (11) shows that all summands are smooth iny. By the choice of theε j

the series converges in allCk-norms. Henceu± is smooth.

We summarize

2.6. Solving the inhomogeneous equation on small domains 65

Theorem 2.6.4. Let M be a timeoriented Lorentzian manifold. Let P be a normallyhyperbolic operator acting on sections in a vector bundle E over M.Then each point in M possesses a relatively compact causal neighborhoodΩ such thatfor each v∈ D(Ω,E) there exist u± ∈C∞(Ω,E) satisfying

(1)∫

Ω ϕ(x) ·u±(x) dV =∫

Ω FΩ± (x)[ϕ ] ·v(x) dV for eachϕ ∈ D(Ω,E∗),

(2) Pu± = v,

(3) supp(u±) ⊂ JΩ±(supp(v)).

Chapter 3

The global theory

In the previous chapter we developed the local theory. We proved existence of advancedand retarded fundamental solutions on small domainsΩ in the Lorentzian manifold.The restriction to small domains arises from two facts. Firstly, Riesz distributions andHadamard coefficients are defined only in domains on which theRiemannian exponentialmap is a diffeomorphism. Secondly, the analysis in Section 2.4 that allows us to turn theapproximate fundamental solution into a true one requires sufficiently good bounds onvarious functions defined onΩ. Consequently, our ability to solve the wave equation asin Theorem 2.6.4 is so far also restricted to small domains.

In this chapter we will use these local results to understandsolutions to a wave equationdefined on the whole Lorentzian manifold. To obtain a reasonable theory we have to makegeometric assumptions on the manifold. In most cases we willassume that the manifoldis globally hyperbolic. This is the class of manifolds wherewe get a very completeunderstanding of wave equations.

However, in some cases we get global results for more generalmanifolds. We start byshowing uniqueness of fundamental solutions with a suitable condition on their support.The geometric assumptions needed here are weaker than global hyperbolicity. In par-ticular, on globally hyperbolic manifolds we get uniqueness of advanced and retardedfundamental solutions.

Then we show that the Cauchy problem is well-posed on a globally hyperbolic manifold.This means that one can uniquely solvePu= f , u|S= u0 and∇nu= u1 wheref , u0 andu1

are smooth and compactly supported,S is a Cauchy hypersurface and∇n is the covariantnormal derivative alongS. The solution depends continuously on the given dataf , u0 andu1. It is unclear how one could set up a Cauchy problem on a non-globally hyperbolicmanifold because one needs a Cauchy hypersurfaceS to impose the initial conditionsu|S = u0 and∇nu = u1.

Once existence of solutions to the Cauchy problem is established it is not hard to showexistence of fundamental solutions and of Green’s operators. In the last section we showhow one can get fundamental solutions to some operators on certain non-globally hyper-bolic manifolds like anti-deSitter spacetime.

68 Chapter 3. The global theory

3.1 Uniqueness of the fundamental solution

The first global result is uniqueness of solutions to the waveequation with future or pastcompact support. For this to be true the manifold must have certain geometric properties.Recall from Definition 1.3.14 and Proposition 1.3.15 the definition and properties of thetime-separation functionτ. The relation “≤” being closed means thatpi ≤ qi , pi → p,andqi → q imply p≤ q.

Theorem 3.1.1.Let M be a connected timeoriented Lorentzian manifold such that

(1) the causality condition holds, i. e., there are no causalloops,

(2) the relation “≤” is closed,

(3) the time separation functionτ is finite and continuous on M×M.

Let P be a normally hyperbolic operator acting on sections ina vector bundle E over M.

Then any distribution u∈D ′(M,E) with past or future compact support solving the equa-tion Pu= 0 must vanish identically on M,

u≡ 0.

The idea of the proof is very simple. We would like to argue as follows: We want to showu[ϕ ] = 0 for all test sectionsϕ ∈ D(M,E∗). Without loss of generality letϕ be a testsection whose support is contained in a sufficiently small open subsetΩ ⊂ M to whichTheorem 2.6.4 can be applied. SolveP∗ψ = ϕ in Ω. Compute

u[ϕ ] = u[P∗ψ ](∗)= Pu︸︷︷︸

=0

[ψ ] = 0.

The problem is that equation (∗) is not justified becauseψ does not have compact support.The argument can be rectified in case supp(u)∩ supp(ψ) is compact. The geometricconsiderations in the proof have the purpose of getting to this situation.

Proof of Theorem 3.1.1.Without loss of generality letA := supp(u) be future compact.We will show thatA is empty. Assume the contrary and consider somex∈A. We fix somey∈ IM

− (x). Then the intersectionA∩JM+ (y) is compact and nonempty.

3.1. Uniqueness of the fundamental solution 69

M

y

x

A

JM+ (y)

Fig. 14: Uniqueness of fundamental solution; constructionof y

Since the functionM → R, z 7→ τ(y,z), is continuous it attains its maximum on the com-pact setA∩ JM

+ (y) at some pointz∈ A∩ JM+ (y). The setB := A∩ JM

+ (z) is compact andcontainsz. For all z′ ∈ B we haveτ(y,z′) ≥ τ(y,z) from (1.10) sincez′ ≥ z and henceτ(y,z′) = τ(y,z) by maximality ofτ(y,z).

The relation “≤” turns B into an ordered set. Thatz1 ≤ z2 andz2 ≤ z1 implies z1 = z2

follows from nonexistence of causal loops. We check that Zorn’s lemma can be appliedto B. Let B′ be a totally ordered subset ofB. Choose1 a countable dense subsetB′′ ⊂ B′.ThenB′′ is totally ordered as well and can be written asB′′ = ζ1,ζ2,ζ3, . . .. Let zi bethe largest element inζ1 . . . ,ζi. This yields a monotonically increasing sequence(zi)i

which eventually becomes at least as large as any givenζ ∈ B′′.

By compactness ofB a subsequence of(zi)i converges to somez′ ∈ B asi → ∞. Since therelation “≤” is closed one easily sees thatz′ is an upper bound forB′′. SinceB′′ ⊂ B′ isdense and “≤” is closedz′ is also an upper bound forB′. Hence Zorn’s lemma applies andyields a maximal elementz0 ∈ B. Replacingz by z0 we may therefore assume thatτ(y, ·)attains its maximum atzand thatA∩JM

+ (z) = z.

1Every (infinite) subset of a manifold has a countable dense subset. This follows from existence of a count-able basis of the topology.


M

y

A

z

JM+ (z)

Fig. 15: Uniqueness of fundamental solution; constructionof z

We fix a relatively compact causal neighborhoodΩ ⊂⊂ M of zas in Theorem 2.6.4.

M

y

A

zΩ

pi

Fig. 16: Uniqueness of fundamental solution; sequencepii converging toz

Let pi ∈ Ω∩ IM− (z)∩ IM

+ (y) such thatpi → z. We claim that fori sufficiently large we haveJM+ (pi)∩A ⊂ Ω. Suppose the contrary. Then there is for eachi a pointqi ∈ JM

+ (pi)∩Asuch thatqi 6∈ Ω. Sinceqi ∈ JM

+ (y)∩A for all i andJM+ (y)∩A is compact we have, after

passing to a subsequence, thatqi → q∈ JM+ (y)∩A. Fromqi ≥ pi , qi → q, pi → z, and the

fact that “≤” is closed we concludeq≥ z. Thusq∈ JM+ (z)∩A, henceq = z. On the other

hand,q 6∈ Ω since allqi 6∈ Ω, a contradiction.

3.1. Uniqueness of the fundamental solution 71

M

y

A

zΩ

pi

Fig. 17: Uniqueness of fundamental solution;JM+ (pi)∩A⊂ Ω for i ≫ 0

This shows that we can fixi sufficiently large so thatJM+ (pi)∩A⊂ Ω. We choose a cut-off

functionη ∈ D(Ω,R) such thatη |JM+ (pi)∩A ≡ 1. We putΩ := Ω∩ IM

+ (pi) and note thatΩis an open neighborhood ofz.

M

y

A

zΩ

Ω

pi

Fig. 18: Uniqueness of fundamental solution; constructionof the neighborhoodΩ of z

Now we consider some arbitraryϕ ∈ D(Ω,E∗). We will show thatu[ϕ ] = 0. This thenproves thatu|Ω = 0, in particular,z 6∈ A = supp(u), the desired contradiction.By the choice ofΩ we can solve the inhomogeneous equationP∗ψ = ϕ on Ω with ψ ∈C∞(Ω,E∗) and supp(ψ) ⊂ JΩ

+(supp(ϕ)) ⊂ JM+ (pi)∩Ω. Then supp(u)∩ supp(ψ) ⊂ A∩

JM+ (pi)∩Ω = A∩JM

+ (pi). Henceη |supp(u)∩supp(ψ) = 1. Thus

u[ϕ ] = u[P∗ψ ] = u[P∗(ηψ)] = (Pu)[ηψ ] = 0.


Corollary 3.1.2. Let M be a connected timeoriented Lorentzian manifold such that

(1) the causality condition holds, i. e., there are no causalloops,

(2) the relation “≤” is closed,

(3) the time separation functionτ is finite and continuous on M×M.

Let P be a normally hyperbolic operator acting on sections ina vector bundle E over M.Then for every x∈ M there exists at most one fundamental solution for P at x withpastcompact support and at most one with future compact support.

Remark 3.1.3. The requirement in Theorem 3.1.1 and Corollary 3.1.2 thatu have futureor past compact support is crucial. For example, on Minkowski spaceu= R+(2)−R−(2)is a nontrivial solution toPu= 0 despite the fact that Minkowski space satisfies the geo-metric assumptions onM in Theorem 3.1.1 and in Corollary 3.1.2.

These assumptions onM hold for convex Lorentzian manifolds and for globally hyper-bolic manifolds. On a globally hyperbolic manifold the setsJM

± (x) are always futurerespectively past compact. Hence we have

Corollary 3.1.4. Let M be a globally hyperbolic Lorentzian manifold. Let P be anormallyhyperbolic operator acting on sections in a vector bundle E over M.Then for every x∈ M there exists at most one advanced and at most one retarded funda-mental solution for P at x.

Remark 3.1.5. In convex Lorentzian manifolds uniqueness of advanced and retardedfundamental solutions need not hold. For example, ifM is a convex open subset ofMinkowski spaceRn such that there exist pointsx∈ M andy∈ Rn\M with JRn

+ (y)∩M ⊂JM+ (x), then the restrictions toM of R+(x) and ofR+(x) + R+(y) are two different ad-

vanced fundamental solutions forP= atx onM. Corollary 3.1.2 does not apply becauseJM+ (x) is not past compact.

M

x

y

Fig. 19: Advanced fundamental solution atx is not unique onM

3.2. The Cauchy problem 73

3.2 The Cauchy problem

The aim of this section is to show that the Cauchy problem on a globally hyperbolicmanifoldM is well-posed. This means that given a normally hyperbolic operatorP and aCauchy hypersurfaceS⊂ M the problem

Pu = f onM,u = u0 alongS,

∇nu = u1 alongS,

has a unique solution for givenu0,u1 ∈D(S,E) and f ∈D(M,E). Moreover, the solutiondepends continuously on the data.We will also see that the support of the solution is containedin JM(K) whereK :=supp(u0)∪supp(u1)∪supp( f ). This is known as finiteness of propagation speed.We start by identifying the divergence term that appears when one compares the operatorP with its formal adjointP∗. This yields a local formula allowing us to control a solutionof Pu = 0 in terms of its Cauchy data. These local considerations already suffice toestablish uniqueness of solutions to the Cauchy problem on general globally hyperbolicmanifolds.Existence of solutions is first shown locally. After some technical preparation we putthese local solutions together to a global one on a globally hyperbolic manifold. Thisis where the crucial passage from the local to the global theory takes place. Continuousdependence of the solutions on the data is an easy consequence of the open mappingtheorem from functional analysis.

Lemma 3.2.1.Let E be a vector bundle over the timeoriented Lorentzian manifold M. LetP be a normally hyperbolic operator acting on sections in E. Let ∇ be the P-compatibleconnection on E.Then for everyψ ∈C∞(M,E∗) and v∈C∞(M,E),

ψ · (Pv)− (P∗ψ) ·v= div(W),

where the vector field W∈C∞(M,T M⊗R K) is characterized by

〈W,X〉 = (∇Xψ) ·v−ψ · (∇Xv)

for all X ∈C∞(M,TM).

Here we have, as before, writtenK = R if E is a real vector bundle andK = C if E iscomplex.

Proof. The Levi-Civita connection onTM and theP-compatible connection∇ on E in-duce connections onT∗M⊗E and onT∗M⊗E∗ which we also denote by∇ for simplicity.We define a linear differential operatorL : C∞(M,T∗M⊗E∗) →C∞(M,E∗) of first orderby

Ls := −n

∑j=1

ε j(∇ej s)(ej)


wheree1, . . . ,en is a local Lorentz orthonormal frame ofTM andε j = 〈ej ,ej〉. It is easilychecked that this definition does not depend on the choice of orthonormal frame. Writee∗1, . . . ,e

∗n for the dual frame ofT∗M. The metric〈·, ·〉 on TM and the natural pairing

E∗⊗E → K, ψ ⊗ v 7→ ψ · v, induce a pairing(T∗M⊗E∗)⊗ (T∗M⊗E) → K which weagain denote by〈·, ·〉. For allψ ∈C∞(M,E∗) ands∈C∞(M,T∗M⊗E) we obtain

〈∇ψ ,s〉 =n

∑j ,k=1

〈e∗j ⊗∇ej ψ , e∗k ⊗s(ek)〉

=n

∑j ,k=1

〈e∗j , e∗k〉 · (∇ej ψ) ·s(ek)

=n

∑j=1

ε j(∇ej ψ) ·s(ej)

=n

∑j=1

ε j(∂ej (ψ ·s(ej))−ψ · (∇ej s)(ej)−ψ ·s(∇ej ej)

)

= ψ · (Ls)+n

∑j=1

ε j(∂ej (ψ ·s(ej))−ψ ·s(∇ej ej)

). (3.1)

Let V1 be the uniqueK-valued vector field characterized by〈V1,X〉 = ψ · s(X) for everyX ∈C∞(M,TM). Then

div(V1) =n

∑j=1

ε j〈∇ej V1 , ej〉

=n

∑j=1

ε j(∂ej 〈V1 , ej〉− 〈V1 , ∇ej ej〉

)

=n

∑j=1

ε j(∂ej (ψ ·s(ej))−ψ ·s(∇ej ej)

).

Plugging this into (3.1) yields

〈∇ψ ,s〉 = ψ ·Ls+div(V1).

In particular, ifv∈C∞(M,E) we get fors := ∇v∈C∞(M,T∗M⊗E)

〈∇ψ ,∇v〉 = ψ ·L∇v+div(V1) = ψ ·∇v+div(V1),

henceψ ·∇v = 〈∇ψ ,∇v〉−div(V1) (3.2)

where〈V1,X〉 = ψ ·∇Xv for all X ∈C∞(M,TM). Similarly, we obtain

(∇ψ) ·v= 〈∇ψ ,∇v〉−div(V2)

whereV2 is the vector field characterized by〈V2,X〉 = (∇Xψ) ·v for all X ∈C∞(M,TM).Thus

ψ ·∇v = (∇ψ) ·v−div(V1)+div(V2) = (∇ψ) ·v+div(W)


whereW = V2−V1. Since∇ is theP-compatible connection onE we haveP = ∇ + B

for someB∈C∞(M,End(E)), see Lemma 1.5.5. Thus

ψ ·Pv= ψ ·∇v+ ψ ·Bv= (∇ψ) ·v+div(W)+ (B∗ψ) ·v.

If ψ or v has compact support, then we can integrateψ ·Pv and the divergence termvanishes. Therefore

∫

Mψ ·Pv dV =

∫

M

((∇ψ) ·v+(B∗ψ) ·v

)dV.

Thus∇ψ +B∗ψ = P∗ψ andψ ·Pv= P∗ψ ·v+div(W) as claimed.

Lemma 3.2.2. Let E be a vector bundle over a timeoriented Lorentzian manifold Mand let P be a normally hyperbolic operator acting on sections in E. Let∇ be the P-compatible connection on E. LetΩ ⊂ M be a relatively compact causal domain satisfyingthe conditions of Lemma 2.4.8. Let S be a smooth spacelike Cauchy hypersurface inΩ.Denote byn the future directed (timelike) unit normal vector field along S.For every x∈ Ω let FΩ

± (x) be the fundamental solution for P∗ at x with support in JΩ±(x)constructed in Proposition 2.4.11.Let u∈C∞(Ω,E) be a solution of Pu= 0 on Ω. Set u0 := u|S and u1 := ∇nu.Then for everyϕ ∈ D(Ω,E∗),

∫

Ωϕ ·u dV =

∫

S

((∇n(FΩ[ϕ ])) ·u0− (FΩ[ϕ ]) ·u1

)dA,

where FΩ[ϕ ] ∈C∞(Ω,E∗) is defined as a distribution by

(FΩ[ϕ ])[w] :=∫

Ωϕ(x) · (FΩ

+ (x)[w]−FΩ− (x)[w]) dV(x)

for every w∈ D(Ω,E).

Proof. Fix ϕ ∈ D(Ω,E∗). We consider the distributionψ defined byψ [w] :=∫

Ω ϕ(x) ·FΩ

+ (x)[w] dV for everyw ∈ D(Ω,E). By Theorem 2.6.4 we know thatψ ∈ C∞(Ω,E∗),has its support contained inJΩ

+(supp(ϕ)) and satisfiesP∗ψ = ϕ .LetW be the vector field from Lemma 3.2.1 withu instead ofv. Since by Corollary A.5.4the subsetJΩ

+ (supp(ϕ))∩JΩ− (S) of Ω is compact, Theorem 1.3.16 applies toD := IΩ

−(S)and the vector fieldW:

∫

D((P∗ψ) ·u−ψ · (Pu)) dV = −

∫

Ddiv(W) dV

= −〈n,n〉︸︷︷︸=−1

∫

∂D〈W,n〉dA

=

∫

∂D((∇nψ) ·u−ψ · (∇nu))dA

=

∫

S((∇nψ) ·u−ψ · (∇nu))dA .


On the other hand,∫

D((P∗ψ) ·u−ψ · (Pu)) dV =

∫

IΩ− (S)

((P∗ψ︸︷︷︸=ϕ

) ·u−ψ · ( Pu︸︷︷︸=0

)) dV =∫

IΩ− (S)

ϕ ·u dV.

Thus ∫

IΩ− (S)

ϕ ·u dV =

∫

S((∇nψ) ·u−ψ · (∇nu))dA . (3.3)

Similarly, usingD = IΩ+ (S) andψ ′[w] :=

∫Ω ϕ(x) ·FΩ

− (x)[w] dV for anyw∈ D(Ω,E) onegets ∫

IΩ+ (S)

ϕ ·u dV =∫

S

(ψ ′ · (∇nu)− (∇nψ ′) ·u

)dA . (3.4)

The different sign is caused by the fact thatn is theinterior unit normal toIΩ+(S). Adding

(3.3) and (3.4) we get∫

Ωϕ ·u dV =

∫

S

((∇n(ψ −ψ ′)) ·u− (ψ −ψ ′) · (∇nu)

)dA,

which is the desired result.

Corollary 3.2.3. Let Ω, u, u0, and u1 be as in Lemma 3.2.2. Then

supp(u) ⊂ JΩ(K)

where K= supp(u0)∪supp(u1).

Proof. Let ϕ ∈ D(Ω,E∗). From Theorem 2.6.4 we know that supp(FΩ[ϕ ]) ⊂JΩ(supp(ϕ)). Hence if, under the hypotheses of Lemma 3.2.2,

supp(u j)∩JΩ(supp(ϕ)) = /0 (3.5)

for j = 0,1, then∫

Ω ϕ ·u dV = 0. Equation (3.5) is equivalent to

supp(ϕ)∩JΩ(supp(u j)) = /0.

Thus∫

Ω ϕ ·u dV = 0 whenever the support of the test sectionϕ is disjoint fromJΩ(K).We conclude thatu must vanish outsideJΩ(K).

Corollary 3.2.4. Let E be a vector bundle over a globally hyperbolic Lorentzian manifoldM. Let ∇ be a connection on E and let P=

∇ + B be a normally hyperbolic operatoracting on sections in E. Let S be a smooth spacelike Cauchy hypersurface in M, and letnbe the future directed (timelike) unit normal vector field along S.If u ∈C∞(M,E) solves

Pu = 0 on M,u = 0 along S,

∇nu = 0 along S,

then u= 0 on M.


Proof. By Theorem 1.3.13 there is a foliation ofM by spacelike smooth Cauchy hyper-surfacesSt (t ∈ R) with S0 = S. Extendn smoothly to all ofM such thatn|St

is the unitfuture directed (timelike) normal vector field onSt for everyt ∈ R. Let p∈ M. We showthatu(p) = 0.

Let T ∈ R be such thatp ∈ ST . Without loss of generality letT > 0 and letp be in thecausal future ofS. Set

t0 := sup

t ∈ [0,T]∣∣∣u vanishes onJM

− (p)∩ ( ∪0≤τ≤t

Sτ).

ST

JM− (p)

b

p

St0

S0 = S

u = 0

M

Fig. 20: Uniqueness of solution to Cauchy problem; domain whereu vanishes

We will show thatt0 = T which implies in particularu(p) = 0.

Assumet0 < T. For eachx∈ JM− (p)∩St0 we may, according to Lemma A.5.6, choose a

relatively compact causal neighborhoodΩ of x in M satisfying the hypotheses of Lemma2.4.8 and such thatSt0 ∩Ω is a Cauchy hypersurface ofΩ.


JM− (p)

St0

Ω

JM− (p)∩JΩ

+(St0 ∩Ω)

Fig. 21: Uniqueness of solution to Cauchy problem;u vanishes onJM− (p)∩JΩ

+ (St0 ∩Ω)

Put u0 := u|St0andu1 := (∇nu)|St0

. If t0 = 0, thenu0 = u1 = 0 on S= S0 by assump-

tion. If t0 > 0, thenu0 = u1 = 0 on St0 ∩ JM− (p) becauseu ≡ 0 on JM

− (p)∩ ( ∪0≤τ≤t

Sτ).

Corollary 3.2.3 impliesu = 0 onJM− (p)∩JΩ

+(St0 ∩Ω).

By Corollary A.5.4 the intersectionSt0 ∩ JM− (p) is compact. Hence it can be covered by

finitely many open subsetsΩi , 1≤ i ≤ N, satisfying the conditions ofΩ above. Thusuvanishes identically on(Ω1∪·· ·∪ΩN)∩JM

− (p)∩JM+ (St0). Since(Ω1∪·· ·∪ΩN)∩JM

− (p)is an open neighborhood of the compact setSt0 ∩ JM

− (p) in JM− (p) there exists anε > 0

such thatSt ∩JM− (p) ⊂ Ω1∪·· ·∪ΩN for everyt ∈ [t0,t0 + ε).


ST

JM− (p)

b

p

StSt0

M

Ωi

Fig. 22: Uniqueness of solution to Cauchy problem;St ∩JM− (p) is contained in

⋃i

Ωi for t ∈ [t0,t0 + ε)

Henceu vanishes onSt ∩JM− (p) for all t ∈ [t0,t0 + ε). This contradicts the maximality of

t0.

Next we prove existence of solutions to the Cauchy problem onsmall domains. LetΩ⊂Msatisfy the hypotheses of Lemma 2.4.8. In particular,Ω is relatively compact, causal, andhas “small volume”. Such domains will be referred to asRCCSV(for “Relatively CompactCausal with Small Volume”). Note that each point in a Lorentzian manifold possesses abasis of RCCSV-neighborhoods. Since causal domains are contained in convex domainsby definition and convex domains are contractible, the vector bundleE is trivial over anyRCCSV-domainΩ. We shall show that one can uniquely solve the Cauchy problemonevery RCCSV-domain with Cauchy data on a fixed Cauchy hypersurface inΩ.

Proposition 3.2.5. Let M be a timeoriented Lorentzian manifold and let S⊂ M be aspacelike hypersurface. Letn be the future directed timelike unit normal field along S.Then for each RCCSV-domainΩ ⊂ M such that S∩Ω is a (spacelike) Cauchy hypersur-face inΩ, the following holds:For each u0,u1 ∈ D(S∩ Ω,E) and for each f∈ D(Ω,E) there exists a unique u∈C∞(Ω,E) satisfying

Pu = f on M,u = u0 along S,

∇nu = u1 along S.

Moreover,supp(u) ⊂ JM(K) where K= supp(u0)∪supp(u1)∪supp( f ).


Proof. Let Ω ⊂ M be an RCCSV-domain such thatS∩Ω is a Cauchy hypersurface inΩ. Corollary 3.2.4 can then be applied onΩ: If u andu are two solutions of the Cauchyproblem, thenP(u− u) = 0, (u− u)|S = 0, and∇n(u− u) = 0. Corollary 3.2.4 impliesu− u= 0 which shows uniqueness. It remains to show existence.Since causal domains are globally hyperbolic we may apply Theorem 1.3.13 and find anisometryΩ = R× (S∩Ω) where the metric takes the form−βdt2+gt . Hereβ : Ω → R∗

+

is smooth, eacht× (S∩Ω) is a smooth spacelike Cauchy hypersurface inΩ, andS∩Ωcorresponds to0× (S∩Ω). Note that the future directed unit normal vector fieldn

alongt× (S∩Ω) is given byn(·) = 1√β (t,·)

∂∂ t .

Now let u0,u1 ∈ D(S∩Ω,E) and f ∈ D(Ω,E). We trivialize the bundleE overΩ andidentify sections inE with Kr -valued functions wherer is the rank ofE.Assume for a moment thatu were a solution to the Cauchy problem of the formu(t,x) =

∑∞j=0 t ju j(x) wherex ∈ S∩ Ω. Write P = 1

β∂ 2

∂ t2+Y whereY is a differential operator

containingt-derivatives only up to order 1. Equation

f = Pu=

(1β

∂ 2

∂ t2 +Y

)u =

1β (t, ·)

∞

∑j=2

j( j −1)t j−2u j +Yu (3.6)

evaluated att = 0 gives

2β (0,x)

u2(x) = −Y(u0 + tu1)(0,x)+ f (0,x)

for everyx∈ S∩Ω. Thusu2 is determined byu0, u1, and f |S. Differentiating (3.6) withrespect to∂

∂ t and repeating the procedure shows that eachu j is recursively determined byu0, . . . ,u j−1 and the normal derivatives off alongS.Now we drop the assumption that we have at-power seriesu solving the problem but wedefinetheu j , j ≥ 2, by these recursive relations. Then supp(u j) ⊂ supp(u0)∪supp(u1)∪(supp( f )∩S) for all j.Let σ : R → R be a smooth function such thatσ |[−1/2,1/2] ≡ 1 andσ ≡ 0 outside[−1,1].We claim that we can find a sequence ofε j ∈ (0,1) such that

u(t,x) :=∞

∑j=0

σ(t/ε j)tju j(x) (3.7)

defines a smooth section that can be differentiated termwise.By Lemma 1.1.11 we have forj > k

‖σ(t/ε j)tju j(x)‖Ck(Ω) ≤ c(k) ·

∥∥σ(t/ε j)tj∥∥

Ck(R)· ‖u j‖Ck(S).

Here and in the followingc(k), c′(k, j), andc′′(k, j) denote universal constants dependingonly onk and j. By Lemma 2.4.1 we have forl ≤ k and 0< ε j ≤ 1

∥∥∥∥dl

dtl(σ(t/ε j)t

j)

∥∥∥∥C0(R)

≤ ε j c′(l , j)‖σ‖Cl (R),


thus‖σ(t/ε j)t

ju j(x)‖Ck(Ω) ≤ ε j c′′(k, j)‖σ‖Ck(R) ‖u j‖Ck(S).

Now we choose 0< ε j ≤ 1 so thatε j c′′(k, j)‖σ‖Ck(R) ‖u j‖Ck(S) ≤ 2− j for all k < j. Then

the series (3.7) defining û converges absolutely in theCk-norm for all k. Hence û isa smooth section with compact support and can be differentiated termwise. From theconstruction of û one sees that supp(u) ⊂ JM(K).By the choice of theu j the sectionPu− f vanishes to infinite order alongS. Therefore

w(t,x) :=

(Pu− f )(t,x), if t ≥ 0,

0, if t < 0,

defines a smooth section with compact support. By Theorem 2.6.4 (which can be appliedsince the hypotheses of Lemma 2.4.8 are fulfilled) we can solve the equationPu= w witha smooth section ˜u having past compact support. Moreover, supp(u) ⊂ JM

+ (supp(w)) ⊂JM+ (supp(u)∪supp( f )) ⊂ JM(K).

Now u+ := u− u is a smooth section such thatPu+ = Pu−Pu = w+ f −w = f onJΩ+(S∩Ω) = t ≥ 0.

The restriction of ˜u to IΩ−(S) has past compact support and satisfiesPu = 0 onIΩ

−(S), thusby Theorem 3.1.1 ˜u = 0 on IΩ

−(S). Thusu+ coincides with û to infinite order alongS.In particular,u+|S = u|S = u0 and∇nu+ = ∇nu = u1. Moreover, supp(u+) ⊂ supp(u)∪supp(u) ⊂ JM(K). Thusu+ has all the required properties onJM

+ (S).Similarly, one constructsu− on JM

− (S). Since bothu+ andu− coincide to infinite orderwith u alongSwe obtain the smooth solution by setting

u(t,x) :=

u+(t,x), if t ≥ 0,u−(t,x), if t ≤ 0.

Remark 3.2.6. It follows from Lemma A.5.6 that every pointp on a spacelike hyper-surfaceSpossesses a RCCSV-neighborhoodΩ such thatS∩Ω is a Cauchy hypersurfacein Ω. Hence Proposition 3.2.5 guarantees the local existence ofsolutions to the Cauchyproblem.

In order to show existence of solutions to the Cauchy problemon globally hyperbolicmanifolds we need some preparation. LetM be globally hyperbolic. We writeM = R×Sand suppose the metric is of the form−βdt2 + gt as in Theorem 1.3.10. HenceM isfoliated by the smooth spacelike Cauchy hypersurfacest×S=: St , t ∈ R. Let p∈ M.Then there exists a uniquet such thatp∈ St . For anyr > 0 we denote byBr(p) the openball in St of radiusr aboutp with respect to theRiemannianmetricgt on St . ThenBr(p)is open as a subset ofSt but not as a subset ofM.Recall thatD(A) denotes the Cauchy development of a subsetA of M (see Defini-tion 1.3.5).

Lemma 3.2.7. The functionρ : M → (0,∞] defined by

ρ(p) := supr > 0|D(Br(p)) is RCCSV,

is lower semi-continuous on M.


Proof. First note thatρ is well-defined since every point has a RCCSV-neighborhood.Let p∈ M andr > 0 be such thatρ(p) > r. Let ε > 0. We want to showρ(p′) > r − εfor all p′ in a neighborhood ofp.For any pointp′ ∈ D(Br(p)) consider

λ (p′) := supr ′ > 0|Br ′(p′) ⊂ D(Br(p)).

Claim: There exists a neighborhood V of p such that for every p′ ∈ V one hasλ (p′) >r − ε.

b

p

D(Br(p))

V

b

p′

D(Br ′(p′))

Fig. 23: Construction of the neighborhoodV of p

Let us assume the claim for a moment. Letp′ ∈V. Pickr ′ with r −ε < r ′ < λ (p′). HenceBr ′(p′) ⊂ D(Br(p)). By Remark 1.3.6 we knowD(Br ′(p′)) ⊂ D(Br(p)). SinceD(Br(p))is RCCSV the subsetD(Br ′(p′)) is RCCSV as well. Thusρ(p′) ≥ r ′ > r − ε. This thenconcludes the proof.It remains to show the claim. Assume the claim is false. Then there is a sequence(pi)i ofpoints inM converging top such thatλ (pi) ≤ r − ε for all i. Hence forr ′ := r − ε/2 wehaveBr ′(pi) 6⊂ D(Br(p)). Choosexi ∈ Br ′(pi)\D(Br(p)).The closed setBr(p) is contained in the compact setD(Br(p)) and therefore compactitself. Thus[−1,1]×Br(p) is compact. Fori sufficiently largeBr ′(pi) ⊂ [−1,1]×Br(p)and thereforexi ∈ [−1,1]×Br(p). We pass to a convergent subsequencexi → x. Sincepi → p andxi ∈ Br ′(pi) we havex∈ Br ′(p). Hencex∈ Br(p). SinceD(Br(p)) is an openneighborhood ofx we must havexi ∈ D(Br(p)) for sufficiently largei. This contradictsthe choice of thexi .

For everyr > 0 andq∈ M = R×Sconsider

θr(q) := supη > 0|JM(Br/2(q))∩ ([t0−η ,t0 + η ]×S)⊂ D(Br(q)).


B r2(q)

D(Br(q))

t0×St0 +η×S

t0−η×S

JM(B r2(q))∩ ([t0−η ,t0 +η ]×S)

b

q

Fig. 24: Definition ofθr(q)

Remark 3.2.8. There existη > 0 with JM(Br/2(q))∩ ([t0−η ,t0 + η ]×S) ⊂ D(Br(q)).Henceθr(q) > 0.

One can see this as follows. If no suchη existed, then there would be pointsxi ∈JM(Br/2(q))∩ ([t0 − 1

i , t0 + 1i ]×S) but xi 6∈ D(Br(q)), i ∈ N. All xi lie in the compact

setJM(Br/2(q))∩ ([t0−1, t0 + 1]×S). Hence we may pass to a convergent subsequencexi → x. Thenx∈ JM(Br/2(q))∩ (t0×S) = Br/2(q). SinceD(Br(q)) is an open neigh-borhood ofBr/2(q) we must havexi ∈ D(Br(q0)) for sufficiently largei in contradictionto the choice of thexi .

Lemma 3.2.9. The functionθr : M → (0,∞] is lower semi-continuous.

Proof. Fix q ∈ M. Let ε > 0. We need to find a neighborhoodU of q such that for allq′ ∈U we haveθr(q′) ≥ θr(q)− ε.

Putη := θr(q) and chooset0 such thatq∈ St0. Assume no such neighborhoodU exists.Then there is a sequence(qi)i in M such thatqi → q andθr(qi) < η − ε for all i. Allpoints to be considered will be contained in the compact set([−T,T]×S)∩ JM(Br(q))for sufficiently bigT and sufficiently largei. Let qi ∈ Sti . Thenti → t0 asi → ∞.

Choosexi ∈ JM(Br/2(qi))∩ ([ti −η + ε,ti + η − ε]×S) butxi 6∈ D(Br(qi)). This is possi-ble because ofθr(qi) < η − ε. Chooseyi ∈ Br/2(qi) such thatxi ∈ JM(yi).


B r2(qi)

Sti+η−ε

Sti−η+ε

Sti

D(Br(qi))

b

qi

b

yi

bxi

JM(B r2(qi))

Fig. 25: Construction of the sequence(xi )i

After passing to a subsequence we may assumexi → x andyi → y. Fromqi → q andyi ∈Br/2(qi) we deducey∈ Br/2(q). Since the causal relation “≤” on a globally hyperbolicmanifold is closed we conclude fromxi → x, yi → y, andxi ∈ JM(yi) that x ∈ JM(y).Thusx ∈ JM(Br/2(q)). Obviously, we also havex ∈ [t0 −η + ε,t0 + η − ε]×S. Fromθr(q) = η > η − ε we concludex∈ D(Br(q)).Sincexi 6∈ D(Br(qi)) there is an inextendible causal curveci throughxi which does notintersectBr(qi). Let zi be the intersection ofci with the Cauchy hypersurfaceSti . Afteragain passing to a subsequence we havezi → zwith z∈ St0. Fromzi 6∈ Br(qi) we concludez 6∈ Br(q). Moreover, sinceci is causal we havexi ∈ JM(zi). The causal relation “≤” isclosed, hencex∈ JM(z). Thus there exists an inextendible causal curvec throughx andz.This curve does not meetBr(q) in contradiction tox∈ D(Br(q)).

Lemma 3.2.10. For each compact subset K⊂ M there existsδ > 0 such that for eacht ∈ R and any u0,u1 ∈ D(St ,E) with supp(u j) ⊂ K, j = 1,2, there is a smooth solutionu of Pu= 0 defined on(t − δ , t + δ )×S satisfying u|St = u0 and∇nu|St = u1. Moreover,supp(u) ⊂ JM(K∩St).

Proof. By Lemma 3.2.7 the functionρ admits a minimum on the compact setK. Hencethere is a constantr0 > 0 such thatρ(q) > 2r0 for all q ∈ K. Chooseδ > 0 such thatθ2r0 > δ onK. This is possible by Lemma 3.2.9.Now fix t ∈ R. Cover the compact setSt ∩K by finitely many ballsBr0(q1), . . . ,Br0(qN),q j ∈ St ∩K. Let u0,u1 ∈ D(St ,E) with supp(u j) ⊂ K. Using a partition of unity writeu0 = u0,1 + . . .+u0,N with supp(u0, j) ⊂ Br0(q j) and similarlyu1 = u1,1 + . . .+u1,N. ThesetD(B2r0(q j)) is RCCSV. By Proposition 3.2.5 we can find a solutionwj of Pwj = 0 onD(B2r0(q j)) with wj |St = u0, j and∇nwj |St = u1, j . Moreover, supp(wj ) ⊂ JM(Br0(q j)).From JM(Br0(q j)) ∩ (t − δ , t + δ ) × S ⊂ D(B2r0(q j)) we see thatwj is defined onJM(Br0(q j))∩ (t − δ , t + δ )×S. Extendwj smoothly by zero to all of(t − δ ,t + δ )×S.Now u := w1 + . . .+wN is a solution defined on(t − δ ,t + δ )×Sas required.


Now we are ready for the main theorem of this section.

Theorem 3.2.11.Let M be a globally hyperbolic Lorentzian manifold and let S⊂ M bea spacelike Cauchy hypersurface. Letn be the future directed timelike unit normal fieldalong S. Let E be a vector bundle over M and let P be a normally hyperbolic operatoracting on sections in E.Then for each u0,u1 ∈ D(S,E) and for each f∈ D(M,E) there exists a unique u∈C∞(M,E) satisfying Pu= f , u|S = u0, and∇nu|S = u1.Moreover,supp(u) ⊂ JM(K) where K= supp(u0)∪supp(u1)∪supp( f ).

Proof. Uniqueness of the solution follows directly from Corollary3.2.4. We have to showexistence of a solution and the statement on its support.Let u0,u1 ∈ D(S,E) and f ∈ D(M,E). Using a partition of unity(χ j) j=1,...,m we canwrite u0 = u0,1+ . . .+u0,m, u1 = u1,1+ . . .+u1,m and f = f1+ . . .+ fm whereu0, j = χ ju0,u1, j = χ ju1, and f j = χ j f . We may assume that eachχ j (and hence eachui, j and f j ) havesupport in an open set as in Proposition 3.2.5. If we can solvethe Cauchy problem onMfor the data(u0, j ,u1, j , f j ), then we can add these solutions to obtain one foru0, u1, and f .Hence we can without loss of generality assume that there is an Ω as in Proposition 3.2.5such thatK := supp(u0)∪supp(u1)∪supp( f ) ⊂ Ω.By Theorem 1.3.13 the spacetimeM is isometric toR×Swith a Lorentzian metric of theform −βdt2 + gt whereS corresponds to0×S, and eachSt := t×S is a spacelikeCauchy hypersurface inM. Let u be the solution onΩ as asserted by Proposition 3.2.5.In particular, supp(u)⊂ JM(K). By choosing the partition of unity(χ j) j appropriately wecan assume thatK is so small that there exists anε > 0 such that((−ε,ε)×S)∩JM(K)⊂Ω andK ⊂ (−ε,ε)×S.

K

JM+ (K)

JM− (K)

Ω

0×S

ε×S

−ε×S

Fig. 26: Construction ofΩ andε

Hence we can extendu by 0 to a smooth solution on all of(−ε,ε)×S. Now letT+ be thesupremum of allT for whichu can be extended to a smooth solution on(−ε,T)×Swith


support contained inJM(K). On [ε,T)×Sthe equation to be solved is simplyPu= 0 be-cause supp( f ) ⊂ K. If we have two extensionsu andu for T < T, then the restriction of ˜uto (−ε,T)×Smust coincide withu by uniqueness. Note here that Corollary 3.2.4 appliesbecause(−ε,T)×S is a globally hyperbolic manifold in its own right. Thus if weshowT+ = ∞ we obtain a solution on(−ε,∞)×S. Similarly considering the correspondinginfimumT− then yields a solution on all ofM = R×S.

Assume thatT+ < +∞. PutK := ([−ε,T+]×S)∩JM(K). By Lemma A.5.4K is compact.Apply Lemma 3.2.10 toK and getδ > 0 as in the Lemma. Fixt < T+ such thatT+− t < δand stillK ⊂ (−ε, t)×S.

On (t −δ , t +δ )×SsolvePw= 0 with w|St = u|St and∇nw|St = ∇nu|St . This is possibleby Lemma 3.2.10. On(t −η , t + δ )×Sthe sectionf vanishes withη > 0 small enough.Thusw coincides withu on (t −η ,t)×S. Here again, Corollary 3.2.4 applies because(t −η , t + δ )×S is a globally hyperbolic manifold in its own right. Hencew extends thesolutionu smoothly to(−ε, t + δ )×S. The support of this extension is still contained inJM(K) because

supp(w|[t,t+δ )×S

)⊂ JM

+ (supp(u|St )∪supp(∇nu|St ))⊂ JM+ (K∩St)⊂ JM

+ (JM+ (K))= JM

+ (K).

SinceT+ < t + δ this contradicts the maximality ofT+. ThereforeT+ = +∞. Similarly,one seesT− = −∞ which concludes the proof.

The solution to the Cauchy problem depends continuously on the data.

Theorem 3.2.12.Let M be a globally hyperbolic Lorentzian manifold and let S⊂ M bea spacelike Cauchy hypersurface. Letn be the future directed timelike unit normal fieldalong S. Let E be vector bundle over M and let P be a normally hyperbolic operatoracting on sections in E.

Then the mapD(M,E)⊕D(S,E)⊕D(S,E)→C∞(M,E) sending( f ,u0,u1) to the uniquesolution u of the Cauchy problem Pu= f , u|S = u|0, ∇nu = u1 is linear continuous.

Proof. The mapP : C∞(M,E) →C∞(M,E)⊕C∞(S,E)⊕C∞(S,E), u 7→ (Pu,u|S,∇nu),is obviously linear and continuous. Fix a compact subsetK ⊂ M. Write DK(M,E) := f ∈ D(M,E) | supp( f ) ⊂ K, DK(S,E) := v∈D(S,E) | supp(v) ⊂ K∩S, andVK :=P−1(DK(M,E) ⊕DK(S,E)⊕ DK(S,E)). SinceDK(M,E) ⊕ DK(S,E)⊕ DK(S,E) ⊂C∞(M,E) ⊕C∞(S,E)⊕C∞(S,E) is a closed subset so isVK ⊂ C∞(M,E). Both VK

and DK(M,E) ⊕ DK(S,E) ⊕ DK(S,E) are therefore Frechet spaces andP : VK →DK(M,E) ⊕ DK(S,E) ⊕ DK(S,E) is linear, continuous and bijective. By the openmapping theorem [Reed-Simon1980, Thm. V.6, p. 132] the inverse mappingP−1 :DK(M,E)⊕DK(S,E)⊕DK(S,E) → VK ⊂C∞(M,E) is continuous as well.

Thus if ( f j ,u0, j ,u1, j) → ( f ,u0,u1) in D(M,E)⊕D(S,E)⊕D(S,E), then we can choosea compact subsetK ⊂ M such that( f j ,u0, j ,u1, j) → ( f ,u0,u1) in DK(M,E)⊕DK(S,E)⊕DK(S,E) and we concludeP−1( f j ,u0, j ,u1, j) → P−1( f ,u0,u1).

3.3. Fundamental solutions 87

3.3 Fundamental solutions on globally hyperbolic mani-folds

Using the knowledge about the Cauchy problem which we obtained in the previous sec-tion it is now not hard to find global fundamental solutions ona globally hyperbolicmanifold.

Theorem 3.3.1.Let M be a globally hyperbolic Lorentzian manifold. Let P be anormallyhyperbolic operator acting on sections in a vector bundle E over M.Then for every x∈ M there is exactly one fundamental solution F+(x) for P at x withpast compact support and exactly one fundamental solution F−(x) for P at x with futurecompact support. They satisfy

(1) supp(F±(x)) ⊂ JM± (x),

(2) for eachϕ ∈D(M,E∗) the maps x7→ F±(x)[ϕ ] are smooth sections in E∗ satisfyingthe differential equation P∗(F±(·)[ϕ ]) = ϕ .

Proof. Uniqueness of the fundamental solutions is a consequence ofCorollary 3.1.2. Toshow existence fix a foliation ofM by spacelike Cauchy hypersurfacesSt , t ∈ R as inTheorem 1.3.10. Letn be the future directed unit normal field along the leavesSt . Letϕ ∈ D(M,E∗). Chooset so large that supp(ϕ) ⊂ IM

− (St). By Theorem 3.2.11 there existsa uniqueχϕ ∈C∞(M,E∗) such thatP∗χϕ = ϕ andχϕ |St = (∇nχϕ)|St = 0.We check thatχϕ does not depend on the choice oft. Let t < t ′ be such that supp(ϕ) ⊂IM− (St) ⊂ IM

− (St′). Let χϕ andχ ′ϕ be the corresponding solutions. Chooset− < t so that

still supp(ϕ) ⊂ IM− (St−). The open subsetM :=

⋃τ>t− Sτ ⊂ M is a globally hyperbolic

Lorentzian manifold itself. Nowχ ′ϕ satisfiesP∗χ ′

ϕ = 0 onM with vanishing Cauchy dataon St′ . By Corollary 3.2.4χ ′

ϕ = 0 onM. In particular,χ ′ϕ has vanishing Cauchy data on

St as well. Thusχϕ − χ ′ϕ has vanishing Cauchy data onSt and solvesP∗(χϕ − χ ′

ϕ) = 0on all ofM. Again by Corollary 3.2.4 we concludeχϕ − χ ′

ϕ = 0 onM.Fix x∈ M. By Theorem 3.2.12χϕ depends continuously onϕ . Since the evaluation mapC∞(M,E) → Ex is continuous, the mapD(M,E∗) → E∗

x , ϕ 7→ χϕ(x), is also continuous.ThusF+(x)[ϕ ] := χϕ(x) defines a distribution. By definitionP∗(F+(·)[ϕ ]) = P∗χϕ = ϕ .Now P∗χP∗ϕ = P∗ϕ , henceP∗(χP∗ϕ −ϕ) = 0. Since bothχP∗ϕ andϕ vanish alongSt weconclude from Corollary 3.2.4χP∗ϕ = ϕ . Thus

(PF+(x))[ϕ ] = F+(x)[P∗ϕ ] = χP∗ϕ (x) = ϕ(x) = δx[ϕ ].

HenceF+(x) is a fundamental solution ofP atx.It remains to show supp(F+(x)) ⊂ JM

+ (x). Let y ∈ M \ JM+ (x). We have to construct a

neighborhood ofy such that for each test sectionϕ ∈ D(M,E∗) whose support is con-tained in this neighborhood we haveF+(x)[ϕ ] = χϕ(x) = 0. SinceM is globally hy-perbolicJM

+ (x) is closed and thereforeJM+ (x)∩ JM

− (y′) = /0 for all y′ sufficiently closeto y. We choosey′ ∈ IM

+ (y) and y′′ ∈ IM− (y) so close thatJM

+ (x) ∩ JM− (y′) = /0 and(

JM+ (y′′)∩⋃t≤t′ St

)∩JM

+ (x) = /0 wheret ′ ∈ R is such thaty′ ∈ St′ .


St′

byb

y′

JM− (y′)

b

y′′

JM+ (y′′)∩ (∪t≤t ′St )

b

x

JM+ (x)

b

Fig. 27: Global fundamental solution; construction ofy, y′ andy′′

Now K := JM− (y′)∩ JM

+ (y′′) is a compact neighborhood ofy. Let ϕ ∈ D(M,E∗) be suchthat supp(ϕ) ⊂ K. By Theorem 3.2.11 supp(χϕ ) ⊂ JM

+ (K)∪ JM− (K) ⊂ JM

+ (y′′)∪ JM− (y′).

By the independence ofχϕ of the choice oft > t ′ we have thatχϕ vanishes on⋃

t>t′ St .Hence supp(χϕ) ⊂

(JM+ (y′′)∩⋃t≤t′ St

)∪ JM

− (y′) and is therefore disjoint fromJM+ (x).

ThusF+(x)[ϕ ] = χϕ(x) = 0 as required.

3.4 Green’s operators

Now we want to find “solution operators” for a given normally hyperbolic operatorP.More precisely, we want to find operators which are inverses of P when restricted to suit-able spaces of sections. We will see that existence of such operators is basically equivalentto the existence of fundamental solutions.

Definition 3.4.1. Let M be a timeoriented connected Lorentzian manifold. LetP be anormally hyperbolic operator acting on sections in a vectorbundleE overM. A linearmapG+ : D(M,E) →C∞(M,E) satisfying

(i) PG+ = idD(M,E),

(ii) G+ P|D(M,E) = idD(M,E),

(iii) supp(G+ϕ) ⊂ JM+ (supp(ϕ)) for all ϕ ∈ D(M,E),

3.4. Green’s operators 89

is called anadvanced Green’s operator forP. Similarly, a linear mapG− : D(M,E) →C∞(M,E) satisfying (i), (ii), and

(iii’) supp(G−ϕ) ⊂ JM− (supp(ϕ)) for all ϕ ∈ D(M,E)

instead of (iii) is called aretarded Green’s operator forP.

Fundamental solutions and Green’s operators are closely related.

Proposition 3.4.2. Let M be a timeoriented connected Lorentzian manifold. Let Pbe anormally hyperbolic operator acting on sections in a vectorbundle E over M.If F±(x) is a family of advanced or retarded fundamental solutions for the adjoint operatorP∗ and if F±(x) depend smoothly on x in the sense that x7→ F±(x)[ϕ ] is smooth for eachtest sectionϕ and satisfies the differential equation P(F±(·)[ϕ ]) = ϕ , then

(G±ϕ)(x) := F∓(x)[ϕ ] (3.8)

defines advanced or retarded Green’s operators for P respectively. Conversely, givenGreen’s operators G± for P, then (3.8) defines fundamental solutions for P∗ dependingsmoothly on x and satisfying P(F±(·)[ϕ ]) = ϕ for each test sectionϕ .

Proof. Let F±(x) be a family of advanced and retarded fundamental solutions for theadjoint operatorP∗ respectively. LetF±(x) depend smoothly onx and suppose the differ-ential equationP(F±(·)[ϕ ]) = ϕ holds. By definition we have

P(G±ϕ) = P(F∓(·)[ϕ ]) = ϕ

thus showing (i). Assertion (ii) follows from the fact that the F±(x) are fundamentalsolutions,

G±(Pϕ)(x) = F∓(x)[Pϕ ] = P∗F∓(x)[ϕ ] = δx[ϕ ] = ϕ(x).

To show (iii) letx∈ M such that(G+ϕ)(x) 6= 0. Since supp(F−(x)) ⊂ JM− (x) the support

of ϕ must hitJM− (x). Hencex∈ JM

+ (supp(ϕ)) and therefore supp(G+ϕ) ⊂ JM+ (supp(ϕ)).

The argument forG− is analogous.The converse is similar.

Theorem 3.3.1 immediately yields

Corollary 3.4.3. Let M be a globally hyperbolic Lorentzian manifold. Let P be anormallyhyperbolic operator acting on sections in a vector bundle E over M.Then there exist unique advanced and retarded Green’s operators G± : D(M,E) →C∞(M,E) for P.

Lemma 3.4.4. Let M be a globally hyperbolic Lorentzian manifold. Let P be anormallyhyperbolic operator acting on sections in a vector bundle E over M. Let G± be theGreen’s operators for P and G∗± the Green’s operators for the adjoint operator P∗. Then

∫

M(G∗

±ϕ) ·ψ dV =

∫

Mϕ · (G∓ψ) dV (3.9)

holds for allϕ ∈ D(M,E∗) andψ ∈ D(M,E).


Proof. For the Green’s operators we havePG± = idD(M,E) andP∗G∗± = idD(M,E∗) and

hence∫

M(G∗

±ϕ) ·ψ dV =

∫

M(G∗

±ϕ) · (PG∓ψ) dV

=

∫

M(P∗G∗

±ϕ) · (G∓ψ) dV

=∫

Mϕ · (G∓ψ) dV.

Notice that supp(G±ϕ)∩ supp(G∓ψ) ⊂ JM± (supp(ϕ)) ∩ JM

∓ (supp(ψ)) is compact in aglobally hyperbolic manifold so that the partial integration in the second equation is jus-tified.

Notation 3.4.5. We writeC∞sc(M,E) for the set of allϕ ∈C∞(M,E) for which there exists

a compact subsetK ⊂ M such that supp(ϕ) ⊂ JM(K). Obviously,C∞sc(M,E) is a vector

subspace ofC∞(M,E).The subscript “sc” should remind the reader of “spacelike compact”. Namely, ifM isglobally hyperbolic andϕ ∈ C∞

sc(M,E), then for every Cauchy hypersurfaceS⊂ M thesupport ofϕ |S is contained inS∩JM(K) hence compact by Corollary A.5.4. In this sensesections inC∞

sc(M,E) have spacelike compact support.

Definition 3.4.6. We say a sequence of elementsϕ j ∈C∞sc(M,E) converges in C∞sc(M,E)

to ϕ ∈C∞sc(M,E) if there exists a compact subsetK ⊂ M such that

supp(ϕ j),supp(ϕ) ⊂ JM(K)

for all j and

‖ϕ j −ϕ‖Ck(K′,E) → 0

for all k∈ N and all compact subsetsK′ ⊂ M.

If G+ andG− are advanced and retarded Green’s operators forP respectively, then we geta linear map

G := G+ −G− : D(M,E) →C∞sc(M,E).

Theorem 3.4.7. Let M be a connected timeoriented Lorentzian manifold. Let Pbe anormally hyperbolic operator acting on sections in a vectorbundle E over M. Let G+and G− be advanced and retarded Green’s operators for P respectively.Then the sequence of linear maps

0→ D(M,E)P−→ D(M,E)

G−→C∞sc(M,E)

P−→C∞sc(M,E) (3.10)

is a complex, i. e., the composition of any two subsequent maps is zero. The complex isexact at the firstD(M,E). If M is globally hyperbolic, then the complex is exact every-where.

3.4. Green’s operators 91

Proof. Properties (i) and (ii) in Definition 3.4.1 of Green’s operators directly yieldGP=0 andPG = 0, both onD(M,E). Properties (iii) and (iii’) ensure thatG mapsD(M,E)to C∞

sc(M,E). Hence the sequence of linear maps forms a complex.Exactness at the firstD(M,E) means that

P : D(M,E) → D(M,E)

is injective. To see injectivity letϕ ∈D(M,E) with Pϕ = 0. Thenϕ = G+Pϕ = G+0= 0.From now on letM be globally hyperbolic. Letϕ ∈ D(M,E) with Gϕ = 0, i. e.,G+ϕ = G−ϕ . We put ψ := G+ϕ = G−ϕ ∈ C∞(M,E) and we see supp(ψ) =supp(G+ϕ)∩ supp(G−ϕ) ⊂ JM

+ (supp(ϕ))∩ JM− (supp(ϕ)). Since(M,g) is globally hy-

perbolicJM+ (supp(ϕ))∩ JM

− (supp(ϕ)) is compact, henceψ ∈ D(M,E). From P(ψ) =P(G+(ϕ)) = ϕ we see thatϕ ∈ P(D(M,E)). This shows exactness at the secondD(M,E).Finally, let ϕ ∈ C∞

sc(M,E) such thatPϕ = 0. Without loss of generality we may as-sume that supp(ϕ) ⊂ IM

+ (K)∪ IM− (K) for a compact subsetK of M. Using a partition of

unity subordinated to the open coveringIM+ (K), IM

− (K) write ϕ asϕ = ϕ1 + ϕ2 wheresupp(ϕ1)⊂ IM

− (K) ⊂ JM− (K) and supp(ϕ2)⊂ IM

+ (K) ⊂ JM+ (K). Forψ := −Pϕ1 = Pϕ2 we

see that supp(ψ) ⊂ JM− (K)∩JM

+ (K), henceψ ∈ D(M,E).We check thatG+ψ = ϕ2. For all χ ∈ D(M,E∗) we have∫

Mχ · (G+Pϕ2) dV =

∫

M(G∗

−χ) · (Pϕ2) dV =

∫

M(P∗G∗

−χ) ·ϕ2 dV =

∫

Mχ ·ϕ2 dV

whereG∗− is the Green’s operator for the adjoint operatorP∗ according to Lemma 3.4.4.

Notice that for the second equation we use the fact that supp(ϕ2)∩supp(G∗−χ)⊂ JM

+ (K)∩JM− (supp(χ)) is compact. Similarly, one showsG−ψ = −ϕ1.

Now Gψ = G+ψ −G−ψ = ϕ2 + ϕ1 = ϕ , henceϕ is in the image ofG.

Proposition 3.4.8. Let M be a globally hyperbolic Lorentzian manifold, let P be anor-mally hyperbolic operator acting on sections in a vector bundle E over M. Let G+ andG− be the advanced and retarded Green’s operators for P respectively.Then G± : D(M,E) → C∞

sc(M,E) and all maps in the complex (3.10) are sequentiallycontinuous.

Proof. The mapsP : D(M,E) → D(M,E) andP : C∞sc(M,E) → C∞

sc(M,E) are sequen-tially continuous simply becauseP is a differential operator. It remains to show thatG : D(M,E) →C∞

sc(M,E) is sequentially continuous.Let ϕ j ,ϕ ∈D(M,E) andϕ j → ϕ in D(M,E) for all j. Then there exists a compact subsetK ⊂ M such that supp(ϕ j),supp(ϕ) ⊂ K. Hence supp(Gϕ j),supp(Gϕ) ⊂ JM(K) for allj. From the proof of Theorem 3.3.1 we know thatG+ϕ coincides with the solutionu tothe Cauchy problemPu= ϕ with initial conditionsu|S− = (∇nu)|S− = 0 whereS− ⊂ Mis a spacelike Cauchy hypersurface such thatK ⊂ IM

+ (S−). Theorem 3.2.12 tells us that ifϕ j → ϕ in D(M,E), then the solutionsG+ϕ j → G+ϕ in C∞(M,E). The proof forG− isanalogous and the statement forG follows.

Remark 3.4.9. Green’s operators need not exist for any normally hyperbolic operatoron any spacetime. For example consider a compact spacetimeM and the d’Alembert


operator acting on real functions. Note that in this caseD(M,R) = C∞(M). If thereexisted Green’s operators the d’Alembert operator would beinjective. But any constantfunction belongs to the kernel of the operator.

3.5 Non-globally hyperbolic manifolds

Globally hyperbolic Lorentzian manifolds turned out to form a good class for the solutiontheory of normally hyperbolic operators. We have unique advanced and retarded funda-mental solutions and Green’s operators. The Cauchy problemis well-posed. Some ofthese analytical features survive when we pass to more general Lorentzian manifolds. Wewill see that we still have existence (but not uniqueness) offundamental solutions andGreen’s operators if the manifold can be embedded in a suitable way as an open subsetinto a globally hyperbolic manifold such that the operator extends. Moreover, we will seethat conformal changes of the Lorentzian metric do not alterthe basic analytical prop-erties. To illustrate this we construct Green’s operators for the Yamabe operator on theimportant anti-deSitter spacetime which is not globally hyperbolic.

Proposition 3.5.1. Let M be a timeoriented connected Lorentzian manifold. Let Pbe anormally hyperbolic operator acting on sections in a vectorbundle E over M. Let G± beGreen’s operators for P. LetΩ ⊂ M be a causally compatible connected open subset.DefineG± : D(Ω,E) →C∞(Ω,E) by

G±(ϕ) := G±(ϕext)|Ω.

HereD(Ω,E) → D(M,E), ϕ 7→ ϕext, denotes extension by zero.ThenG+ andG− are advanced and retarded Green’s operators for the restriction of P toΩ respectively.

Proof. Denote the restriction ofP to Ω by P. To show (i) in Definition 3.4.1 we check forϕ ∈ D(Ω,E)

PG±ϕ = P(G±(ϕext)|Ω) = P(G±(ϕext))|Ω = ϕext|Ω = ϕ .

Similarly, we see for (ii)

G±Pϕ = G±((Pϕ)ext)|Ω = G±(Pϕext)|Ω = ϕext|Ω = ϕ .

For (iii) we need thatΩ is a causally compatible subset ofM.

supp(G±ϕ) = supp(G±(ϕext)|Ω)

= supp(G±(ϕext))∩Ω⊂ JM

± (supp(ϕext))∩Ω= JM

± (supp(ϕ))∩Ω= JΩ

±(supp(ϕ)).

3.5. Non-globally hyperbolic manifolds 93

Example 3.5.2. In Minkowski space every convex open subsetΩ is causally compatible.Proposition 3.5.1 shows the existence of an advanced and a retarded Green’s operator forany normally hyperbolic operator onΩ which extends to a normally hyperbolic operatoronM.On the other hand, we have already noticed in Remark 3.1.5 that on convex domainsthe advanced and retarded fundamental solutions need not beunique. Thus the Green’soperatorsG± are not unique in general.

The proposition fails if we drop the condition onΩ to be a causally compatible subset ofM.

Example 3.5.3. For non-convex domainsΩ in Minkowski spaceM = Rn causal com-patibility does not hold in general, see Figure 7 on page 19. For anyϕ ∈ D(Ω,E) theproof of Proposition 3.5.1 shows that supp(G±ϕ) ⊂ JM

± (supp(ϕ))∩Ω. Now, if JΩ±(p) is

a proper subset ofJM± (p)∩Ω there is no reason why supp(G±ϕ) should be a subset of

JΩ±(supp(ϕ)). HenceG± are not Green’s operators in general.

Example 3.5.4.We consider theEinstein cylinderM = R×Sn−1 equipped with the prod-uct metricg = −dt2+canSn−1 where canSn−1 denotes the canonical Riemannian metric ofconstant sectional curvature 1 on the sphere. SinceSn−1 is compact, the Einstein cylinderis globally hyperbolic, compare Example 1.3.11.We putΩ := R×Sn−1

+ whereSn−1+ := (z1, . . . ,zn) ∈ Sn−1 | zn > 0 denotes the north-

ern hemisphere. Letp andq be two points inΩ which can be joined by a causal curvec : [0,1] → M in M. We writec(s) = (t(s),x(s)) with x(s) ∈ Sn−1. After reparametriza-tion we may assume that the curvex in Sn−1 is parametrized proportionally to arclength,canSn−1(x′,x′) ≡ ξ whereξ is a nonnegative constant.SinceSn−1

+ is a geodesically convex subset of the Riemannian manifoldSn−1 there is acurvey : [0,1] → Sn−1

+ with the same end points asx and of length at most the length ofx. If we parametrizey proportionally to arclength this means canSn−1(y′,y′)≡ η ≤ ξ . Thecurvec being causal means 0≥ g(c′,c′) = −(t ′)2 +canSn−1(x′,x′), i. e.,

(t ′)2 ≥ ξ .

This implies(t ′)2 ≥ η which in turn is equivalent to the curve ˜c := (t,y) being causal.Thusp andq can be joined by a causal curve which stays inΩ. ThereforeΩ is a causallycompatible subset of the Einstein cylinder.

Next we study conformal changes of the metric. LetM be a timeoriented connectedLorentzian manifold. Denote the Lorentzian metric byg. Let f : M → R be a positivesmooth function. Denote the conformally related metric by ˜g := f · g. This means thatg(X,Y) = f (p) ·g(X,Y) for all X,Y ∈ TpM. The causal type of tangent vectors and curvesis unaffected by this change of metric. Therefore all causalconcepts such as the chrono-logical or causal future and past remain unaltered by a conformal change of the metric.Similarly, the causality conditions are unaffected. Hence(M,g) is globally hyperbolic ifand only if(M, g) is globally hyperbolic.Let us denote byg∗ andg∗ the metrics on the cotangent bundleT∗M induced byg andgrespectively. Then we have ˜g∗ = 1

f g∗.


Let P be a normally hyperbolic operator with respect to ˜g. PutP := f · P, more precisely,

P(ϕ) = f · P(ϕ) (3.11)

for all ϕ . Since the principal symbol ofP is given byg∗, the principal symbol ofP isgiven byg∗,

σP(ξ ) = f ·σP(ξ ) = − f · g∗(ξ ,ξ ) · id = −g∗(ξ ,ξ ) · id.

ThusP is normally hyperbolic forg. Now suppose we have an advanced or a retardedGreen’s operatorG+ or G− for P. We defineG± : D(M,E) →C∞(M,E) by

G±ϕ := G± ( f ·ϕ) . (3.12)

We see thatG±(Pϕ) = G±( f · 1

f ·Pϕ) = G±(Pϕ) = ϕ

andP(G±ϕ) = 1

f ·P(G±( f ·ϕ)) = 1f · f ·ϕ = ϕ .

Multiplication by a nowhere vanishing function does not change supports, hence

supp(G±ϕ) = supp(G±( f ϕ)) ⊂ JM± (supp( f ϕ)) = JM

± (supp(ϕ)).

Notice again thatJM± is the same forg and forg. We have thus shown thatG± is a Green’s

operator forP. We summarize:

Proposition 3.5.5. Let M be a timeoriented connected Lorentzian manifold withLorentzian metric g. Let f: M → R be a positive smooth function and denote the confor-mally related metric byg := f ·g.Then (3.11) yields a 1-1-correspondence P↔ P between normally hyperbolic operatorsfor g and such operators forg. Similarly, (3.12) yields a 1-1-correspondence G± ↔ G±for their Green’s operators.

This discussion can be slightly generalized.

Remark 3.5.6. Let (M,g) be a timeoriented connected Lorentzian manifold. LetP be anormally hyperbolic operator onM for which advanced and retarded Green’s operatorsG+ andG− exist. Let f1, f2 : M → R be positive smooth functions. Then the operatorP := 1

f1·P· 1

f2, given by

P(ϕ) = 1f1·P( 1

f2·ϕ) (3.13)

for all ϕ , possesses advanced and retarded Green’s operatorsG±. They can be defined inanalogy to (3.12):

G±(ϕ) := f2 ·G±( f1 ·ϕ).

As above one getsPG±(ϕ) = ϕ andG±(Pϕ) = ϕ for all ϕ ∈ D(M,E). OperatorsP ofthe form (3.13) are normally hyperbolic with respect to the conformally related metricg = f1 · f2 ·g.


Combining Propositions 3.5.1 and 3.5.5 we get:

Corollary 3.5.7. Let (M, g) be timeoriented connected Lorentzian manifold which canbe conformally embedded as a causally compatible open subset Ω into the globally hy-perbolic manifold(M,g). Hence onΩ we haveg = f · g for some positive functionf ∈C∞(Ω,R).Let P be a normally hyperbolic operator on(M, g) and let P be the operator onΩ definedas in (3.11). Assume that P can be extended to a normally hyperbolic operator on thewhole manifold(M,g). Then the operatorP possesses advanced and retarded Green’soperators. Uniqueness is lost in general.

In the remainder of this section we will show that the preceding considerations can be ap-plied to an important example in general relativity:anti-deSitter spacetime. We will showthat it can be conformally embedded into the Einstein cylinder. The image of this embed-ding is the setΩ in Example 3.5.4. Hence we realize anti-deSitter spacetimeconformallyas a causally compatible subset of a globally hyperbolic Lorentzian manifold.For an integern≥ 2, one defines then-dimensionalpseudohyperbolic space

Hn1 := x∈ R

n+1 | 〈〈x,x〉〉 = −1,

where 〈〈x,y〉〉 := −x0y0 − x1y1 + ∑nj=2x jy j for all x = (x0,x1, . . . ,xn) and y =

(y0,y1, . . . ,yn) in Rn+1. With the induced metric (also denoted by〈〈· , ·〉〉) Hn

1 be-comes a connected Lorentzian manifold with constant sectional curvature−1, see e. g.[O’Neill1983, Chap. 4, Prop. 29].

Lemma 3.5.8. There exists a conformal diffeomorphism

Ψ :(

S1×Sn−1+ ,−canS1 +canSn−1

+

)→ (Hn

1 ,〈〈· , ·〉〉)

such that for any(p,x) ∈ S1×Sn−1+ ⊂ S1×Rn one has

(ψ∗〈〈· , ·〉〉)(p,x) =1x2

n

(−canS1 +canSn−1

+

).

Proof. We first construct an isometry between the pseudohyperbolicspace and

(S1×Hn−1,−y2

1canS1 +canHn−1

),

whereHn−1 := (y1, . . . ,yn) ∈ Rn | y1 > 0 and− y21 + ∑n

j=2y2j = −1 is the (n− 1)-

dimensional hyperbolic space. The hyperbolic metric canHn−1 is induced by theMinkowski metric onR

n. Then(Hn−1,canHn−1) is a Riemannian manifold with constantsectional curvature−1. Define the map

Φ : S1×Hn−1 → Hn1 ,

(p = (p0, p1),y = (y1, . . . ,yn)) 7→ (y1p0,y1p1,y2, . . . ,yn) ∈ Rn+1.


This map is clearly well-defined because−y21(p2

0 + p21︸︷︷︸

=1

)+ y22 + . . .y2

n = −y21 + ∑n

j=2y2j =

−1. The inverse map is given by

Φ−1(x) =

x0√

x20 +x2

1

,x1√

x20 +x2

1

,

(√x2

0 +x21,x2, . . . ,xn

) .

Geometrically, the mapΦ can be interpreted as follows: For any pointp = (p0, p1) ∈ S1,consider the hyperplaneHp of Rn+1 defined by

Hp := R · (p0, p1,0, . . . ,0)⊕Rn−1,

where (p0, p1,0, . . . ,0) ∈ Rn+1 and Rn−1 is identified with the subspace(0,0,w2, . . . ,wn) | wj ∈ R ⊂ Rn+1. If e2, . . . ,en is the canonical basis of thisRn−1, then

Bp := e1 := (p0, p1,0. . . ,0),e2, . . . ,enis a Lorentz orthonormal basis ofHp with respect to the metric induced by〈〈· , ·〉〉. DefineHn−1(p) as the hyperbolic space of(Hp,〈〈· , ·〉〉) in this basis. More precisely,Hn−1(p) =∑n

j=1η jej ∈ Hp |η1 > 0,−η21 + ∑n

j=2η2j = −1. Theny 7→ Φ(p,y) yields an isometry

from Minkowski space toHp which restricts to an isometryHn−1 → Hn−1(p).

H (p)

Hn1

S1bp

Hn−1(p)

b

Φ(p,y)

Rn−1

Fig. 28: Pseudohyperbolic space; construction ofΦ(p,y)


Let now(p,y) ∈ S1×Hn−1 andX = (X1,Xn−1) ∈ TpS1⊕TyHn−1. Then the differentialof Φ at (p,y) is given by

d(p,y)Φ(X) =(y1X1 +Xn−1

1 p,Xn−12 , . . . ,Xn−1

n

).

Therefore the pull-back of the metric onHn1 via Φ can be computed to yield

(Φ∗〈〈· , ·〉〉)(p,y)(X,X) = 〈〈d(p,y)Φ(X),d(p,y)Φ(X)〉〉

= y21〈〈X1,X1〉〉+(Xn−1

1 )2〈〈p, p〉〉+n

∑j=2

(Xn−1j )2

= −y21(X1

0 )2 +(X11 )2− (Xn−1

1 )2 +n

∑j=2

(Xn−1j )2

= −y21canS1(X1,X1)+canHn−1(Xn−1,Xn−1).

HenceΦ is an isometry(S1×Hn−1,−y2

1canS1 +canHn−1

)→ (Hn

1 ,〈〈· , ·〉〉) .

The stereographic projection from the south pole

π : Sn−1+ → Hn−1,

x = (x1, . . . ,xn) 7→ 1xn

(1,x1, . . . ,xn−1),

is a conformal diffeomorphism. It is easy to check thatπ is a well-defined diffeomorphismwith inverse given byy= (y1, . . . ,yn) 7→ 1

y1(y2, . . . ,yn,1). For anyx∈Sn−1

+ andX ∈TxSn−1+

the differential ofπ atx is given by

dxπ(X) =1xn

(0,X1, . . . ,Xn−1)−Xn

x2n(1,x1, . . . ,xn−1)

=1x2

n(−Xn,xnX1−x1Xn, . . . ,xnXn−1−xn−1Xn).

Therefore we get for the pull-back of the hyperbolic metric

(π∗canHn−1)x(X,X) =1x4

n

−X2

n +n−1

∑j=1

(xnXj −x jXn)2

=1x4

n

−X2

n +x2n

n−1

∑j=1

X2j −2xnXn

n−1

∑j=1

x jXj

︸︷︷︸=−xnXn

+X2n

n−1

∑j=1

x2j

︸︷︷︸=1−x2

n

=1x2

n

n−1

∑j=1

X2j +

X2n

x2n

=1x2

n

n

∑j=1

X2j ,


that is,(π∗canHn−1)x = 1x2n(canSn−1

+)x. We obtain an explicit diffeomorphism

Ψ := Φ (id×π) : S1×Sn−1+ → Hn

1 ,

(p = (p0, p1),x = (x1, . . . ,xn)) 7→ 1xn

(p0, p1,x1, . . . ,xn−1),

satisfying, for every(p,x) ∈ S1×Sn−1+ ,

(ψ∗〈〈· , ·〉〉)(p,x) = ((id×π)∗(Φ∗〈〈· , ·〉〉))x

= ((id×π)∗(−π(x)21canS1 +canHn−1))x

= −π(x)21canS1 +

1x2

ncanSn−1

+

=1x2

n


+

). (3.14)

This concludes the proof.

Following [O’Neill1983, Chap. 8, p. 228f], one defines then-dimensionalanti-deSitterspacetimeHn

1 to be the universal covering manifold of the pseudohyperbolic spaceHn1 .

For Hn1 the sectional curvature is identically−1 and the scalar curvature equals−n(n−

1). In physics,H41 is important because it provides a vacuum solution to Einstein’s field

equation with cosmological constantΛ = −3.The causality properties ofHn

1 are discussed in [O’Neill1983, Chap. 14, Example 41]. Itturns out thatHn

1 is not globally hyperbolic. The conformal diffeomorphism constructedin Lemma 3.5.8 lifts to a conformal diffeomorphism of the universal covering manifolds:

Ψ :(R×Sn−1

+ ,−dt2 +canSn−1+

)→(

Hn1 ,〈〈·, ·〉〉

)

such that for any(t,x) ∈ R1×Sn−1+ ⊂ R1×Rn one has

(ψ∗〈〈· , ·〉〉)(t,x) =1x2

n

(−dt2+canSn−1

+

).

Then Hn1 is conformally diffeomorphic to the causally compatible subsetR×Sn−1

+ ofthe globally hyperbolic Einstein cylinder. From the considerations above we will deriveexistence of Green’s operators for the Yamabe operatorYg on anti-deSitter spacetimeHn

1 .

Definition 3.5.9. Let (M,g) be a Lorentzian manifold of dimensionn ≥ 3. Then theYamabe operatorYg acting on functions onM is given by

Yg = 4n−1n−2

g +scalg (3.15)

whereg denotes the d’Alembert operator and scalg is the scalar curvature taken withrespect tog.


We perform a conformal change of the metric. To simplify formulas we write the confor-mally related metric asg = ϕ p−2g wherep = 2n

n−2 andϕ is a positive smooth function onM. The Yamabe operators for the metricsg andg are related by

Ygu = ϕ1−p ·Yg (ϕu), (3.16)

whereu∈C∞(M), see [Lee-Parker1987, p. 43, Eq. (2.7)]. MultiplyingYg with n−24·(n−1) we

obtain a normally hyperbolic operator

Pg = g +n−2

4 · (n−1)·scalg .

Equation (3.16) gives for this operator

Pgu = ϕ1−p (Pg(ϕu)) . (3.17)

Now we consider this operatorPg on the Einstein cylinderR×Sn−1. Since the Einsteincylinder is globally hyperbolic we get unique advanced and retarded Green’s operatorsG±for Pg. From Example 3.5.4 we know thatR×Sn−1

+ is a causally compatible subset of theEinstein cylinderR×Sn−1. By Proposition 3.5.1 we have advanced and retarded Green’soperators forPg onR×Sn−1

+ . From Equation (3.17) and Remark 3.5.6 we conclude

Corollary 3.5.10. On the anti-deSitter spacetimeHn1 the Yamabe operator possesses ad-

vanced and retarded Green’s operators.

Remark 3.5.11. It should be noted that the precise form of the zero order termof theYamabe operator given by the scalar curvature is crucial forour argument. On(Hn

1 , g) thescalar curvature is constant, scalg = −n(n−1). Hence the rescaled Yamabe operator isPg = g− 1

4n(n−2) = g−c with c := 14n · (n−2). For the d’Alembert operatorg on

(Hn1 , g) we have for anyu∈C∞(Hn

1)

gu = Pgu+c ·u= ϕ1−pPg(ϕu)+c ·u= ϕ1−p(Pg +c ·ϕ p−2)(ϕu).

The conformal factorϕ p−2 tends to infinity as one approaches the boundary ofR×Sn−1+

in R×Sn−1. Namely, for(t,x) ∈ R×Sn−1+ one has by (3.14)ϕ p−2(t,x) = x−2

n wherexn

denotes the last component ofx ∈ Sn−1 ⊂ Rn. Hence if one approaches the boundary,thenxn → 0 and thereforeϕ p−2 = x−2

n → ∞. Therefore one cannot extend the operatorPg+c·ϕ p−2 to an operator defined on the whole Einstein cylinderR×Sn−1. Thus we can-not establish existence of Green’s operators for the d’Alembert operator on anti-deSitterspacetime with the methods developed here.

How about uniqueness of fundamental solutions for normallyhyperbolic operators onanti-deSitter spacetime? We note that Theorem 3.1.1 cannotbe applied for anti-deSitterspacetime because the time separation functionτ is not finite. This can be seen as follows:We fix two pointsx,y∈ R×Sn−1

+ with x < y sufficiently far apart such that there exists atimelike curve connecting them in(p,x)∈R×Sn−1 |xn ≥ 0 having a nonempty segmenton the boundary(p,x) ∈ R×Sn−1|xn = 0.


x y

R×Sn−1+

R×Sn−1

Fig. 29: The time separation function is not finite on anti-deSitter spacetime

By sliding the segment on the boundary slightly we obtain a timelike curve in the upperhalf of the Einstein cylinder connectingx andy whose length with respect to the metric1x2n


+

)in (3.14) can be made arbitrarily large. This is due to the factor 1

x2n

which is large if the segment is chosen so thatxn is small along it.

x y

R×Sn−1+

R×Sn−1

A discussion as in Remark 3.1.5 considering supports (see picture below) shows thatfundamental solutions for normally hyperbolic operators are not unique on the upper halfR×Sn−1

+ of the Einstein cylinder. The fundamental solution of a point y in the lower halfof the Einstein cylinder can be added to a given fundamental solution of x in the upperhalf thus yielding a second fundamental solution ofx with the same support in the upperhalf.


xb

b

y

Fig. 30: Advanced fundamental solution inx on the (open) upper half-cylinder is not unique

Since anti-deSitter spacetime andR×Sn−1+ are conformally equivalent we obtain dis-

tinct fundamental solutions for operators on anti-deSitter spacetime as described in Corol-lary 3.5.7.

Chapter 4

Quantization

We now want to apply the analytical theory of wave equations and develop some math-ematical basics of field (or second) quantization. We do not touch the so-called firstquantization which is concerned with replacing point particles by wave functions. As inthe preceding chapters we look at fields (sections in vector bundles) which have to satisfysome wave equation (specified by a normally hyperbolic operator) and now we want toquantize such fields.We will explain two approaches. In the more traditional approach one constructs a quan-tum field which is a distribution satisfying the wave equation in the distributional sense.This quantum field takes its values in selfadjoint operatorson Fock space which is themulti-particle space constructed out of the single-particle space of wave functions. Thisconstruction will however crucially depend on the choice ofa Cauchy hypersurface.It seems that for quantum field theory on curved spacetimes the approach of local quan-tum physics is more appropriate. The idea is to associate to each (reasonable) spacetimeregion the algebra of observables that can be measured in this region. We will find con-firmed the saying that “quantization is a mystery, but secondquantization is a functor”by mathematical physicist Edward Nelson. One indeed constructs a functor from the cat-egory of globally hyperbolic Lorentzian manifolds equipped with a formally selfadjointnormally hyperbolic operator to the category ofC∗-algebras. We will see that this functorobeys the Haag-Kastler axioms of a local quantum field theory. This functorial interpre-tation of local covariant quantum field theory on curved spacetimes was introduced in[Hollands-Wald2001], [Verch2001], and [Brunetti-Fredenhagen-Verch2003].It should be noted that in contrast to what is usually done in the physics literature there isno need to fix a wave equation and then quantize the corresponding fields (e. g. the Klein-Gordon field). In the present book, both the underlying manifold as well as the normallyhyperbolic operator occur as variables in one single functor.In Sections 4.1 and 4.2 we develop the theory ofC∗-algebras and CCR-representations infull detail to the extent that we need. In the next three sections we construct the quanti-zation functors and check the Haag-Kastler axioms. The lasttwo sections are devoted tothe construction of the Fock space and the quantum field. We will see that the quantumfield determines the CCR-algebras up to isomorphism. This relates the two approaches to

104 Chapter 4. Quantization

quantum field theory on curved backgrounds.

4.1 C∗-algebras

In this section we will collect those basic concepts and facts related toC∗-algebras that wewill need when we discuss the canonical commutator relations in the subsequent section.We give complete proofs. Readers familiar withC∗-algebras may skip this section. Formore information onC∗-algebras see e. g. [Bratteli-Robinson2002-I].

Definition 4.1.1. Let A be an associativeC-algebra, let‖ · ‖ be a norm on theC-vectorspaceA, and let∗ : A → A, a 7→ a∗, be aC-antilinear map. Then(A,‖ · ‖,∗) is called aC∗-algebra, if (A,‖ · ‖) is complete and we have for alla, b∈ A:

(1) a∗∗ = a (∗ is an involution)

(2) (ab)∗ = b∗a∗

(3) ‖ab‖ ≤ ‖a‖‖b‖ (submultiplicativity)

(4) ‖a∗‖ = ‖a‖ (∗ is an isometry)

(5) ‖a∗a‖ = ‖a‖2 (C∗–property).

A (not necessarily complete) norm onA satisfying conditions (1) to (5) is called aC∗-norm.

Example 4.1.2.Let (H,(·, ·)) be a complex Hilbert space, letA = L (H) be the algebraof bounded operators onH. Let ‖ · ‖ be theoperator norm, i. e.,

‖a‖ := supx∈H‖x‖=1

‖ax‖.

Let a∗ be the operator adjoint toa, i. e.,

(ax,y) = (x,a∗y) for all x, y∈ H.

Axioms 1 to 4 are easily checked. Using Axioms 3 and 4 and the Cauchy-Schwarz in-equality we see

‖a‖2 = sup‖x‖=1

‖ax‖2 = sup‖x‖=1

(ax,ax) = sup‖x‖=1

(x,a∗ax)

≤ sup‖x‖=1

‖x‖ · ‖a∗ax‖ = ‖a∗a‖Axiom 3≤ ‖a∗‖ · ‖a‖ Axiom 4

= ‖a‖2.

This shows Axiom 5.

Example 4.1.3.Let X be a locally compact Hausdorff space. Put

A := C0(X) := f : X → C continuous| ∀ε > 0∃K ⊂ X compact, so that

∀x∈ X \K : | f (x)| < ε.

4.1.C∗-algebras 105

We callC0(X) thealgebra of continuous functions vanishing at infinity. If X is compact,thenA = C0(X) = C(X). All f ∈C0(X) are bounded and we may define:

‖ f‖ := supx∈X

| f (x)|.

Moreover letf ∗(x) := f (x).

Then(C0(X),‖ · ‖,∗) is a commutativeC∗–algebra.

Example 4.1.4.Let X be a differentiable manifold. Put

A := C∞0 (X) := C∞(X)∩C0(X).

We call C∞0 (X) the algebra of smooth functions vanishing at infinity. Norm and∗ are

defined as in the previous example. Then(C∞0 (X),‖ · ‖,∗) satisfies all axioms of a com-

mutativeC∗-algebra except that(A,‖ · ‖)) is not complete. If we complete this normedvector space, then we are back to the previous example.

Definition 4.1.5. A subalgebraA0 of a C∗-algebraA is called aC∗-subalgebraif it is aclosed subspace anda∗ ∈ A0 for all a∈ A0.

Any C∗-subalgebra is aC∗-algebra in its own right.

Definition 4.1.6. Let S be a subset of aC∗-algebraA. Then the intersection of allC∗-subalgebras ofA containingS is called theC∗-subalgebra generated by S.

Definition 4.1.7. An elementa of aC∗-algebra is calledselfadjointif a = a∗.

Remark 4.1.8. Like any algebra aC∗-algebraA has at most one unit 1. Namely, let 1′ beanother unit, then

1 = 1 ·1′ = 1′.

Now we have for alla∈ A

1∗a = (1∗a)∗∗ = (a∗1∗∗)∗ = (a∗1)∗ = a∗∗ = a

and similarly one seesa1∗ = a. Thus 1∗ is also a unit. By uniqueness 1= 1∗, i. e., theunit is selfadjoint. Moreover,

‖1‖ = ‖1∗1‖ = ‖1‖2,

hence‖1‖ = 1 or‖1‖ = 0. In the second case 1= 0 and thereforeA = 0. Hence we may(and will) from now on assume that‖1‖ = 1.

Example 4.1.9. (1) In Example 4.1.2 the algebraA = L (H) has a unit 1= idH .

(2) The algebraA = C0(X) has a unitf ≡ 1 if and only ifC0(X) = C(X), i. e., if andonly if X is compact.


Let A be aC∗–algebra with unit 1. We writeA× for the set of invertible elements inA. Ifa∈ A×, then alsoa∗ ∈ A× because

a∗ · (a−1)∗ = (a−1a)∗ = 1∗ = 1,

and similarly(a−1)∗ ·a∗ = 1. Hence(a∗)−1 = (a−1)∗.

Lemma 4.1.10.Let A be a C∗-algebra. Then the maps

A×A→ A, (a,b) 7→ a+b,

C×A→ A, (α,a) 7→ αa,

A×A→ A, (a,b) 7→ a ·b,

A× → A×, a 7→ a−1,

A→ A, a 7→ a∗,

are continuous.

Proof. (a) The first two maps are continuous for all normed vector spaces. This easilyfollows from the triangle inequality and from homogeneity of the norm.(b) Continuity of multiplication. Let a0, b0 ∈ A. Then we have for alla, b ∈ A with‖a−a0‖ < ε and‖b−b0‖ < ε:

‖ab−a0b0‖ = ‖ab−a0b+a0b−a0b0‖≤ ‖a−a0‖ · ‖b‖+‖a0‖ · ‖b−b0‖≤ ε

(‖b−b0‖+‖b0‖

)+‖a0‖ · ε

≤ ε(ε +‖b0‖

)+‖a0‖ · ε.

(c) Continuity of inversion. Let a0 ∈ A×. Then we have for alla∈ A× with ‖a−a0‖ <ε < ‖a−1

0 ‖−1:

‖a−1−a−10 ‖ = ‖a−1(a0−a)a−1

0 ‖≤ ‖a−1‖ · ‖a0−a‖ · ‖a−1

0 ‖≤

(‖a−1−a−1

0 ‖+‖a−10 ‖)· ε · ‖a−1

0 ‖.

Thus (1− ε‖a−1

0 ‖)

︸︷︷︸>0, sinceε<‖a−1

0 ‖−1

‖a−1−a−10 ‖ ≤ ε · ‖a−1

0 ‖2

and therefore‖a−1−a−1

0 ‖ ≤ ε1− ε‖a−1

0 ‖· ‖a−1

0 ‖2.

(d) Continuity of∗ is clear because∗ is an isometry.

Remark 4.1.11. If (A,‖ ·‖,∗) satisfies the axioms of aC∗-algebra except that(A,‖ ·‖) isnot complete, then the above lemma still holds because completeness has not been usedin the proof. LetA be the completion ofA with respect to the norm‖ · ‖. By the abovelemma+, ·, and∗ extend continuously toA thus makingA into aC∗-algebra.


Definition 4.1.12. Let A be aC∗–algebra with unit 1. Fora∈ A we call

rA(a) := λ ∈ C | λ ·1−a∈ A×

theresolvent set of aandσA(a) := C\ rA(a)

thespectrum of a. Forλ ∈ rA(a)

(λ ·1−a)−1 ∈ A

is called theresolvent of a atλ . Moreover, the number

ρA(a) := sup|λ | | λ ∈ σA(a)

is called thespectral radius of a.

Example 4.1.13.Let X be a compact Hausdorff space and letA = C(X). Then

A× = f ∈C(X) | f (x) 6= 0 for all x∈ X,σC(X)( f ) = f (X) ⊂ C,

rC(X)( f ) = C\ f (X),

ρC(X)( f ) = ‖ f‖∞ = maxx∈X | f (x)|.

Proposition 4.1.14.Let A be a C∗–algebra with unit1 and let a∈ A. ThenσA(a) ⊂ C isa nonempty compact subset and the resolvent

rA(a) → A, λ 7→ (λ ·1−a)−1,

is continuous. Moreover,

ρA(a) = limn→∞

‖an‖ 1n = inf

n∈N

‖an‖ 1n ≤ ‖a‖.

Proof. (a) Letλ0 ∈ rA(a). Forλ ∈ C with

|λ −λ0| < ‖(λ01−a)−1‖−1 (4.1)

the Neumann series∞

∑m=0

(λ0−λ )m(λ01−a)−m−1

converges absolutely because

‖(λ0−λ )m(λ01−a)−m−1‖ ≤ |λ0−λ |m · ‖(λ01−a)−1‖m+1

= ‖(λ01−a)−1‖ ·( ‖(λ01−a)−1‖

|λ0−λ |−1︸︷︷︸

<1 by (4.1)

)m.


SinceA is complete the Neumann series converges inA. It converges to the resolvent(λ1−a)−1 because

(λ1−a)∞

∑m=0

(λ0−λ )m(λ01−a)−m−1

= [(λ −λ0)1+(λ01−a)]∞

∑m=0

(λ0−λ )m(λ01−a)−m−1

= −∞

∑m=0

(λ0−λ )m+1(λ01−a)−m−1+∞

∑m=0

(λ0−λ )m(λ01−a)−m

= 1.

Thus we have shownλ ∈ rA(a) for all λ satisfying (4.1). HencerA(a) is open andσA(a)is closed.(b) Continuity of the resolvent.We estimate the difference of the resolvent ofa at λ0 andat λ using the Neumann series. Ifλ satisfies (4.1), then

∥∥(λ1−a)−1− (λ01−a)−1∥∥=

∥∥∥∞

∑m=0

(λ0−λ )m(λ01−a)−m−1− (λ01−a)−1∥∥∥

≤∞

∑m=1

|λ0−λ |m‖(λ01−a)−1‖m+1

= ‖(λ01−a)−1‖ · |λ0−λ | · ‖(λ01−a)−1‖1−|λ0−λ | · ‖(λ01−a)−1‖

= |λ0−λ | · ‖(λ01−a)−1‖2

1−|λ0−λ | · ‖(λ01−a)−1‖→ 0 for λ → λ0.

Hence the resolvent is continuous.(c) We showρA(a) ≤ infn‖an‖ 1

n ≤ lim infn→∞ ‖an‖ 1n . Let n∈ N be fixed and let|λ |n >

‖an‖. Eachm∈ N0 can be written uniquely in the formm= pn+q, p, q∈ N0, 0≤ q≤n−1. The series

1λ

∞

∑m=0

( aλ

)m=

1λ

n−1

∑q=0

( aλ

)q ∞

∑p=0

( an

λ n︸︷︷︸‖·‖<1

)p

converges absolutely. Its limit is(λ1−a)−1 because

(λ1−a

)·( ∞

∑m=0

λ−m−1am)

=∞

∑m=0

λ−mam−∞

∑m=0

λ−m−1am+1 = 1

and similarly ( ∞

∑m=0

λ−m−1am)·(λ1−a

)= 1.

Hence for|λ |n > ‖an‖ the element(λ1−a) is invertible and thusλ ∈ rA(a). Therefore

ρA(a) ≤ infn∈N

‖an‖ 1n ≤ lim inf

n→∞‖an‖ 1

n .


(d) We showρA(a) ≥ limsupn→∞ ‖an‖ 1n . We abbreviateρ(a) := limsupn→∞ ‖an‖ 1

n .Case 1:ρ(a) = 0. If a were invertible, then

1 = ‖1‖ = ‖ana−n‖ ≤ ‖an‖ · ‖a−n‖

would imply 1≤ ρ(a) · ρ(a−1) = 0, which yields a contradiction. Thereforea 6∈A×. Thus0∈ σA(a). In particular, the spectrum ofa is nonempty. Hence the spectral radiusρA(a)is bounded from below by 0 and thus

ρ(a) = 0≤ ρA(a).

Case 2:ρ(a) > 0. If an ∈ A are elements for whichRn := (1−an)−1 exist, then

an → 0 ⇔ Rn → 1.

This follows from the fact that the mapA× → A×, a 7→ a−1, is continuous byLemma 4.1.10. Put

S:= λ ∈ C | |λ | ≥ ρ(a).We want to show thatS 6⊂ rA(a) since then there existsλ ∈ σA(a) such that|λ | ≥ ρ(a)and hence

ρA(a) ≥ |λ | ≥ ρ(a).

Assume in the contrary thatS⊂ rA(a). Let ω ∈ C be ann–th root of unity, i. e.,ωn = 1.For λ ∈ Swe also haveλ

ωk ∈ S⊂ rA(a). Hence there exists

( λωk 1−a

)−1=

ωk

λ

(1− ωka

λ

)−1

and we may define

Rn(a,λ ) :=1n

n

∑k=1

(1− ωka

λ

)−1.

We compute

(1− an

λ n

)Rn(a,λ ) =

1n

n

∑k=1

n

∑l=1

(ωk(l−1)al−1

λ l−1 − ωklal

λ l

)(1− ωka

λ

)−1

=1n

n

∑k=1

n

∑l=1

ωk(l−1)al−1

λ l−1

=1n

n

∑l=1

al−1

λ l−1

n

∑k=1

(ω l−1)k

︸︷︷︸

=

0 if l ≥ 2

n if l = 1

= 1.

Similarly one seesRn(a,λ )(1− an

λ n

)= 1. Hence

Rn(a,λ ) =(

1− an

λ n

)−1


for anyλ ∈ S⊂ rA(a). Moreover forλ ∈ Swe have∥∥∥(

1− an

ρ(a)n

)−1−(

1− an

λ n

)−1∥∥∥

≤ 1n

n

∑k=1

∥∥∥(

1− ωkaρ(a)

)−1−(

1− ωkaλ

)−1∥∥∥

=1n

n

∑k=1

∥∥∥(

1− ωkaρ(a)

)−1(1− ωka

λ−1+

ωkaρ(a)

)(1− ωka

λ

)−1∥∥∥

=1n

n

∑k=1

∥∥∥( ρ(a)

ωk 1−a)−1(

− ρ(a)aωk +

λaωk

)( λωk 1−a

)−1∥∥∥

≤ |ρ(a)−λ | · ‖a‖ ·supz∈S

‖(z1−a)−1‖2.

The supremum is finite sincez 7→ (z1−a)−1 is continuous onrA(a)⊃ Sby part (b) of theproof and since for|z| ≥ 2 · ‖a‖ we have

‖(z1−a)−1‖ ≤ 1|z|

∞

∑n=0

‖a‖n

|z|n︸︷︷︸≤( 1

2 )n

≤ 2|z| ≤

1‖a‖ .

Outside the annulusB2‖a‖(0) −Bρ(a)(0) the expression‖(z1 −a)−1‖ is bounded by 1

‖a‖ and on thecompact annulus it is bounded bycontinuity.

0

2‖a‖

ρ(a)

Fig. 31:‖(z1−a)−1‖ is bounded

PutC := ‖a‖ ·sup

z∈S‖(z1−a)−1‖2.

We have shown‖Rn(a, ρ(a))−Rn(a,λ )‖ ≤C · |ρ(a)−λ |

for all n∈ N and allλ ∈ S. Puttingλ = ρ(a)+ 1j we obtain

∥∥∥(

1− an

ρ(a)n

)−1−(

1− an

(ρ(a)+ 1j )

n

︸︷︷︸→0 for n→∞

)−1

︸︷︷︸→1 for n→∞

∥∥∥≤ Cj,


thus

limsupn→∞

∥∥∥(

1− an

ρ(a)n

)−1−1∥∥∥≤ C

j

for all j ∈ N and hence

limsupn→∞

∥∥∥(

1− an

ρ(a)n

)−1−1∥∥∥= 0.

Forn→ ∞ we get (1− an

ρ(a)n

)−1→ 1

and thus‖an‖ρ(a)n → 0. (4.2)

On the other hand we have

‖an+1‖ 1n+1 ≤ ‖a‖ 1

n+1 · ‖an‖ 1n+1

= ‖a‖ 1n+1 · ‖an‖−

1n(n+1) · ‖an‖ 1

n

≤ ‖a‖ 1n+1 · ‖a‖−

nn(n+1) · ‖an‖ 1

n

= ‖an‖ 1n .

Hence the sequence(‖an‖ 1

n

)n∈N

is monotonically nonincreasing and therefore

ρ(a) = limsupk→∞

‖ak‖ 1k ≤ ‖an‖ 1

n for all n∈ N.

Thus 1≤ ‖an‖ρ(a)n for all n∈ N, in contradiction to (4.2).

(e) The spectrum is nonempty.If σ(a) = /0, thenρA(a) = −∞ contradictingρA(a) =

limn→∞ ‖an‖ 1n ≥ 0.

Definition 4.1.15. Let A be aC∗-algebra with unit. Thena∈ A is called

• normal, if aa∗ = a∗a,

• an isometry, if a∗a = 1, and

• unitary, if a∗a = aa∗ = 1.

Remark 4.1.16. In particular, selfadjoint elements are normal. In a commutative algebraall elements are normal.

Proposition 4.1.17. Let A be a C∗–algebra with unit and let a∈ A. Then the followingholds:

(1) σA(a∗) = σA(a) = λ ∈ C |λ ∈ σA(a).


(2) If a∈ A×, thenσA(a−1) = σA(a)−1.

(3) If a is normal, thenρA(a) = ‖a‖.

(4) If a is an isometry, thenρA(a) = 1.

(5) If a is unitary, thenσA(a) ⊂ S1 ⊂ C.

(6) If a is selfadjoint, thenσA(a) ⊂ [−‖a‖,‖a‖] and moreoverσA(a2) ⊂ [0,‖a‖2].

(7) If P(z) is a polynomial with complex coefficients and a∈ A is arbitrary, then

σA(P(a)

)= P

(σA(a)

)= P(λ ) |λ ∈ σA(a).

Proof. We start by showing assertion (1). A numberλ does not lie in the spectrum ofaif and only if (λ1−a) is invertible, i. e., if and only if(λ1−a)∗ = λ1−a∗ is invertible,i. e., if and only ifλ does not lie in the spectrum ofa∗.To see (2) leta be invertible. Then 0 lies neither in the spectrumσA(a) of a nor in thespectrumσA(a−1) of a−1. Moreover, we have forλ 6= 0

λ1−a= λa(a−1−λ−11)

andλ−11−a−1 = λ−1a−1(a−λ1).

Henceλ1−a is invertible if and only ifλ−11−a−1 is invertible.To show (3) leta be normal. Thena∗a is selfadjoint, in particular normal. Using theC∗–property we obtain inductively

‖a2n‖2 = ‖(a2n)∗a2n‖ = ‖(a∗)2n

a2n‖ = ‖(a∗a)2n‖= ‖(a∗a)2n−1

(a∗a)2n−1‖ = ‖(a∗a)2n−1‖2

= · · · = ‖a∗a‖2n= ‖a‖2n+1

.

ThusρA(a) = lim

n→∞‖a2n‖ 1

2n = limn→∞

‖a‖ = ‖a‖.

To prove (4) leta be an isometry. Then

‖an‖2 = ‖(an)∗an‖ = ‖(a∗)nan‖ = ‖1‖ = 1.

HenceρA(a) = lim

n→∞‖an‖ 1

n = 1.

For assertion (5) leta be unitary. On the one hand we have by (4)

σA(a) ⊂ λ ∈ C | |λ | ≤ 1.

On the other hand we have

σA(a)(1)= σA(a∗) = σA(a−1)

(2)= σA(a)

−1.


Both combined yieldσA(a) ⊂ S1.To show (6) leta be selfadjoint. We need to showσA(a)⊂R. Let λ ∈ R with λ−1 > ‖a‖.Then|− iλ−1| = λ−1 > ρ(a) and hence 1+ iλa= iλ (−iλ−1+a) is invertible. Put

U := (1− iλa)(1+ iλa)−1.

ThenU∗ = ((1+ iλa)−1)∗(1− iλa)∗ = (1− iλa∗)−1 ·(1+ iλa∗) = (1− iλa)−1·(1+ iλa)and therefore

U∗U = (1− iλa)−1 · (1+ iλa)(1− iλa)(1+ iλa)−1

= (1− iλa)−1(1− iλa)(1+ iλa)(1+ iλa)−1

= 1.

Similarly UU∗ = 1, i. e.,U is unitary. By (5)σA(U) ⊂ S1. A simple computation withcomplex numbers shows that

|(1− iλ µ)(1+ iλ µ)−1| = 1 ⇔ µ ∈ R.

Thus(1− iλ µ)(1+ iλ µ)−1 ·1−U is invertible if µ ∈ C\R. From

(1− iλ µ)(1+ iλ µ)−1 ·1−U

= (1+ iλ µ)−1((1− iλ µ)(1+ iλa)1− (1+ iλ µ)(1− iλa))(1+ iλa)−1

= 2iλ (1+ iλ µ)−1(a− µ1)(1+ iλ µ)−1

we see thata− µ1 is invertible for allµ ∈ C \R. Thusµ ∈ rA(a) for all µ ∈ C \R andhenceσA(a) ⊂ R. The statement aboutσA(a2) now follows from part (7).Finally, to prove (7) decompose the polynomialP(z)−λ into linear factors

P(z)−λ = α ·n

∏j=1

(α j −z), α,α j ∈ C.

We insert an algebra elementa∈ A:

P(a)−λ1= α ·n

∏j=1

(α j 1−a).

Since the factors in this product commute the product is invertible if and only if all factorsare invertible.1 In our case this means

λ ∈ σA(P(a)

)⇔ at least one factor is noninvertible

⇔ α j ∈ σA(a) for somej

⇔ λ = P(α j ) ∈ P(σA(a)

).

1This is generally true in algebras with unit. Letb = a1 · · ·an with commuting factors. Thenb is invertibleif all factors are invertible:b−1 = a−1

n · · ·a−11 . Conversely, ifb is invertible, thena−1

i = b−1 ·∏ j 6=i aj where wehave used that the factors commute.


Corollary 4.1.18. Let (A,‖ · ‖,∗) be a C∗–algebra with unit. Then the norm‖ · ‖ isuniquely determined by A and∗.

Proof. Fora∈ A the elementa∗a is selfadjoint and hence

‖a‖2 = ‖a∗a‖ 4.1.17(3)= ρA(a∗a)

depends only onA and∗.

Definition 4.1.19. Let A andB beC∗–algebras. An algebra homomorphism

π : A→ B

is called∗–morphismif for all a∈ A we have

π(a∗) = π(a)∗.

A mapπ : A→ A is called∗–automorphismif it is an invertible∗-morphism.

Corollary 4.1.20. Let A and B be C∗–algebras with unit. Each unit-preserving∗–morphismπ : A→ B satisfies

‖π(a)‖ ≤ ‖a‖for all a ∈ A. In particular,π is continuous.

Proof. Fora∈ A×

π(a)π(a−1) = π(aa−1) = π(1) = 1

holds and similarlyπ(a−1)π(a) = 1. Henceπ(a) ∈ B× with π(a)−1 = π(a−1). Now ifλ ∈ rA(a), then

λ1−π(a) = π(λ1−a)∈ π(A×) ⊂ B×,

i. e., λ ∈ rB(π(a)). HencerA(a) ⊂ rB(π(a)) andσB(π(a)) ⊂ σA(a). This implies theinequality

ρB(π(a)) ≤ ρA(a).

Sinceπ is a∗–morphism anda∗a andπ(a)∗π(a) are selfadjoint we can estimate the normas follows:

‖π(a)‖2 = ‖π(a)∗π(a)‖ = ρB(π(a)∗π(a)

)= ρB

(π(a∗a)

)

≤ ρA(a∗a) = ‖a‖2.

Corollary 4.1.21. Let A be a C∗–algebra with unit. Then each unit-preserving∗-automorphismπ : A→ A satisfies for all a∈ A:

‖π(a)‖= ‖a‖

Proof.‖π(a)‖ ≤ ‖a‖ = ‖π−1(π(a)

)‖ ≤ ‖π(a)‖.


We extend Corollary 4.1.21 to the case whereπ is injective but not necessarily onto. Thisis not a direct consequence of Corollary 4.1.21 because it isnot a priori clear that theimage of a∗-morphism is closed and hence aC∗-algebra in its own right.

Proposition 4.1.22.Let A and B be C∗–algebras with unit. Each injective unit-preserving∗–morphismπ : A→ B satisfies

‖π(a)‖= ‖a‖

for all a ∈ A.

Proof. By Corollary 4.1.20 we only have to show‖π(a)‖ ≥ ‖a‖. Once we know thisinequality for selfadjoint elements it follows for alla∈ A because

‖π(a)‖2 = ‖π(a)∗π(a)‖ = ‖π(a∗a)‖ ≥ ‖a∗a‖ = ‖a‖2.

Assume there exists a selfadjoint elementa ∈ A such that‖π(a)‖ < ‖a‖. By Proposi-tion 4.1.17σA(a) ⊂ [−‖a‖,‖a‖] andρA(a) = ‖a‖, hence‖a‖ ∈ σA(a) or −‖a‖ ∈ σA(a).Similarly, σB(π(a)) ⊂ [−‖π(a)‖,‖π(a)‖].Choose a continuous functionf : [−‖a‖,‖a‖] → R such that f vanishes on[−‖π(a)‖,‖π(a)‖] and f (−‖a‖) = f (‖a‖) = 1. By the Stone-Weierstrass theorem wecan find polynomialsPn such that‖ f −Pn‖C0([−‖a‖,‖a‖]) → 0 asn → ∞. In particular,‖Pn‖C0([−‖π(a)‖,‖π(a)‖]) = ‖ f −Pn‖C0([−‖π(a)‖,‖π(a)‖]) → 0 asn→ ∞. We may and will as-sume that the polynomialsPn are real.FromσB(Pn(π(a))) = Pn(σB(π(a))) ⊂ Pn([−‖π(a)‖,‖π(a)‖]) we see

‖Pn(π(a))‖ = ρB(Pn(π(a))) ≤ max|Pn([−‖π(a)‖,‖π(a)‖])| n→∞−→ 0

and thus

limn→∞

Pn(π(a)) = 0.

The sequence(Pn(a))n is a Cauchy sequence because

‖Pn(a)−Pm(a)‖ = ρA(Pn(a)−Pm(a))

≤ max|(Pn−Pm)([−‖a‖,‖a‖])|= ‖Pn−Pm‖C0([−‖a‖,‖a‖])≤ ‖Pn− f‖C0([−‖a‖,‖a‖]) +‖ f −Pm‖C0([−‖a‖,‖a‖]).

Denote its limit byf (a) ∈ A. Since‖a‖ ∈ σA(a) or−‖a‖ ∈ σA(a) and sincef (±‖a‖) = 1we have

‖ f (a)‖ = limn→∞

‖Pn(a)‖ = limn→∞

ρB(Pn(a)) ≥ limn→∞

|Pn(±‖a‖)|= 1.

Hence f (a) 6= 0. But π( f (a)) = π(limn→∞ Pn(a)) = limn→∞ π(Pn(a)) =limn→∞ Pn(π(a)) = 0. This contradicts the injectivity ofπ .


4.2 The canonical commutator relations

In this section we introduce Weyl systems and CCR-representations. They formalize the“canonical commutator relations” from quantum field theoryin an “exponentiated form”as we shall see later. The main result of the present section is Theorem 4.2.9 which saysthat for each symplectic vector space there is an essentially unique CCR-representation.Our approach follows ideas in [Manuceau1968]. A different proof of this result may befound in [Bratteli-Robinson2002-II, Sec. 5.2.2.2].Let (V,ω) be asymplectic vector space, i. e.,V is a real vector space of finite or infinitedimension andω : V ×V → R is an antisymmetric bilinear map such thatω(ϕ ,ψ) = 0for all ψ ∈V impliesϕ = 0.

Definition 4.2.1. A Weyl systemof (V,ω) consists of aC∗-algebraA with unit and a mapW : V → A such that for allϕ ,ψ ∈V we have

(i) W(0) = 1,

(ii) W(−ϕ) = W(ϕ)∗,

(iii) W(ϕ) ·W(ψ) = e−iω(ϕ,ψ)/2W(ϕ + ψ).

Condition (iii) says thatW is a representation of the additive groupV in A up to the“twisting factor” e−iω(ϕ,ψ)/2. Note that sinceV is not given a topology there is no re-quirement onW to be continuous. In fact, we will see that even in the case when V isfinite-dimensional and soV carries a canonical topologyW will in general not be contin-uous.

Example 4.2.2. We construct a Weyl system for an arbitrary symplectic vector space(V,ω). Let H = L2(V,C) be the Hilbert space of square-integrable complex-valued func-tions onV with respect to the counting measure, i. e.,H consists of those functionsF : V → C that vanish everywhere except for countably many points andsatisfy

‖F‖2L2 := ∑

ϕ∈V|F(ϕ)|2 < ∞.

The Hermitian product onH is given by

(F,G)L2 = ∑ϕ∈V

F(ϕ) ·G(ϕ).

Let A := L (H) be theC∗-algebra of bounded linear operators onH as in Example 4.1.2.We define the mapW : V → A by

(W(ϕ)F)(ψ) := eiω(ϕ,ψ)/2F(ϕ + ψ).

Obviously,W(ϕ) is a bounded linear operator onH for anyϕ ∈V andW(0) = idH = 1.

4.2. The canonical commutator relations 117

We check (ii) by making the substitutionχ = ϕ + ψ :

(W(ϕ)F,G)L2 = ∑ψ∈V

(W(ϕ)F)(ψ)G(ψ)

= ∑ψ∈V

eiω(ϕ,ψ)/2F(ϕ + ψ)G(ψ)

= ∑χ∈V

eiω(ϕ,χ−ϕ)/2F(χ)G(χ −ϕ)

= ∑χ∈V

eiω(ϕ,χ)/2 ·F(χ) ·G(χ −ϕ)

= ∑χ∈V

F(χ) ·eiω(−ϕ,χ)/2 ·G(χ −ϕ)

= (F,W(−ϕ)G)L2.

HenceW(ϕ)∗ = W(−ϕ). To check (iii) we compute

(W(ϕ)(W(ψ)F))(χ) = eiω(ϕ,χ)/2(W(ψ)F)(ϕ + χ)

= eiω(ϕ,χ)/2eiω(ψ,ϕ+χ)/2F(ϕ + χ + ψ)

= eiω(ψ,ϕ)/2eiω(ϕ+ψ,χ)/2F(ϕ + χ + ψ)

= e−iω(ϕ,ψ)/2 (W(ϕ + ψ)F)(χ).

Thus W(ϕ)W(ψ) = e−iω(ϕ,ψ)/2W(ϕ + ψ). Let CCR(V,ω) be theC∗-subalgebra ofL (H) generated by the elementsW(ϕ), ϕ ∈V. Then CCR(V,ω) together with the mapW forms a Weyl-system for(V,ω).

Proposition 4.2.3.Let(A,W) be a Weyl system of a symplectic vector space(V,ω). Then

(1) W(ϕ) is unitary for eachϕ ∈V,

(2) ‖W(ϕ)−W(ψ)‖ = 2 for all ϕ ,ψ ∈V, ϕ 6= ψ ,

(3) The algebra A is not separable unless V= 0,

(4) the familyW(ϕ)ϕ∈V is linearly independent.

Proof. From W(ϕ)∗W(ϕ) = W(−ϕ)W(ϕ) = eiω(−ϕ,ϕ)W(0) = 1 and similarlyW(ϕ)W(ϕ)∗ = 1 we see thatW(ϕ) is unitary.To show (2) letϕ ,ψ ∈V with ϕ 6= ψ . For arbitraryχ ∈V we have

W(χ)W(ϕ −ψ)W(χ)−1 = W(χ)W(ϕ −ψ)W(χ)∗

= e−iω(χ ,ϕ−ψ)/2W(χ + ϕ −ψ)W(−χ)

= e−iω(χ ,ϕ−ψ)/2e−iω(χ+ϕ−ψ,−χ)/2W(χ + ϕ −ψ − χ)

= e−iω(χ ,ϕ−ψ)W(ϕ −ψ).

Hence the spectrum satisfies

σA(W(ϕ −ψ)) = σA(W(χ)W(ϕ −ψ)W(χ)−1) = e−iω(χ ,ϕ−ψ) σA(W(ϕ −ψ)).


Sinceϕ −ψ 6= 0 the real numberω(χ ,ϕ −ψ) runs through all ofR asχ runs throughV. Therefore the spectrum ofW(ϕ −ψ) is U(1)-invariant. By Proposition 4.1.17 (5) thespectrum is contained inS1 and by Proposition 4.1.14 it is nonempty. HenceσA(W(ϕ −ψ)) = S1 and therefore

σA(eiω(ψ,ϕ)/2W(ϕ −ψ)) = S1.

ThusσA(eiω(ψ,ϕ)/2W(ϕ −ψ)−1) is the circle of radius 1 centered at−1. Now Proposi-tion 4.1.17 (3) says

‖eiω(ψ,ϕ)/2W(ϕ −ψ)−1‖= ρA

(eiω(ψ,ϕ)/2W(ϕ −ψ)−1

)= 2.

FromW(ϕ)−W(ψ) = W(ψ)(W(ψ)∗W(ϕ)−1) = W(ψ)(eiω(ψ,ϕ)/2W(ϕ −ψ)−1) weconclude

‖W(ϕ)−W(ψ)‖2

= ‖(W(ϕ)−W(ψ))∗(W(ϕ)−W(ψ))‖= ‖(eiω(ψ,ϕ)/2W(ϕ −ψ)−1)∗W(ψ)∗W(ψ)(eiω(ψ,ϕ)/2W(ϕ −ψ)−1)‖= ‖(eiω(ψ,ϕ)/2W(ϕ −ψ)−1)∗ (eiω(ψ,ϕ)/2W(ϕ −ψ)−1)‖= ‖eiω(ψ,ϕ)/2W(ϕ −ψ)−1‖2

= 4.

This shows (2). Assertion (3) now follows directly since theballs of radius 1 centered atW(ϕ), ϕ ∈V, form an uncountable collection of mutually disjoint open subsets.

We show (4). Letϕ j ∈ V, j = 1, . . . ,n, be pairwise different and let∑nj=1α jW(ϕ j) = 0.

We showα1 = . . .αn = 0 by induction onn. The casen = 1 is trivial by (1). Without lossof generality assumeαn 6= 0. Hence

W(ϕn) =n−1

∑j=1

−α j

αnW(ϕ j )

and therefore

1 = W(ϕn)∗W(ϕn)

=n−1

∑j=1

−α j

αnW(−ϕn)W(ϕ j)

=n−1

∑j=1

−α j

αne−iω(−ϕn,ϕ j )/2W(ϕ j −ϕn)

=n−1

∑j=1

β jW(ϕ j −ϕn)


where we have putβ j :=−α jαn

eiω(ϕn,ϕ j )/2. For an arbitraryψ ∈V we obtain

1 = W(ψ) ·1 ·W(−ψ)

=n−1

∑j=1

β jW(ψ)W(ϕ j −ϕn)W(−ψ)

=n−1

∑j=1

β je−iω(ψ,ϕ j−ϕn)W(ϕ j −ϕn).

Fromn−1

∑j=1

β jW(ϕ j −ϕn) =n−1

∑j=1

β je−iω(ψ,ϕ j−ϕn)W(ϕ j −ϕn)

we conclude by the induction hypothesis

β j = β je−iω(ψ,ϕ j−ϕn)

for all j = 1, . . . ,n−1. If someβ j 6= 0, thene−iω(ψ,ϕ j−ϕn) = 1, hence

ω(ψ ,ϕ j −ϕn) = 0

for all ψ ∈ V. Sinceω is nondegenerateϕ j −ϕn = 0, a contradiction. Therefore allβ j

and thus allα j are zero, a contradiction.

Remark 4.2.4. Let (A,W) be a Weyl system of the symplectic vector space(V,ω).Then the linear span of theW(ϕ), ϕ ∈ V, is closed under multiplication and under∗.This follows directly from the properties of a Weyl system. We denote this linear spanby 〈W(V)〉 ⊂ A. Now if (A′,W′) is another Weyl system of the same symplectic vec-tor space(V,ω), then there is a unique linear mapπ : 〈W(V)〉 → 〈W′(V)〉 determinedby π(W(ϕ)) = W′(ϕ). Sinceπ is given by a bijection on the basesW(ϕ)ϕ∈V andW′(ϕ)ϕ∈V it is a linear isomorphism. By the properties of a Weyl systemπ is a ∗-isomorphism. In other words, there is a unique∗-isomorphism such that the followingdiagram commutes

〈W′(V)〉

VW1 //

W2

<<xxxxxxxxx〈W(V)〉

π

OO

Remark 4.2.5. On 〈W(V)〉 we can define the norm∥∥∥∑

ϕaϕW(ϕ)

∥∥∥1

:= ∑ϕ|aϕ |.

This norm is not aC∗-norm but for everyC∗-norm‖·‖0 on〈W(V)〉we have by the triangleinequality and by Proposition 4.2.3 (1)

‖a‖0 ≤ ‖a‖1 (4.3)

for all a∈ 〈W(V)〉.


Lemma 4.2.6. Let (A,W) be a Weyl system of a symplectic vector space(V,ω). Then

‖a‖max := sup‖a‖0 |‖ · ‖0 is a C∗-norm on〈W(V)〉

defines a C∗-norm on〈W(V)〉.

Proof. The givenC∗-norm onA restricts to one on〈W(V)〉, so the supremum is not takenon the empty set. Estimate (4.3) shows that the supremum is finite. The properties of aC∗-norm are easily checked. E. g. the triangle inequality follows from

‖a+b‖max = sup‖a+b‖0|‖ · ‖0 is aC∗-norm on〈W(V)〉≤ sup‖a‖0+‖b‖0|‖ · ‖0 is aC∗-norm on〈W(V)〉≤ sup‖a‖0 |‖ · ‖0 is aC∗-norm on〈W(V)〉

+sup‖b‖0 |‖ · ‖0 is aC∗-norm on〈W(V)〉= ‖a‖max+‖b‖max.

The other properties are shown similarly.

Lemma 4.2.7. Let (A,W) be a Weyl system of a symplectic vector space(V,ω). Thenthe completion〈W(V)〉max

of 〈W(V)〉 with respect to‖ · ‖max is simple, i. e., it has nonontrivial closed twosided∗-ideals.

Proof. By Remark 4.2.4 we may assume that(A,W) is the Weyl system constructed inExample 4.2.2. In particular,〈W(V)〉 carries theC∗-norm‖ ·‖Op, the operator norm givenby 〈W(V)〉 ⊂ L (H) whereH = L2(V,C).Let I ⊂ 〈W(V)〉max

be a closed twosided∗-ideal. ThenI0 := I ∩C ·W(0) is a (complex)vector subspace inC ·W(0) = C · 1 ∼= C and thusI0 = 0 or I0 = C ·W(0). If I0 =

C ·W(0), thenI contains 1 and thereforeI = 〈W(V)〉max. Hence we may assumeI0 = 0.

Now we look at the projection mapP : 〈W(V)〉 → C ·W(0), P(∑ϕ aϕW(ϕ)) = a0W(0).

We check thatP extends to a bounded operator on〈W(V)〉max. Let δ0 ∈ L2(V,C) denote

the function given byδ0(0) = 1 andδ0(ϕ) = 0 otherwise. Fora= ∑ϕ aϕW(ϕ) andψ ∈Vwe have

(a·δ0)(ψ) = (∑ϕ

aϕW(ϕ)δ0)(ψ) = (∑ϕ

aϕ eiω(ϕ,ψ)/2δ0(ϕ +ψ) = a−ψ eiω(−ψ,ψ)/2 = a−ψ

and therefore

(δ0,a ·δ0)L2 = ∑ψ∈V

δ0(ψ)(a ·δ0)(ψ) = (a ·δ0)(0) = a0.

Moreover,‖δ0‖ = 1. Thus

‖P(a)‖max = ‖a0W(0)‖max = |a0| = |(δ0,a ·δ0)L2| ≤ ‖a‖Op ≤ ‖a‖max

which shows thatP extends to a bounded operator on〈W(V)〉max.


Now let a∈ I ⊂ 〈W(V)〉max. Fix ε > 0. We write

a = a0W(0)+n

∑j=1

a j W(ϕ j)+ r

where theϕ j 6= 0 are pairwise different and the remainder termr satisfies‖r‖max< ε. Foranyψ ∈V we have

I ∋W(ψ)aW(−ψ) = a0W(0)+n

∑j=1

a j e−iω(ψ,ϕ j )W(ϕ j)+ r(ψ)

where‖r(ψ)‖max = ‖W(ψ)rW(−ψ)‖max ≤ ‖r‖max < ε. If we chooseψ1 andψ2 suchthate−iω(ψ1,ϕn) = −e−iω(ψ2,ϕn), then adding the two elements

a0W(0)+n

∑j=1

a j e−iω(ψ1,ϕ j )W(ϕ j)+ r(ψ1) ∈ I

a0W(0)+n

∑j=1

a j e−iω(ψ2,ϕ j )W(ϕ j)+ r(ψ2) ∈ I

yields

a0W(0)+n−1

∑j=1

a′j W(ϕ j )+ r1 ∈ I

where‖r1‖max = ‖ r(ψ1)+r(ψ2)2 ‖max < ε+ε

2 = ε. Repeating this procedure we eventuallyget

a0W(0)+ rn ∈ I

where‖rn‖max < ε. Sinceε is arbitrary andI is closed we conclude

P(a) = a0W(0) ∈ I0,

thusa0 = 0.For a = ∑ϕ aϕ W(ϕ) ∈ I and arbitraryψ ∈ V we haveW(ψ)a ∈ I as well, henceP(W(ψ)a) = 0. This meansa−ψ = 0 for all ψ , thusa = 0. This showsI = 0.

Definition 4.2.8. A Weyl system(A,W) of a symplectic vector space(V,ω) is called aCCR-representationof (V,ω) if A is generated as aC∗-algebra by the elementsW(ϕ),ϕ ∈V. In this case we callA aCCR-algebraof (V,ω)

Of course, for any Weyl system(A,W) we can simply replaceA by theC∗-subalgebragenerated by the elementsW(ϕ), ϕ ∈V, and we obtain a CCR-representation.Existence of Weyl systems and hence CCR-representations has been established in Ex-ample 4.2.2. Uniqueness also holds in the appropriate sense.

Theorem 4.2.9.Let (V,ω) be a symplectic vector space and let(A1,W1) and(A2,W2) betwo CCR-representations of(V,ω).Then there exists a unique∗-isomorphismπ : A1 → A2 such that the diagram


A2

VW1 //

W2

??A1

π

OO

commutes.

Proof. We have to show that the∗-isomorphismπ : 〈W1(V)〉 → 〈W2(V)〉 as constructedin Remark 4.2.4 extends to an isometry(A1,‖ · ‖1) → (A2,‖ · ‖2). Since the pull-back ofthe norm‖ · ‖2 on A2 to 〈W1(V)〉 via π is aC∗-norm we have‖π(a)‖2 ≤ ‖a‖max for alla ∈ 〈W1(V)〉. Henceπ extends to a∗-morphism〈W1(V)〉max → A2. By Lemma 4.2.7the kernel ofπ is trivial, henceπ is injective. Proposition 4.1.22 implies thatπ :(〈W1(V)〉max

,‖ · ‖max) → (A2,‖ · ‖2) is an isometry.In the special case(A1,‖ · ‖1) = (A2,‖ · ‖2) whereπ is the identity this yields‖ · ‖max =‖·‖1. Thus for arbitraryA2 the mapπ extends to an isometry(A1,‖·‖1)→ (A2,‖·‖2).

From now on we will call CCR(V,ω) as defined in Example 4.2.2the CCR-algebra of(V,ω).

Corollary 4.2.10. CCR-algebras of symplectic vector spaces are simple, i. e.,all unitpreserving∗-morphisms to other C∗-algebras are injective.

Proof. Direct consequence of Corollary 4.1.20 and Lemma 4.2.7.

Corollary 4.2.11. Let (V1,ω1) and(V2,ω2) be two symplectic vector spaces and let S:V1 →V2 be asymplectic linear map, i. e.,ω2(Sϕ ,Sψ) = ω1(ϕ ,ψ) for all ϕ ,ψ ∈V1.Then there exists a unique injective∗-morphismCCR(S) : CCR(V1,ω1) → CCR(V2,ω2)such that the diagram

V1S //

W1

V2

W2

CCR(V1,ω1)CCR(S) // CCR(V2,ω2)

commutes.

Proof. One immediately sees that(CCR(V2,ω2),W2 S) is a Weyl system of(V1,ω1).Theorem 4.2.9 yields the result.

From uniqueness of the map CCR(S) we conclude that CCR(idV) = idCCR(V,ω) andCCR(S2S1) = CCR(S2)CCR(S1). In other words, we have constructed a functor

CCR :SymplVec → C ∗−Alg

whereSymplVec denotes the category whose objects are symplectic vector spaces andwhose morphisms are symplectic linear maps, i. e., linear maps A : (V1,ω1) → (V2,ω2)with A∗ω2 = ω1. By C ∗−Alg we denote the category whose objects areC∗-algebras andwhose morphisms areinjectiveunit preserving∗-morphisms. Observe that symplecticlinear maps are automatically injective.In the caseV1 = V2 the induced∗-automorphisms CCR(S) are calledBogoliubov trans-formationin the physics literature.

4.3. Quantization functors 123

4.3 Quantization functors

In the preceding section we introduced the functor CCR from the categorySymplVec ofsymplectic vector spaces (with symplectic linear maps as morphisms) to the categoryC ∗−Alg of C∗-algebras (with unit preserving∗-monomorphisms as morphisms). We wantto link these considerations to Lorentzian manifolds and the analysis of normally hyper-bolic operators. In order to achieve this we introduce two further categories which are ofgeometric-analytical nature.So far we have treated real and complex vector bundlesE over the manifoldM on anequal footing. From now on we will restrict ourselves to realbundles. This is not veryrestrictive since we can always forget a complex structure and regard complex bundles asreal bundles. We will have to give the real bundleE another piece of additional structure.We will assume thatE comes with anondegenerate inner product〈·, ·〉. This meansthat each fiberEx is equipped with a nondegenerate symmetric bilinear form dependingsmoothly on the base pointx. In other words,〈·, ·〉 is like a Riemannian metric except thatit need not be positive definite.We say that a differential operatorP acting on sections inE is formally selfadjointwithrespect to to the inner product〈·, ·〉 of E, if

∫

M〈Pϕ ,ψ〉 dV =

∫

M〈ϕ ,Pψ〉 dV

for all ϕ ,ψ ∈ D(M,E).

Example 4.3.1.Let M be ann-dimensional timeoriented connected Lorentzian manifoldwith metricg. Let E be a real vector bundle overM with inner product〈·, ·〉. Let ∇ be aconnection onE. We assume that∇ is metricwith respect to〈·, ·〉, i. e.,

∂X〈ϕ ,ψ〉 = 〈∇Xϕ ,ψ〉+ 〈ϕ ,∇Xψ〉

for all sectionsϕ ,ψ ∈ C∞(M,E). The inner product induces an isomorphism2Z : E →E∗, ϕ 7→ 〈ϕ , ·〉. Since∇ is metric we get

(∇X(Zϕ)) ·ψ = ∂X((Zϕ)ψ)− (Zϕ)(∇Xψ) = ∂X〈ϕ ,ψ〉− 〈ϕ ,∇Xψ〉= 〈∇Xϕ ,ψ〉 = (Z(∇Xϕ)) ·ψ ,

hence∇X(Zϕ) =Z(∇Xϕ).

Equation (3.2) says for allϕ ,ψ ∈C∞(M,E)

(Zϕ) · (∇ψ) =n

∑i=1

εi∇ei (Zϕ) ·∇ei ψ −div(V1),

thus

〈ϕ ,∇ψ〉 =n

∑i=1

εi〈∇ei ϕ ,∇ei ψ〉−div(V1),

2In caseE = TM with the Lorentzian metric as the inner product we write instead ofZ and♯ instead ofZ−1. Hence ifE = T∗M we haveZ= ♯ andZ−1 = .


whereV1 is a smooth vector field with supp(V1) ⊂ supp(ϕ)∩supp(ψ) ande1, . . . ,en is alocal Lorentz orthonormal tangent frame,εi = g(ei ,ei). Interchanging the roles ofϕ andψ we get

〈∇ϕ ,ψ〉 =n

∑i=1

εi〈∇ei ϕ ,∇ei ψ〉−div(V2),

and therefore〈ϕ ,∇ψ〉− 〈∇ϕ ,ψ〉 = div(V2−V1).

If supp(ϕ)∩supp(ψ) is compact we obtain∫

M〈ϕ ,∇ψ〉 dV−

∫

M〈∇ϕ ,ψ〉 dV =

∫

Mdiv(V2−V1) dV = 0,

thus∇ is formally selfadjoint. If, moreover,B ∈ C∞(M,End(E)) is selfadjoint with

respect to〈·, ·〉, then the normally hyperbolic operatorP= ∇ +B is formally selfadjoint.

As a special case letE be the trivial real line bundle. In other words, sections inE aresimply real-valued functions. The inner product is given bythe pointwise product. Inthis case the inner product is positive definite. Then the above discussion shows that thed’Alembert operator is formally selfadjoint and, more generally,P= +B is formallyselfadjoint where the zero-order termB is a smooth real-valued function onM. Thisincludes the (normalized) Yamabe operatorPg discussed in Section 3.5, theKlein-GordonoperatorP = + m2 and thecovariant Klein-Gordon operatorP = + m2 + κ scal,wheremandκ are real constants.

Example 4.3.2. Let M be ann-dimensional timeoriented connected Lorentzian man-ifold. Let ΛkT∗M be the bundle ofk-forms on M. The Lorentzian metric inducesa nondegenerate inner product〈·, ·〉 on ΛkT∗M, which is indefinite if 1≤ k ≤ n− 1.Let d : C∞(M,ΛkT∗M) → C∞(M,Λk+1T∗M) denote exterior differentiation. Letδ :C∞(M,ΛkT∗M)→C∞(M,Λk−1T∗M) be thecodifferential. This is the unique differentialoperator formally adjoint tod, i. e.,

∫

M〈dϕ ,ψ〉 dV =

∫

M〈ϕ ,δψ〉 dV

for all ϕ ∈ D(M,ΛkT∗M) andψ ∈ D(M,Λk+1T∗M). Then the operator

P = dδ + δd : C∞(M,ΛkT∗M) →C∞(M,ΛkT∗M)

is obviously formally selfadjoint. The Levi-Civita connection induces a metric connection∇ on the bundleΛkT∗M. TheWeitzenbock formularelatesP and

∇, P = ∇ +B where

B is a certain expression in the curvature tensor ofM, see [Besse1987, Eq. (12.92’)]. Inparticular,P and

∇ have the same principal symbol, henceP is normally hyperbolic.The operatorP appears in physics in different contexts. LetM be of dimensionn = 4.Let us first look at theProca equationdescribing a spin-1 particle of massm > 0. Thequantum mechanical wave function of such a particle is givenby A∈C∞(M,Λ1T∗M) andsatisfies

δdA+m2A = 0. (4.4)


Applying δ to this equation and usingδ 2 = 0 andm 6= 0 we concludeδA = 0. Thus theProca equation (4.4) is equivalent to

(P+m2)A = 0

together with the constraintδA = 0.Now we discusselectrodynamics. Let M be a 4-dimensional globally hyperbolicLorentzian manifold and assume that the second deRham cohomology vanishes,H2(M;R) = 0. By Poincare duality, the second cohomology with compact supportsalso vanishes,H2

c (M;R) = 0. See [Warner1983] for details on deRham cohomology.The electric and the magnetic fields can be combined to thefield strengthF ∈D(M,Λ2T∗M). TheMaxwell equationsare

dF = 0 and δF = J

whereJ∈D(M,Λ1T∗M) is thecurrent density. FromH2c (M;R) = 0 we have thatdF = 0

implies the existence of avector potentialA ∈ D(M,Λ1T∗M) with dA= F . Now δA∈D(M,R) and by Theorem 3.2.11 we can findf ∈ C∞

sc(M,R) with f = δA. We putA′ := A−d f . We see thatdA′ = dA= F andδA′ = δA− δd f = δA− f = 0. A vectorpotential satisfying the last equationδA′ = 0 is said to be inLorentz gauge. From theMaxwell equations we concludeδdA′ = δF = J. Hence

PA′ = J.

Example 4.3.3. Next we look at spinors and the Dirac operator. These concepts arestudied in much detail on general semi-Riemannian manifolds in [Baum1981], see also[Bar-Gauduchon-Moroianu2005, Sec. 2] for an overview.Let M be ann-dimensional oriented and timeoriented connected Lorentzian manifold.Furthermore, we assume thatM carries a spin structure. Then we can form thespinorbundleΣM overM. This is a complex vector bundle of rank 2n/2 or 2(n−1)/2 dependingon whethern is even or odd. This bundle carries an indefinite Hermitian producth.The Dirac operatorD : C∞(M,ΣM) → C∞(M,ΣM) is a formally selfadjoint differentialoperator of first order. The Levi-Civita connection inducesa metric connection∇ onΣM.The Weitzenbock formula says

D2 = ∇ +

14

scal.

ThusD2 is normally hyperbolic. SinceD is formally selfadjoint so isD2.If we forget the complex structure onΣM, i. e., we regardΣM as a real bundle, and ifwe let 〈·, ·〉 be given by the real part ofh, then the operatorP = D2 is of the type underconsideration in this section.

Now we define the category of globally hyperbolic manifolds equipped with normallyhyperbolic operators:

Definition 4.3.4. The categoryGlobHyp is defined as follows: The objects ofGlobHyp aretriples(M,E,P) whereM is a globally hyperbolic Lorentzian manifold,E → M is a real


vector bundle with nondegenerate inner product, andP is a formally selfadjoint normallyhyperbolic operator acting on sections inE.

A morphism(M1,E1,P1) → (M2,E2,P2) in GlobHyp is a pair( f ,F) where f : M1 → M2

is a timeorientation preserving isometric embedding so that f (M1) ⊂ M2 is a causallycompatible open subset. Moreover,F : E1 → E2 is a vector bundle homomorphism overf which is fiberwise an isometry. In particular,

E1F //

E2

M1

f // M2

commutes. Furthermore,F has to preserve the normally hyperbolic operators, i. e.,

D(M1,E1)P1 //

ext

D(M1,E1)

ext

D(M2,E2)P2 // D(M2,E2)

commutes where ext(ϕ) denotes the extension ofF ϕ f−1 ∈D( f (M1),E2) to all of M2

by 0.

Notice that a morphism between two objects(M1,E1,P1) and(M2,E2,P2) can exist onlyif M1 andM2 have equal dimension and ifE1 andE2 have the same rank. The conditionthat f (M1) ⊂ M2 is causally compatible does not follow from the fact thatM1 andM2 areglobally hyperbolic. For example, considerM2 = R×S1 with metric−dt2+canS1 and letM1 ⊂ M2 be a small strip about a spacelike helix. ThenM1 andM2 are both intrinsicallyglobally hyperbolic butM1 is not a causally compatible subset ofM2.


M2

M1 p

JM2+ (p)

Fig. 32: The spacelike helixM1 is not causally compatible inM2

Lemma 4.3.5. Let M be a globally hyperbolic Lorentzian manifold. Let E→ M be a realvector bundle with nondegenerate inner product〈· , ·〉. Consider a formally selfadjointnormally hyperbolic operator P with advanced and retarded Green’s operators G± as inCorollary 3.4.3. Then

∫

M〈G±ϕ ,ψ〉 dV =

∫

M〈ϕ ,G∓ψ〉 dV (4.5)

holds for allϕ ,ψ ∈ D(M,E).

Proof. The proof is basically the same as that of Lemma 3.4.4. Namely, for Green’soperators we havePG± = idD(M,E) and therefore we get

∫

M〈G±ϕ ,ψ〉 dV =

∫

M〈G±ϕ ,PG∓ψ〉 dV =

∫

M〈PG±ϕ ,G∓ψ〉 dV =

∫

M〈ϕ ,G∓ψ〉 dV

where we have made use of the formal selfadjointness ofP in the second equality. Noticethat supp(G±ϕ)∩ supp(G∓ψ) ⊂ JM

± (supp(ϕ))∩ JM∓ (supp(ψ)) is compact in a globally

hyperbolic manifold so that the partial integration is justified.Alternatively, we can also argue as follows. The inner product yields the vector bundleisomorphismZ : E → E∗, e 7→ 〈e, ·〉, as noted in Example 4.3.1. Formal selfadjoint-ness now means that the operatorP corresponds to the dual operatorP∗ underZ. NowLemma 4.3.5 is a direct consequence of Lemma 3.4.4.


If we want to deal with Lorentzian manifolds which are not globally hyperbolic we havethe problem that Green’s operators need not exist and if theydo they are in general nolonger unique. In this case we have to provide the Green’s operators as additional data.This motivates the definition of a category of Lorentzian manifolds with normally hyper-bolic operators and global fundamental solutions.

Definition 4.3.6. Let LorFund denote the category whose objects are 5-tuples(M,E,P,G+,G−) where M is a timeoriented connected Lorentzian manifold,E is areal vector bundle overM with nondegenerate inner product,P is a formally selfad-joint normally hyperbolic operator acting on sections inE, andG± are advanced andretarded Green’s operators forP respectively. Moreover, weassumethat (4.5) holds forall ϕ ,ψ ∈ D(M,E).Let X = (M1,E1,P1,G1,+,G1,−) and Y = (M2,E2,P2,G2,+,G2,−) be two objects inLorFund . If M1 is not globally hyperbolic, then we let the set of morphisms from Xto Y be empty unlessX = Y in which case we put Mor(X,Y) := (idM1, idE1).

If M1 is globally hyperbolic, then Mor(X,Y) consists of all pairs( f ,F) with the sameproperties as those of the morphisms inGlobHyp . It then follows from Proposition 3.5.1and Corollary 3.4.3 that we automatically have compatibility of the Green’s operators,i. e., the diagram

D(M1,E1)ext //

G1,±

D(M2,E2)

G2,±

C∞(M1,E1) C∞(M2,E2)resoo

commutes. Here res stands for “restriction”. More precisely, res(ϕ) = F−1ϕ f . Com-position of morphisms is given by the usual composition of maps.The definition of the categoryLorFund is such that nontrivial morphisms exist only if thesource manifoldM1 is globally hyperbolic while there is no such restriction onthe targetmanifoldM2. It will become clear in the proof of Lemma 4.3.8 why we restrict to globallyhyperbolicM1.By Corollary 3.4.3 there exist unique advanced and retardedGreen’s operatorsG± for anormally hyperbolic operator on a globally hyperbolic manifold. Hence we can define

SOLVE(M,E,P) := (M,E,P,G+,G−)

on objects ofGlobHyp andSOLVE( f ,F) := ( f ,F)

on morphisms.

Lemma 4.3.7. This defines a functorSOLVE :GlobHyp → LorFund .

Proof. We only need to check that SOLVE( f ,F) = ( f ,F) is actually a morphism inLorFund , i. e., that( f ,F) is compatible with the Green’s operators. By uniqueness ofGreen’s operators on globally hyperbolic manifolds it suffices to show that resG2,+ extis an advanced Green’s operator onM1 and similarly forG2,−. Since f (M1) ⊂ M2 is acausally compatible connected open subset this follows from Proposition 3.5.1.


Next we would like to use the Green’s operators in order to construct a symplectic vectorspace to which we can then apply the functor CCR. Let(M,E,P,G+,G−) be an object ofLorFund . UsingG = G+−G− : D(M,E) →C∞(M,E) we define

ω : D(M,E)×D(M,E) → R

by

ω(ϕ ,ψ) :=∫

M〈Gϕ ,ψ〉 dV. (4.6)

Obviously, ω is R-bilinear and by (4.5) it is skew-symmetric. Butω does not makeD(M,E) a symplectic vector space becauseω is degenerate. The null space is given by

ker(G) = ϕ ∈ D(M,E) |Gϕ = 0 = ϕ ∈ D(M,E) |G+ϕ = G−ϕ.

This null space is infinite dimensional because it certainlycontainsP(D(M,E)) by The-orem 3.4.7. In the globally hyperbolic case this is precisely the null space,

ker(G) = P(D(M,E)),

again by Theorem 3.4.7. On the quotient spaceV(M,E,G) := D(M,E)/ker(G) the de-generate bilinear formω induces a symplectic form which we denote byω .

Lemma 4.3.8. Let X = (M1,E1,P1,G1,+,G1,−) and Y= (M2,E2,P2,G2,+,G2,−) be twoobjects inLorFund . Let( f ,F) ∈ Mor(X,Y) be a morphism.Thenext :D(M1,E1)→D(M2,E2) maps the null spaceker(G1) to the null spaceker(G2)and hence induces a symplectic linear map

V(M1,E1,G1) →V(M2,E2,G2).

Proof. If the morphism is the identity, then there is nothing to show. Thus we may assumethatM1 is globally hyperbolic. Letϕ ∈ ker(G1). Thenϕ = P1ψ for someψ ∈ D(M1,E1)becauseM1 is globally hyperbolic. FromG2(extϕ) = G2(ext(P1ψ)) = G2(P2(extψ)) = 0we see that ext(ker(G1)) ⊂ ker(G2). Hence ext induces a linear mapV(M1,E1,G1) →V(M2,E2,G2). From

ω2(extϕ ,extψ) =

∫

M2

〈G2extϕ ,extψ〉 dV

=∫

M1

〈resG2extϕ ,ψ〉 dV

=

∫

M1

〈G1ϕ ,ψ〉 dV

= ω1(ϕ ,ψ)

we see that this linear map is symplectic.

We have constructed a functor from the categoryLorFund to the categorySymplVec bymapping each object(M,E,P,G+,G−) to V(M,E,G+ −G−) and each morphism( f ,F)to the symplectic linear map induced by ext. We denote this functor by SYMPL.We summarize the categories and functors we have defined so far in the following scheme:


GlobHyp

LorFund

SymplVec

C ∗−Alg

CCRSOLVE SYMPL

Fig. 33: Quantization functors

4.4 Quasi-localC∗-algebras

The composition of the functors CCR and SYMPL constructed inthe previous sectionsallows us to assign aC∗-algebra to each timeoriented connected Lorentzian manifoldequipped with a formally selfadjoint normally hyperbolic operator and Green’s operators.Further composing with the functor SOLVE we no longer need toprovide Green’s opera-tors if we are willing to restrict ourselves to globally hyperbolic manifolds. The elementsof this algebra are physically interpreted as the observables related to the field whose waveequation is given by the normally hyperbolic operator.“Reasonable” open subsets ofM are timeoriented Lorentzian manifolds in their ownright and come equipped with the restriction of the normallyhyperbolic operator overM. Hence each such open subsetO yields an algebra whose elements are considered asthe observables which can be measured in the spacetime region O. This gives rise to theconcept of nets of algebras or quasi-local algebras. A systematic exposition of quasi-localalgebras can be found in [Baumgartel-Wollenberg1992].Before we define quasi-local algebras we characterize the systems that parametrize the“local algebras”. For this we need the notion of directed sets with orthogonality relation.

Definition 4.4.1. A set I is called adirected set with orthogonality relationif it carries apartial order≤ and a symmetric relation⊥ between its elements such that

(1) for all α,β ∈ I there exists aγ ∈ I with α ≤ γ andβ ≤ γ,

(2) for everyα ∈ I there is aβ ∈ I with α⊥β ,

(3) if α ≤ β andβ⊥γ, thenα⊥γ,

(4) if α⊥β andα⊥γ, then there exists aδ ∈ I such thatβ ≤ δ , γ ≤ δ andα⊥δ .

4.4. Quasi-localC∗-algebras 131

In order to handle non-globally hyperbolic manifolds we need to relax this definitionslightly:

Definition 4.4.2. A setI is called adirected set with weak orthogonality relationif it car-ries a partial order≤ and a symmetric relation⊥ between its elements such that conditions(1), (2), and (3) in Definition 4.4.1 are fulfilled.

Obviously, directed sets with orthogonality relation are automatically directed sets withweak orthogonality relation. We use such sets in the following as index sets for nets ofC∗-algebras.

Definition 4.4.3. A (bosonic) quasi-local C∗-algebra is a pair(A,Aαα∈I

)of a C∗-

algebraA and a familyAαα∈I of C∗-subalgebras, whereI is a directed set with orthog-onality relation such that the following holds:

(1) Aα ⊂ Aβ wheneverα ≤ β ,

(2) A =⋃α

Aα where the bar denotes the closure with respect to the norm ofA.

(3) the algebrasAα have a common unit 1,

(4) if α⊥β the commutator ofAα andAβ is trivial:[Aα ,Aβ

]= 0.

Remark 4.4.4. This definition is a special case of the one in [Bratteli-Robinson2002-I,Def. 2.6.3] where there is in addition an involutive automorphismσ of A. In our caseσ = id which physically corresponds to a bosonic theory. This iswhy one might call ourversion of quasi-localC∗-algebrasbosonic.

Definition 4.4.5. A morphismbetween two quasi-localC∗-algebras(A,Aαα∈I

)and(

B,Bββ∈J

)is a pair(ϕ ,Φ) whereΦ : A → B is a unit-preservingC∗-morphism and

ϕ : I → J is a map such that:

(1) ϕ is monotonic, i. e., ifα1 ≤ α2 in I thenϕ(α1) ≤ ϕ(α2) in J,

(2) ϕ preserves orthogonality, i. e., ifα1⊥α2 in I , thenϕ(α1)⊥ϕ(α2) in J,

(3) Φ(Aα) ⊂ Bϕ(α) for all α ∈ I .

The composition of morphisms of quasi-localC∗-algebras is just the composition of maps,and we obtain the categoryQuasiLocA lg of quasi-localC∗-algebras.

Definition 4.4.6. A weak quasi-local C∗-algebrais a pair(A,Aαα∈I

)of aC∗-algebra

A and a familyAαα∈I of C∗-subalgebras, whereI is a directed set with weak orthog-onality relation such that the same conditions as in Definition 4.4.3 hold. Morphismsbetween weak quasi-localC∗-algebras are defined in exactly the same way as morphismsbetween quasi-localC∗-algebras.

This yields another category, the category of weak quasi-local C∗-algebrasQuasiLocA lg

weak. We note thatQuasiLocA lg is a full subcategory ofQuasiLocA lg

weak.


Next we want to associate to any object(M,E,P,G+,G−) in LorFund a weak quasi-localC∗-algebra. For this we set

I := O⊂M | O is open, relatively compact, causally compatible, globally hyperbolic∪ /0,M.

On I we take the inclusion⊂ as the partial order≤ and define the orthogonality relationby

O⊥ O′ :⇔ JM(O)∩O′ = /0.

This means that two elements ofI are orthogonal if and only if they arecausally indepen-dentsubsets ofM in the sense that there are no causal curves connecting a point in O witha point inO′. Of course, this relation is symmetric.

Lemma 4.4.7. The set I defined above is a directed set with weak orthogonality relation.

Proof. Condition (1) in Definition 4.4.1 holds withγ = M and (2) withβ = /0. Property(3) is also clear becauseO⊂ O′ impliesJM(O) ⊂ JM(O′).

Lemma 4.4.8. Let M be globally hyperbolic. Then the set I is a directed set with (non-weak) orthogonality relation.

Proof. In addition to Lemma 4.4.7 we have to show Property (4) of Definition 4.4.1. LetO1,O2,O3 ∈ I with JM(O1)∩O2 = /0 andJM(O1)∩O3 = /0. We want to findO4 ∈ I withO2∪O3 ⊂ O4 andJM(O1)∩O4 = /0.Without loss of generality letO1,O2,O3 be non-empty. Now none ofO1, O2, andO3 canequalM. In particular,O1, O2, andO3 are relatively compact. SetΩ := M \ JM(O1). ByLemma A.5.11 the subsetΩ of M is causally compatible and globally hyperbolic. ThehypothesisJM(O1)∩O2 = /0 = JM(O1)∩O3 implies O2 ∪O3 ⊂ Ω. Applying Proposi-tion A.5.13 withK := O2∪O3 in the globally hyperbolic manifoldΩ, we obtain a rela-tively compact causally compatible globally hyperbolic open subsetO4 ⊂ Ω containingO2 andO3. SinceΩ is itself causally compatible inM, the subsetO4 is causally compati-ble inM as well. By definition ofΩ we haveO4 ⊂Ω = M\JM(O1), i. e.,JM(O1)∩O4 = /0.This shows Property (4) and concludes the proof of Lemma 4.4.8.

Remark 4.4.9. If M is globally hyperbolic, the proof of Proposition A.5.13 shows thatthe index setI would also be directed if we removedM from it in its definition. Namely,for all elementsO1,O2 ∈ I different from /0 andM, the elementO from Proposition A.5.13applied toK := O2∪O3 belongs toI .

Now we are in the situation to associate a weak quasi-localC∗-algebra to any object(M,E,P,G+,G−) in LorFund .We consider the index setI as defined above. For any non-emptyO ∈ I we take therestrictionE|O and the corresponding restriction of the operatorP to sections ofE|O. Dueto the causal compatibility ofO⊂ M the restrictions of the Green’s operatorsG+, G− tosections overO yield the Green’s operatorsGO

+, GO− for P on O, see Proposition 3.5.1.

Therefore we get an object(O,E|O,P,GO+,GO

−) for eachO∈ I , O 6= /0.For /0 6= O1 ⊂ O2 the inclusion induces a morphismιO2,O1 in the categoryLorFund . Thismorphism is given by the embeddingsO1 → O2 andE|O1 → E|O2. Let αO2,O1 denote


the morphism CCRSYMPL(ιO2,O1) in C ∗−Alg . Recall thatαO2,O1 is an injective unit-preserving∗-morphism.We set for /06= O∈ I

(VO,ωO) := SYMPL(O,E|O,P,GO+,GO

−)

and forO∈ I , O 6= /0, O 6= M,

AO := αM,O (CCR(VO,ωO)) .

Obviously, for anyO ∈ I , O 6= /0, O 6= M the algebraAO is a C∗-subalgebra ofCCR(VM,ωM). For O = M we defineAM as theC∗-subalgebra of CCR(VM,ωM) gen-erated by allAO,

AM := C∗( ⋃

O∈IO6= /0,O6=M

AO

).

Finally, for O = /0 we setA /0 = C · 1. We have thus defined a familyAOO∈I of C∗-subalgebras ofAM.

Lemma 4.4.10.Let (M,E,P,G+,G−) be an object inLorFund . Then(AM,AOO∈I

)is

a weak quasi-local C∗-algebra.

Proof. We know from Lemma 4.4.7 thatI is a directed set with weak orthogonality rela-tion.It is clear thatAM =

⋃O∈I

AO becauseM belongs toI . By construction it is also clear that

all algebrasAO have the common unitW(0), 0∈ VM. Hence Conditions (2) and (3) inDefinition 4.4.3 are obvious.By functoriality we have the following commutative diagram

CCR(VO,ωO)αM,O //

αO′,O

CCR(VM,ωM)

CCR(VO′ ,ωO′)

αM,O′

66mmmmmmmmmmmm

SinceαO′ ,O is injective we haveAO ⊂ AO′ . This proves Condition (1) in Definition 4.4.3.Let nowO,O′ ∈ I be causally independent. Letϕ ∈D(O,E) andψ ∈D(O′,E). It followsfrom supp(Gϕ) ⊂ JM(O) that supp(Gϕ)∩supp(ψ) = /0, hence

∫

M〈Gϕ ,ψ〉 dV = 0.

For the symplectic formω onD(M,E)/ker(G) this meansω(ϕ ,ψ) = 0, where we denotethe equivalence class inD(M,E)/ker(G) of the extension toM by zero ofϕ again byϕand similarly forψ . This yields by Property (iii) of a Weyl-system

W(ϕ) ·W(ψ) = W(ϕ + ψ) = W(ψ) ·W(ϕ),

i. e., the generators ofAO commute with those ofAO′ . Therefore[AO,AO′ ] = 0. Thisproves (4) in Definition 4.4.3.


Next we associate a morphism inQuasiLocA lg weak to any morphism( f ,F) in LorFundbeween two objects(M1,E1,P1,G1+,G1−) and(M2,E2,P2,G2+,G2−). Recall that in thecase of distinct objects such a morphism only exists ifM1 is globally hyperbolic. LetI1and I2 denote the index sets for the two objects as above and let

(AM1,AOO∈I1

)and(

BM2,BOO∈I2

)denote the corresponding weak quasi-localC∗-algebras. Thenf maps

anyO1 ∈ I1, O1 6= M1, to f (O1) which is an element ofI2 by definition ofLorFund . Weget a mapϕ : I1 → I2 by M1 7→ M2 andO1 7→ f (O1) if O1 6= M1. Sincef is an embeddingsuch thatf (M1) ⊂ M2 is causally compatible, the mapϕ is monotonic and preservescausal independence.

Let Φ : CCR(VM1,ωM1) → CCR(VM2,ωM2) be the morphismΦ = CCRSYMPL( f ,F).From the commutative diagram of inclusions and embeddings

O1

//

f |O1

O2

f |O2

f (O1)

// M2

we see

Φ(AO1) = CCR(SYMPL( f ,F))CCR(SYMPL(ιM,O1))(CCR(VO1,ωO1))

= CCR(SYMPL(ιM2, f (O1)))CCR(SYMPL( f |O1 ,F |E|O1))(CCR(VO1,ωO1))

⊂ αM2, f (O1)(CCR(Vf (O1),ω f (O1)))

= B f (O1).

This also impliesΦ(AM1) ⊂ BM2. Therefore the pair(ϕ ,Φ|AM1) is a morphism in

QuasiLocA lg weak . We summarize

Theorem 4.4.11.The assignments(M,E,P,G+,G−) 7→(AM,AOO∈I

)and ( f ,F) 7→

(ϕ ,Φ|AM1) yield a functorLorFund → QuasiLocA lg

weak.

Proof. If f = idM andF = idE, thenϕ = idI andΦ = idAM . Similarly, the compositionof two morphisms inLorFund is mapped to the composition of the corresponding twomorphisms inQuasiLocA lg

weak.

Corollary 4.4.12. The composition ofSOLVEand the functor from Theorem 4.4.11 yieldsa functorGlobHyp → QuasiLocA lg . One gets the following commutative diagram of func-tors:


GlobHyp

QuasiLocA lg

LorFund

QuasiLocA lgweak

inclusion

SOLVE

Fig. 34: Functors which yield nets of algebras

Proof. Let (M,E,P) be an object inGlobHyp . Then we know from Lemma 4.4.8 thatthe index setI associated to SOLVE(M,E,P) is a directed set with (non-weak) orthog-onality relation, and the corresponding weak quasi-localC∗-algebra is in fact a quasi-local C∗-algebra. This concludes the proof sinceQuasiLocA lg is a full subcategory ofQuasiLocA lg

weak.

Lemma 4.4.13. Let (M,E,P) be an object inGlobHyp , and denote by(AM,

AO

O∈I

)

the corresponding quasi-local C∗-algebra. Then

AM = CCRSYMPLSOLVE(M,E,P

).

Proof. Denote the right hand side byA. By definition ofAM we haveAM ⊂ A.In order to prove the other inclusion write(M,E,P,G+,G−) := SOLVE(M,E,P). ThenSYMPL(M,E,P,G+,G−) is given byVM = D(M,E)/ker(G) with symplectic formωM

induced byG. Now A is generated by

E =W([ϕ ])

∣∣ϕ ∈ D(M,E)

whereW is the Weyl system from Example 4.2.2 and[ϕ ] denotes the equivalence classof ϕ in VM. For givenϕ ∈ D(M,E) there exists a relatively compact globally hyperboliccausally compatible open subsetO⊂ M containing the compact set supp(ϕ) by Proposi-tion A.5.13. For this subsetO we haveW([ϕ ]) ∈ AO. Hence we getE ⊂ ⋃

O∈IAO ⊂ AM

which impliesA ⊂ AM.

Example 4.4.14.Let M be globally hyperbolic. All the operators listed in Examples 4.3.1to 4.3.3 give rise to quasi-localC∗-algebras. These operators include the d’Alembertoperator, the Klein-Gordon operator, the Yamabe operator,the wave operators for theelectro-magnetic potential and the Proca field as well as thesquare of the Dirac operator.


Example 4.4.15.Let M be the anti-deSitter spacetime. ThenM is not globally hyperbolicbut as we have seen in Section 3.5 we can get Green’s operatorsfor the (normalized)Yamabe operatorPg by embeddingM conformally into the Einstein cylinder. This yieldsan object(M,M×R,Pg,G+,G−) in LorFund . Hence there is a corresponding weak quasi-localC∗-algebra overM.

4.5 Haag-Kastler axioms

We now check that the functor that assigns to each object inLorFund a quasi-localC∗-algebra as constructed in the previous section satisfies theHaag-Kastler axioms of a quan-tum field theory. These axioms have been proposed in [Haag-Kastler1964, p. 849] forMinkowski space. Dimock [Dimock1980, Sec. 1] adapted them to the case of globally hy-perbolic manifolds. He also constructed the quasi-localC∗-algebras for the Klein-Gordonoperator.

Theorem 4.5.1. The functorLorFund → QuasiLocA lg weak from Theorem 4.4.11 satisfiestheHaag-Kastler axioms, that is, for every object(M,E,P,G+,G−) in LorFund the cor-responding weak quasi-local C∗-algebra

(AM,AOO∈I

)satisfies:

(1) If O1 ⊂ O2, thenAO1 ⊂ AO2 for all O1,O2 ∈ I.

(2) AM = ∪O∈I

O6= /0,O6=M

AO.

(3) If M is globally hyperbolic, thenAM is simple.

(4) TheAO’s have a common unit1.

(5) For all O1,O2 ∈ I with J(O1)∩O2 = /0 the subalgebrasAO1 andAO2 of AM com-mute:[AO1,AO2] = 0.

(6) (Time-slice axiom) Let O1 ⊂ O2 be nonempty elements of I admitting a commonCauchy hypersurface. ThenAO1 = AO2.

(7) Let O1,O2 ∈ I and let the Cauchy development D(O2) be relatively compact in M.If O1 ⊂ D(O2), thenAO1 ⊂ AO2.

Remark 4.5.2. It can happen that the Cauchy developmentD(O) of a causally compat-ible globally hyperbolic subsetO in a globally hyperbolic manifoldM is not relativelycompact even ifO itself is relatively compact. See the following picture foran examplewhereM andO are “lens-like” globally hyperbolic subsets of Minkowski space:

M

O

D(O)

Fig. 35: Cauchy developmentD(O) is not relatively compact inM

4.5. Haag-Kastler axioms 137

This is why weassumein (7) thatD(O2) is relatively compact.

Remark 4.5.3. Instead of (3) one often finds the requirement thatAM should beprimitivefor globally hyperbolicM. This means that there exists a faithful irreducible representa-tion of AM on a Hilbert space. We know by Lemma 4.4.13 and Corollary 4.2.10 thatAM

is simple, i. e., that (3) holds. Simplicity implies primitivity because eachC∗-algebra hasirreducible representations [Bratteli-Robinson2002-I,Sec. 2.3.4].

Proof of Theorem 4.5.1.Only axioms (6) and (7) require a proof. First note that axiom(7)follows from axioms (1) and (6):Let O1,O2 ∈ I , let the Cauchy developmentD(O2) be relatively compact inM, and letO1 ⊂ D(O2). By Theorem 1.3.10 there is a smooth spacelike Cauchy hypersurfaceΣ ⊂O2. It follows from the definitions thatD(O2) = D(Σ). SinceO2 is causally compatiblein M the hypersurfaceΣ is acausal inM. By Lemma A.5.9D(Σ) is a causally compatibleand globally hyperbolic open subset ofM. SinceD(O2) = D(Σ) is relatively compact byassumption we haveD(O2) ∈ I .Axiom (6) impliesAO2 = AD(O2). By axiom (1)AO1 ⊂ AD(O2) = AO2.It remains to show the time-slice axiom. We prepare the proofby first deriving two lem-mas. The first lemma is of technical nature while the second one is essentially equivalentto the time-slice axiom.

Lemma 4.5.4.Let O be a causally compatible globally hyperbolic open subset of a glob-ally hyperbolic manifold M. Assume that there exists a Cauchy hypersurfaceΣ of O whichis also a Cauchy hypersurface of M. Let h be a Cauchy time-function on O as in Corol-lary 1.3.12 (applied to O). Let K⊂ M be compact. Assume that there exists a t∈ R withK ⊂ IM

+ (h−1(t)).Then there is a smooth functionρ : M → [0,1] such that

• ρ = 1 on a neighborhood of K,

• supp(ρ)∩JM− (K) ⊂ M is compact, and

• x∈ M |0 < ρ(x) < 1∩JM− (K) is compact and contained in O.

Remark 4.5.5. Similarly, if instead ofK ⊂ IM+ (h−1(t)) we haveK ⊂ IM

− (h−1(t)) for somet, then we can find a smooth functionρ : M → [0,1] such that

• ρ = 1 on a neighborhood ofK,

• supp(ρ)∩JM+ (K) ⊂ M is compact, and

• x∈ M |0 < ρ(x) < 1∩JM+ (K) is compact and contained inO.

Proof of Lemma 4.5.4.By assumption there exist real numberst− < t+ in the range ofhsuch thatK ⊂ IM

+ (St+), hence alsoK ⊂ IM+ (St−), whereSt := h−1(t). SinceSt is a Cauchy

hypersurface ofO and sinceO andM admit a common Cauchy hypersurface, it followsfrom Lemma A.5.10 thatSt− andSt+ are also Cauchy hypersurfaces ofM. SinceJM

+ (St+)andJM

− (St−) are disjoint closed subsets ofM there exists a smooth functionρ : M → [0,1]such thatρ|

JM+ (St+)

= 1 andρ|JM− (St− )

= 0.


St+

0 < ρ < 1

ρ = 1

St−

ρ = 0

K M

O

K

Fig. 36: Construction ofρ

We check thatρ has the three properties stated in Lemma 4.5.4. The first one followsfrom K ⊂ IM

+ (St+).Sinceρ|

IM− (St− )= 0, we have supp(ρ)⊂ JM

+ (St−). It follows from Lemma A.5.3 applied to

the past-compact subsetJM+ (St−) of M thatJM

+ (St−)∩JM− (K) is relatively compact, hence

compact by Lemma A.5.1. Therefore the second property holds.The closed set0 < ρ < 1∩ JM

− (K) is contained in the compact set supp(ρ)∩ JM− (K),

hence compact itself.The subset0 < ρ < 1 of M lies in JM

+ (St−) ∩ JM− (St+). We claim thatJM

+ (St−) ∩JM− (St+) ⊂ O which will then imply 0 < ρ < 1∩ JM

− (K) ⊂ O and hence conclude theproof.Assume that there existsp∈ JM

+ (St−)∩JM− (St+) but p 6∈O. Choose a future directed causal

curvec : [s−,s+] → M from St− to St+ throughp. Extend this curve to an inextendiblefuture directed causal curvec : R → M. Let I ′ be the connected component ofc−1(O)containings−. ThenI ′ ⊂ R is an open interval andc|I ′ is an inextendible causal curvein O. Sincep 6∈ O the curve leavesO before it reachesSt+ , hences+ 6∈ I ′. But St+ is aCauchy hypersurface inO and so there must be ans∈ I ′ with c(s) ∈ St+ . Therefore thecurvec meetsSt+ at least twice (namely ins and ins+) in contradiction toSt+ being aCauchy hypersurface inM.

Lemma 4.5.6. Let (M,E,P) be an object inGlobHyp and let O be a causally compatibleglobally hyperbolic open subset of M. Assume that there exists a Cauchy hypersurfaceΣof O which is also a Cauchy hypersurface of M. Letϕ ∈ D(M,E).

4.6. Fock space 139

Then there existψ ,χ ∈ D(M,E) such thatsupp(ψ) ⊂ O and

ϕ = ψ +Pχ .

Proof of Lemma 4.5.6.Let h be a time-function onO as in Corollary 1.3.12 (applied toO). Fix t− < t+ ∈ R in the range ofh. By Lemma A.5.10 the subsetsSt− := h−1(t−) andSt+ := h−1(t+) are also Cauchy hypersurfaces ofM. Hence every inextendible timelikecurve inM meetsSt− andSt+ . Sincet− < t+ the setIM

+ (St−), IM− (St+) is a finite open

cover ofM.Let f+, f− be a partition of unity subordinated to this cover. In particular, supp( f±) ⊂IM± (St∓). SetK± := supp( f±ϕ) = supp(ϕ)∩ supp( f±). ThenK± is a compact subset

of M satisfyingK± ⊂ IM± (St∓). Applying Lemma 4.5.4 we obtain two smooth functions

ρ+,ρ− : M → [0,1] satisfying:

• ρ± = 1 in a neighborhood ofK±,

• supp(ρ±)∩JM∓ (K±) ⊂ M is compact, and

• 0 < ρ± < 1∩JM∓ (K±) is compact and contained inO.

Setχ± := ρ±G∓( f±ϕ), χ := χ+ + χ− andψ := ϕ −Pχ . By definition,χ±, χ , andψare smooth sections inE overM. Since supp(G∓( f±ϕ)) ⊂ JM

∓ (supp( f±ϕ)) ⊂ JM∓ (K±),

the support ofχ± is contained in supp(ρ±)∩ JM∓ (K±), which is compact by the second

property ofρ±. Thereforeχ ∈ D(M,E).It remains to show that supp(ψ) is compact and contained inO. By the first property ofρ± one hasχ± = G∓( f±ϕ) in a neighborhood ofK±. Moreover,f±ϕ = 0 onρ± = 0.HencePχ± = f±ϕ on ρ± = 0∪ ρ± = 1. Thereforef±ϕ −Pχ± vanishes outside0 < ρ± < 1, i. e., supp( f±ϕ −Pχ±) ⊂ 0 < ρ± < 1. By the definitions ofχ± and f±one also has supp( f±ϕ −Pχ±) ⊂ K± ∪ JM

∓ (K±) = JM∓ (K±), hence supp( f±ϕ −Pχ±) ⊂

0 < ρ± < 1∩ JM∓ (K±), which is compact and contained inO by the third property of

ρ±. Therefore the support ofψ = f+ϕ −Pχ+ + f−ϕ −Pχ− is compact and contained inO. This shows Lemma 4.5.6.

End of proof of Theorem 4.5.1.The time-slice axiom in Theorem 4.5.1 follows directlyfrom Lemma 4.5.6. Namely, letO1 ⊂ O2 be nonempty causally compatible globallyhyperbolic open subsets ofM admitting a common Cauchy hypersurface. Let[ϕ ]∈VO2 :=D(O2,E)/ker(GO2). Lemma 4.5.6 applied toM := O2 andO := O1 yieldsχ ∈ D(O2,E)andψ ∈D(O1,E) such thatϕ = extψ +Pχ . SincePχ ∈ ker(GO2) we have[ϕ ] = [extψ ],that is,[ϕ ] is the image of the symplectic linear mapVO1 →VO2 induced by the inclusionO1 → O2, compare Lemma 4.3.8. We see that this symplectic map is surjective, hencean isomorphism. Symplectic isomorphisms induce isomorphisms ofC∗-algebras, hencethe inclusionAO1 ⊂ AO2 is actually an equality. This proves the time-slice axiom andconcludes the proof of Theorem 4.5.1.

4.6 Fock space

In quantum mechanics a particle is described by its wave function which mathematicallyis a solutionu to an equationPu= 0. We consider normally hyperbolic operatorsP in


this text. The passage from single particle systems to multiparticle systems is known assecond quantizationin the physics literature. Mathematically it requires the constructionof the quantum field which we will do in the subsequent section. In this section we willdescribe some functional analytical underpinnings, namely the construction of the bosonicFock space.We start by describing the symmetric tensor product of Hilbert spaces. LetH denote acomplex vector space. We will use the convention that the Hermitian scalar product(·, ·)on H is anti-linear in the first argument. LetHn be the vector space freely generated byH ×·· ·×H (n copies), i. e., the space of all finite formal linear combinations of elementsof H×·· ·×H. LetVn be the vector subspace ofHn generated by all elements of the form(v1, . . . ,cvk, . . . ,vn)−c· (v1, . . . ,vk, . . . ,vn), (v1, . . . ,vk +v′k, . . . ,vn)− (v1, . . . ,vk, . . . ,vn)−(v1, . . . ,v′k, . . . ,vn) and (v1, . . . ,vn)− (vσ(1), . . . ,vσ(n)) wherev j ,v′j ∈ H, c ∈ C andσ apermutation.

Definition 4.6.1. The vector space⊙n

algH := Hn/Vn is called thealgebraic nth symmetric

tensor productof H. By convention, we put⊙0

algH := C.

For the equivalence class of(v1, . . . ,vn)∈Hn in⊙n

algH we writev1⊙·· ·⊙vn. The mapγ :H×·· ·×H →⊙n

algH given byγ(v1, . . . ,vn) = v1⊙·· ·⊙vn is multilinear and symmetric.The algebraic symmetric tensor product has the following universal property.

Lemma 4.6.2. For each complex vector space W and each symmetric multilinear mapα : H ×·· ·×H →W there exists one and only one linear mapα :

⊙nalgH →W such that

the diagram

H ×·· ·×H

γ

α

$$IIIIIIIIII

⊙nalgH α // W

commutes.

Proof. Uniqueness ofα is clear because

α(v1⊙·· ·⊙vn) = α(v1, . . . ,vn) (4.7)

and the elementsv1⊙·· ·⊙vn generate⊙n

algH.To show existence one definesα by Equation 4.7 and checks easily that this is well-defined.

The algebraic symmetric tensor product⊙n

algH inherits a scalar product fromH charac-terized by

(v1⊙·· ·⊙vn,w1⊙·· ·⊙wn) = ∑σ

(v1,wσ(1)) · · · (vn,wσ(n))

where the sum is taken over all permutationsσ on1, . . . ,n.

Definition 4.6.3. The completion of⊙n

algH with respect to this scalar product is called

thenth symmetric tensor productof the Hilbert spaceH and is denoted⊙nH. In particu-

lar,⊙0H = C.

4.6. Fock space 141

Remark 4.6.4. If ej j∈J is an orthonormal system ofH whereJ is some orderedindex set, thenej1 ⊙ ·· ·⊙ejn j1≤···≤ jn forms an orthogonal system of

⊙nH. For eachordered multiindexJ = ( j1, . . . , jn) there is a corresponding partition ofn, n= k1+ · · ·+kl ,given by

j1 = · · · = jk1 < jk1+1 = · · · = jk1+k2 < · · · < jk1+···+kl−1+1 = · · · = jn.

We compute

‖ej1 ⊙·· ·⊙ejn‖2 = ∑σ

(ej1,ejσ(1)) · · · (ejn,ejσ(n)

)

= #σ |( jσ(1), . . . , jσ(n)) = ( j1, . . . , jn)= k1! · · ·kl !.

In particular,1≤ ‖ej1 ⊙·· ·⊙ejn‖ ≤

√n!.

The algebraic direct sumFalg(H) :=⊕∞

alg,n=0⊙nH carries a natural scalar product,

namely

((w0,w1,w2, . . .),(u0,u1,u2, . . .)) =∞

∑n=0

(wn,un)

wherewn,un ∈⊙nH.

Definition 4.6.5. We callFalg(H) thealgebraic symmetric Fock spaceof H. The com-pletion ofFalg(H) with respect to this scalar product is denotedF (H) and is called thebosonicor symmetricFock spaceof H. The vectorΩ := 1 ∈ C =

⊙0H ⊂ Falg(H) ⊂F (H) is called thevacuum vector.

The elements of the Hilbert spaceF (H) are therefore sequences(w0,w1,w2, . . .) withwn ∈

⊙nH such that∞

∑n=0

‖wn‖2 < ∞.

Fix v ∈ H. The mapα : H × ·· · ×H →⊙n+1H, α(v1, . . . ,vn) = v⊙ v1 ⊙ ·· · ⊙ vn, issymmetric multilinear and induces a linear mapα :

⊙nalgH →⊙n+1H, v1⊙ ·· · ⊙ vn 7→

v⊙ v1 ⊙ ·· · ⊙ vn, by Lemma 4.6.2. We compute the operator norm ofα . Without lossof generality we can assume‖v‖ = 1. We choose the orthonormal systemej j∈J of Hsuch thatv belongs to it. Ifv is perpendicular to allej1, . . . ,ejn, then

‖α(ej1 ⊙·· ·⊙ejn)‖ = ‖v⊙ej1 ⊙·· ·⊙ejn‖ = ‖ej1 ⊙·· ·⊙ejn‖.

If v is one of theejµ , sayv = ej1, then

‖α(ej1 ⊙·· ·⊙ejn)‖2 = (k1 +1)! k2! · · ·kl !

= (k1 +1)‖ej1 ⊙·· ·⊙ejn‖2.

Thus in any case

‖α(ej1 ⊙·· ·⊙ejn)‖ ≤√

n+1‖ej1 ⊙·· ·⊙ejn‖


and equality holds forej1 = · · · = ejn = v. Dropping the assumption‖v‖ = 1 this shows

‖α‖ =√

n+1‖v‖.

Henceα extends to a bounded linear map

a∗(v) :⊙n

H →⊙n+1

H, a∗(v)(v1⊙·· ·⊙vn) = v⊙v1⊙·· ·⊙vn

with‖a∗(v)‖ =

√n+1‖v‖. (4.8)

For the vacuum vector this meansa∗(v)Ω = v. The mapa∗(v) is naturally defined as alinear mapFalg(H) → Falg(H). By (4.8)a∗(v) is unbounded onFalg(H) unlessv = 0and therefore does not extend continuously toF (H). Writing v = v0 we see

(a∗(v)(v1⊙·· ·⊙vn),w0⊙w1⊙·· ·⊙wn)

= (v0⊙v1⊙·· ·⊙vn,w0⊙w1⊙·· ·⊙wn)

= ∑σ

(v0,wσ(0))(v1,wσ(1)) · · · (vn,wσ(n))

=n

∑k=0

∑σ with

σ(0)=k

(v,wk)(v1,wσ(1)) · · · (vn,wσ(n))

=

(v1⊙·· ·⊙vn,

n

∑k=0

(v,wk)w0⊙·· ·⊙ wk⊙·· ·⊙wn

)

wherewk indicates that the factorwk is left out. Hence if we definea(v) :⊙n+1

alg H →⊙nalgH by

a(v)(w0⊙·· ·⊙wn) =n

∑k=0

(v,wk)w0⊙·· ·⊙ wk⊙·· ·⊙wn

anda(v)Ω = 0, then we have

(a∗(v)ω ,η) = (ω ,a(v)η) (4.9)

for all ω ∈⊙nalgH andη ∈⊙n+1

alg H. The operator norm ofa(v) is easily determined:

‖a(v)‖ = supη∈⊙n+1

alg H

‖η‖=1

‖a(v)η‖ = supη∈⊙n+1

alg H

ω∈⊙nalgH

‖η‖=‖ω‖=1

(ω ,a(v)η)

= supη∈⊙n+1

alg H

ω∈⊙nalgH

‖η‖=‖ω‖=1

(a∗(v)ω ,η) = supω∈⊙n

algH

‖ω‖=1

‖a∗(v)ω‖

= ‖a∗(v)‖ =√

n+1‖v‖.

Thusa(v) extends continuously to a linear operatora(v) :⊙n+1H →⊙nH with

‖a(v)‖ =√

n+1‖v‖. (4.10)

4.6. Fock space 143

We consider botha∗(v) anda(v) as unbounded linear operators in Fock spaceF (H) withFalg(H) as invariant domain of definition. In the physics literature, a(v) is known asannihilation operatoranda∗(v) ascreation operatorfor v∈ H.

Lemma 4.6.6. Let H be a complex Hilbert space and let v,w∈ H. Then the canonicalcommutator relations (CCR) hold, i. e.,

[a(v),a(w)] = [a∗(v),a∗(w)] = 0,

[a(v),a∗(w)] = (v,w)id.

Proof. From

a∗(v)a∗(w)v1⊙·· ·⊙vn = v⊙w⊙v1⊙·· ·⊙vn

= w⊙v⊙v1⊙·· ·⊙vn

= a∗(w)a∗(v)v1⊙·· ·⊙vn

we see directly that[a∗(v),a∗(w)] = 0. By (4.9) we have for allω ,η ∈ Falg(H)

(ω , [a(v),a(w)]η) = ([a∗(w),a∗(v)]ω ,η) = 0.

SinceFalg(H) is dense inF (H) this implies[a(v),a(w)]η = 0. Finally, subtracting

a(v)a∗(w)v1⊙·· ·⊙vn = a(v)w⊙v1⊙·· ·⊙vn

= (v,w)v1⊙·· ·⊙vn

+n

∑k=1

(v,vk)w⊙v1⊙·· ·⊙ vk⊙·· ·⊙vn

and

a∗(w)a(v)v1⊙·· ·⊙vn = w⊙n

∑k=1

(v,vk)v1⊙·· ·⊙ vk⊙·· ·⊙vn

=n

∑k=1

(v,vk)w⊙v1⊙·· ·⊙ vk⊙·· ·⊙vn

yields[a(v),a∗(w)](v1⊙·· ·⊙vn) = (v,w)v1⊙·· ·⊙vn.

Definition 4.6.7. Let H be a complex Hilbert space and letv ∈ H. We define theSegalfieldas the unbounded operator

θ (v) :=1√2(a(v)+a∗(v))

in F (H) with Falg(H) as domain of definition.

Notice thata∗(v) dependsC-linearly onv while v 7→ a(v) is anti-linear. Henceθ (v) isonly R-linear inv.


Lemma 4.6.8. Let H be a complex Hilbert space and let v∈ H. Then the Segal operatorθ (v) is essentially selfadjoint.

Proof. Sinceθ (v) is symmetric by (4.9) and densely defined it is closable. The domainof definitionFalg(H) is invariant forθ (v) and hence all powersθ (v)m are defined on it.By Nelson’s theorem is suffices to show that all vectors inFalg(H) are analytic, see The-orem A.2.18. Since all vectors in the domain of definition arefinite linear combinationsof vectors in

⊙nH for variousn we only need to show thatω ∈⊙nH is analytic. By(4.8), (4.10), and the fact thata∗(v)ω ∈⊙n+1H anda(v)ω ∈⊙n−1H are perpendicularwe have

‖θ (v)ω‖2 =12‖a∗(v)ω +a(v)ω‖2

=12(‖a∗(v)ω‖2 +‖a(v)ω‖2)

≤ 12((n+1)‖v‖2‖ω‖2+n‖v‖2‖ω‖2)

≤ (n+1)‖v‖2‖ω‖2,

hence

‖θ (v)mω‖ ≤√

(n+1)(n+2) · · ·(n+m)‖v‖m‖ω‖. (4.11)

For anyt > 0

∞

∑m=0

tm

m!‖θ (v)mω‖ ≤

∞

∑m=0

tm

m!

√(n+1)(n+2) · · ·(n+m)‖v‖m‖ω‖

=∞

∑m=0

tm√

m!

√n+1

1· n+2

2· · · n+m

m‖v‖m‖ω‖

≤∞

∑m=0

tm√

m!

√n+1

m‖v‖m‖ω‖

< ∞

because the power series∑∞m=0

xm√m!

has infinite radius of convergence. Thusω is ananalytic vector.

Lemma 4.6.9. Let H be a complex Hilbert space and let v,v j ,w ∈ H, j = 1,2, . . . Letη ∈ Falg(H). Then the following holds:

(1) (θ (v)θ (w)−θ (w)θ (v))η = iIm(v,w)η .

(2) If ‖v−v j‖→ 0, then‖θ (v)η −θ (v j)η‖→ 0 as j→ ∞.

(3) The linear span of the vectorsθ (v1) · · ·θ (vn)Ω where vj ∈ H and n∈ N is dense inF (H).

4.6. Fock space 145

Proof. We see (1) by Lemma 4.6.6

θ (v)θ (w)η =12(a∗(v)+a(v))(a∗(w)+a(w))η

=12(a∗(v)a∗(w)+a∗(v)a(w)+a(v)a∗(w)+a(v)a(w))η

=12(a∗(w)a∗(v)+a(w)a∗(v)− (w,v)id+a(w)a∗(w)+ (v,w)id

+a(w)a(v))η= θ (w)θ (v)η + iIm(v,w)η .

For (2) it suffices to prove the statement forη ∈⊙nH. By (4.8) and (4.10) we see

‖θ (v)η −θ (v j)η‖ = 2−1/2‖a∗(v)η +a(v)η −a∗(v j)η −a(v j)η‖≤ 2−1/2(‖a∗(v)η −a∗(v j)η‖+‖a(v)η −a(v j)η‖)= 2−1/2(‖a∗(v−v j)η‖+‖a(v−v j)η‖)≤ 2−1/2(

√n+1+

√n)‖v−v j‖‖η‖

which implies the statement.For (3) one can easily see by induction onN that the span of the vectorsθ (v1) · · ·θ (vn)Ω,n ≤ N, and the span of the vectorsa∗(v1) · · ·a∗(vn)Ω, n ≤ N, both coincide with⊕N

n=0⊙n

algH. This is dense in⊕N

n=0⊙nH and the assertion follows.

Now we can relate this discussion to the canonical commutator relations as studied inSection 4.2. Denote the (selfadjoint) closure ofθ (v) again byθ (v) and denote the domainof this closure by dom(θ (v)). Look at the unitary operatorW(v) := exp(iθ (v)). Recallthat for the analytic vectorsω ∈ Falg(H) we have the series

W(v)ω =∞

∑m=0

im

m!θ (v)mω

converging absolutely.

Proposition 4.6.10.For v,v j ,w∈ H we have

(1) The domaindom(θ (w)) is preserved by W(v), i. e., W(v)(dom(θ (w))) =dom(θ (w)) and

W(v)θ (w)ω = θ (w)W(v)ω −Im(v,w)W(v)ω

for all ω ∈ dom(θ (w)).

(2) The map W: H → L (F (H)) is a Weyl system of the symplectic vector space(H,Im(·, ·)).

(3) If ‖v−v j‖→ 0, then‖(W(v)−W(v j))η‖ → 0 for all η ∈ F (H).


Proof. We first check the formula (1) forω ∈ Falg(H). From Lemma 4.6.9 (1) we getinductively

θ (v)mθ (w)ω = θ (w)θ (v)mω + i ·m·Im(v,w)θ (v)m−1ω .

Sinceθ (w)ω ∈ Falg(H) we have

θ (w)W(v)ω =∞

∑m=0

im

m!θ (w)θ (v)mω

=∞

∑m=0

im

m!

(θ (v)mθ (w)− im Im(v,w)θ (v)m−1)ω

= W(v)θ (w)ω −∞

∑m=1

im+1Im(v,w)

(m−1)!θ (v)m−1ω

= W(v)θ (w)ω +Im(v,w)W(v)ω . (4.12)

In particular,W(v)ω is an analytic vector forθ (w) and we have forω ,η ∈ Falg(H)

‖θ (w)W(v)(ω −η)‖ = ‖(W(v)θ (w)+Im(v,w)W(v))(ω −η)‖≤ ‖W(v)θ (w)(ω −η)‖+ |Im(v,w)|‖W(v)(ω −η)‖≤ (‖θ (w)(ω −η)‖+ |Im(v,w)| · ‖ω −η‖) · ‖v‖. (4.13)

Now let ω ∈ dom(θ (w)). Then there existω j ∈ Falg(H) such that‖ω −ω j‖ → 0 and‖θ (w)(ω)− θ (w)(ω j )‖ → 0 as j → ∞. SinceW(v) is bounded we have‖W(v)(ω)−W(v)(ω j )‖ → 0 and by (4.13)θ (w)W(v)(ω j) j is a Cauchy sequence and thereforeconvergent as well. HenceW(v)(ω) ∈ dom(θ (w)) and the validity of (4.12) extends toall ω ∈ dom(θ (w)).We have shownW(v)(dom(θ (w))) ⊂ dom(θ (w)). ReplacingW(v) byW(−v) = W(v)−1

yieldsW(v)(dom(θ (w))) = dom(θ (w)).For (2) observeW(0) = exp(0) = id and W(−v) = exp(iθ (−v)) = exp(−iθ (v)) =exp(iθ (v))∗ = W(v)∗. We fix ω ∈ Falg(H) and look at the smooth curve

x(t) := W(tv)W(tw)W(−t(v+w))ω

in F (H). We havex(0) = ω and

x(t) =ddt

W(tv)W(tw)W(−t(v+w))ω

= iW(tv)θ (v)W(tw)W(−t(v+w))ω + iW(tv)W(tw)θ (w)W(−t(v+w))ω+iW(tv)W(tw)θ (−v−w)W(−t(v+w))ω

= iW(tv)θ (v)W(tw)W(−t(v+w))ω + iW(tv)W(tw)θ (−v)W(−t(v+w))ω(1)= iW(tv)W(tw)(θ (v)+Im(tw,v))W(−t(v+w))ω

+iW(tv)W(tw)θ (−v)W(−t(v+w))ω= i · t ·Im(w,v) ·x(t).

4.7. The quantum field defined by a Cauchy hypersurface 147

Thusx(t) = eiIm(w,v)t2/2ω andt = 1 yieldsW(v)W(w)W(−(v+w))ω = eiIm(w,v)/2ω . Bycontinuity this equation extends to allω ∈ F (H) and shows thatW is a Weyl system forthe symplectic formIm(·, ·).For (3) letη ∈⊙n H and let‖v−v j‖→ 0 as j → ∞. Then

‖(W(v)−W(v j))η‖ = ‖W(v)(id−W(−v)W(v j))η‖≤ ‖(id−W(−v)W(v j))η‖≤ ‖(1−eiIm(v,vj )/2)η‖+‖(eiIm(v,vj )/2−W(−v)W(v j))η‖(2)= ‖(1−eiIm(v,vj )/2)η‖+‖(eiIm(v,vj )/2−eiIm(v,vj )/2W(v j −v))η‖≤ |1−eiIm(v,vj )/2| · ‖η‖+‖(id−W(v j −v))η‖.

SinceIm(v,v j) = 12Im(v−v j ,v+v j) → 0 it suffices to show‖(id−W(v j −v))η‖ → 0.

This follows from

∞

∑m=1

1m!

‖θ (v j −v)mη‖(4.11)≤

∞

∑m=1

√(n+1)m

m!‖v j −v‖m‖η‖

= ‖v j −v‖∞

∑m=0

√(n+1)m+1

(m+1)!‖v j −v‖m‖η‖

and ∑∞m=0

√(n+1)m+1

(m+1)! ‖v j − v‖m < ∞ uniformly in j. We have seen that‖(W(v) −W(v j))η‖ → 0 for all η ∈⊙nH (n fixed) hence for allη ∈ Falg(H).Finally, letη ∈F (H) be arbitrary. Letε > 0. Chooseη ′ ∈Falg(H) such that‖η−η ′‖<ε. For j ≫ 0 we have‖(W(v)−W(v j))η ′‖ < ε. Hence

‖(W(v)−W(v j))η‖ ≤ ‖(W(v)−W(v j))(η −η ′)‖+‖(W(v)−W(v j))η ′‖≤ 2‖η −η ′‖+ ε< 3ε.

This concludes the proof.

4.7 The quantum field defined by a Cauchy hypersurface

In this final section we construct the quantum field. This yields a formulation of the quan-tized theory on Fock space which is closer to the traditionalpresentations of quantumfield theory than the formulation in terms of quasi-localC∗-algebras given in Sections 4.4and 4.5. It has the disadvantage however of depending on a choice of Cauchy hypersur-face. Even worse from a physical point of view, this quantum field has all the propertiesthat one usually requires except for one, the “microlocal spectrum condition”. We do notdiscuss this condition in the present book, see the remarks at the end of this section andthe references mentioned therein. The construction given here is nevertheless useful be-cause it illustrates how the abstract algebraic formulation of quantum field theory relatesto more traditional ones.


Let (M,E,P) be an object in the categoryGlobHyp , i. e., M is a globally hyperbolicLorentzian manifold,E is a real vector bundle overM with nondegenerate inner prod-uct, andP is a formally selfadjoint normally hyperbolic operator acting on sections inE.We need an additional piece of structure.

Definition 4.7.1. Let k ∈ N. A twist structure of spin k/2 on E is a smooth sectionQ∈C∞(M,Hom(

⊙k TM,End(E))) with the following properties:

(1) Q is symmetric with respect to the inner product onE, i. e.,

〈Q(X1⊙·· ·⊙Xk)e, f 〉 = 〈e,Q(X1⊙·· ·⊙Xk) f 〉

for all Xj ∈ TpM, e, f ∈ Ep, andp∈ M.

(2) If X is future directed timelike, then the bilinear form〈·, ·〉X defined by

〈 f ,g〉X := 〈Q(X⊙·· ·⊙X) f ,g〉

is positive definite.

Note that the bilinear form〈·, ·〉X is symmetric by (1) so that (2) makes sense. From nowon we writeQX := Q(X ⊙ ·· ·⊙X) for brevity. If X is past directed timelike, then〈·, ·〉X

is positive or negative definite depending on the parity ofk. Note furthermore, thatQX

is a field ofisomorphismsof E in case thatX is timelike since otherwise〈·, ·〉X would bedegenerate.

Examples 4.7.2.a) Let E be a real vector bundle overM with Riemannian metric. Inthe case of the d’Alembert, the Klein-Gordon, and the Yamabeoperator we are in thissituation. We takek = 0 andQ :

⊙0TM = R → End(E), t 7→ t · id. By convention,QX = id and hence〈·, ·〉X = 〈·, ·〉 for a timelike vectorX of unit length.b) Let M carry a spin structure and letE = ΣM be the spinor bundle. The Dirac oper-ator and its square act on sections inΣM. As explained in [Baum1981, Sec. 3.3] and[Bar-Gauduchon-Moroianu2005, Sec. 2] there is a natural indefinite Hermitian product(·, ·) on ΣM such that for future directed timelikeX the sesquilinear form(·, ·)X definedby

(ϕ ,ψ)X = (ϕ ,X ·ψ)

is symmetric and positive definite where “·” denotes Clifford multiplication. Hence if weview ΣM as a real bundle and put〈·, ·〉 := Re(·, ·), k := 1, andQ(X)ϕ := X ·ϕ , then wehave a twist structure of spin 1/2 on the spinor bundle.c) On the bundle ofp-formsE = ΛpT∗M there is a natural indefinite inner product〈·, ·〉characterized by

〈α,β 〉 = ∑0≤i1<···<ip≤n

εi1 · · ·εip ·α(ei1, . . . ,eip) ·β (ei1, . . . ,eip)

wheree1, . . . ,en is an orthonormal basis withεi = 〈ei ,ei〉 = ±1. We putk := 2 and

Q(X⊙Y)α := X∧ ιYα +Y∧ ιXα −〈X,Y〉 ·α


whereιX denotes insertion ofX in the first argument,ιXα = α(X, ·, . . . , ·), andX 7→ X

is the natural isomorphismTM → T∗M induced by the Lorentzian metric. It is easy tocheck thatQ is a twist structure of spin 1 onΛpT∗M. Recall that the casep= 1 is relevantfor the wave equation in electrodynamics and for the Proca equation.The physically oriented reader will have noticed that in allthese examplesk/2 indeedcoincides with the spin of the particle under consideration.

Remark 4.7.3. If Q is a twist structure onE, thenQ∗ is a twist structure onE∗ of thesame spin whereQ∗(X1⊙·· ·⊙Xk) = Q(X1⊙·· ·⊙Xk)

∗ is given by the adjoint map. OnE∗, we will always use this induced twist structure without further comment.

Let us return to the construction of the quantum field for the object(M,E,P) in GlobHyp .As an additional data we fix a twist structureQ on E. As usual letE∗ denote the dualbundle andP∗ the adjoint operator acting on sections inE∗. Let G∗

± denote the Green’soperators forP∗ andG∗ = G∗

+ −G∗−.

We choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. We denote byL2(Σ,E∗)thereal Hilbert space of square integrable sections inE∗ overΣ with scalar product

(u,v)Σ :=∫

Σ〈u,v〉n dA =

∫

Σ〈Q∗

nu,v〉dA

wheren denotes the future directed (timelike) unit normal toΣ. Here〈·, ·〉 denotes theinner product onE∗ inherited from the one onE. Let HΣ := L2(Σ,E∗)⊗R C be the com-plexification of this real Hilbert space and extend(·, ·)Σ to a Hermitian scalar product onHΣ thus turningHΣ into a complex Hilbert space. We use the convention that(·, ·)Σ isconjugate linear in the first argument.We construct the symmetric Fock spaceF (HΣ) as in the previous section. Letθ be thecorresponding Segal field.Given f ∈ D(M,E∗) the smooth sectionG∗ f is contained inC∞

sc(M,E∗), i. e., there existsa compact subsetK ⊂ M such that supp(G∗ f ) ⊂ JM(K), see Theorem 3.4.7. It thusfollows from Corollary A.5.4 that the intersection supp(G∗ f )∩Σ is compact andG∗ f |Σ ∈D(Σ,E∗) ⊂ L2(Σ,E∗) ⊂ HΣ. Similarly, ∇n(G∗ f ) ∈ D(Σ,E∗) ⊂ L2(Σ,E∗) ⊂ HΣ. We cantherefore define

ΦΣ( f ) := θ (i(G∗ f )|Σ − (Q∗n)−1∇n(G∗ f )).

Definition 4.7.4. The mapΦΣ from D(M,E∗) to the set of selfadjoint operators on FockspaceF (HΣ) is called thequantum field(or thefield operator) for P defined byΣ.

Notice thatΦΣ depends upon the choice of the Cauchy hypersurfaceΣ. One thinks ofΦΣas an operator-valued distribution onM. This can be made more precise.

Proposition 4.7.5. Let (M,E,P) be an object in the categoryGlobHyp with a twist struc-ture Q. Choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. LetΦΣ be the quantumfield for P defined byΣ.Then for everyω ∈ Falg(HΣ) the map

D(M,E∗) → F (HΣ), f 7→ ΦΣ( f )ω ,


is continuous. In particular, the map

D(M,E∗) → C, f 7→ (η ,ΦΣ( f )ω),

is a distributional section in E for anyη ,ω ∈ Falg(HΣ).

Proof. Let f j → f in D(M,E∗). ThenG∗ f j → G∗ f in C∞sc(M,E∗) by Proposition 3.4.8.

Thus G∗ f j |Σ → G∗ f |Σ and (Q∗n)−1∇nG∗ f j → (Q∗

n)−1∇nG∗ f in D(Σ,E∗). HenceG∗ f j |Σ → G∗ f |Σ and(Q∗

n)−1∇nG∗ f j → (Q∗n)−1∇nG∗ f in HΣ. The proposition now fol-

lows from Lemma 4.6.9 (2).

The quantum field satisfies the equationPΦΣ = 0 in the distributional sense. More pre-cisely, we have

Proposition 4.7.6. Let (M,E,P) be an object in the categoryGlobHyp with a twist struc-ture Q. Choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. LetΦΣ be the quantumfield for P defined byΣ.For every f∈ D(M,E∗) one has

ΦΣ(P∗ f ) = 0.

Proof. This is clear fromG∗P∗ f = 0 andθ (0) = 0.

To proceed we need the following reformulation of Lemma 3.2.2.

Lemma 4.7.7. Let (M,E,P) be an object in the categoryGlobHyp , let G± be the Green’soperators for P and let G= G+ −G−. Furthermore, letΣ ⊂ M be a spacelike Cauchyhypersurface with future directed (timelike) unit normal vector fieldn.Then for all f,g∈ D(M,E),

∫

M〈 f ,Gg〉 dV =

∫

Σ(〈∇n(G f),Gg〉− 〈G f,∇n(Gg)〉)dA .

Proof. SinceJM+ (Σ) is past compact andJM

− (Σ) is future compact Lemma 3.2.2 applies.After identification ofE∗ with E via the inner product〈· , ·〉 the assertion follows fromLemma 3.2.2 withu = Gg.

The quantum field satisfies the following commutator relation.

Proposition 4.7.8. Let (M,E,P) be an object in the categoryGlobHyp with a twist struc-ture Q. Choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. LetΦΣ be the quantumfield for P defined byΣ.Then for all f,g∈ D(M,E∗) and all η ∈ Falg(HΣ) one has

[ΦΣ( f ),ΦΣ(g)]η = i ·∫

M〈G∗ f ,g〉 dV ·η .


Proof. Using Lemma 4.6.9 and the fact that(·, ·)Σ is the complexification of a real scalarproduct we compute

[ΦΣ( f ),ΦΣ(g)]η=

[θ(i(G∗ f )|Σ − (Q∗

n)−1∇n(G∗ f )),θ(i(G∗g)|Σ − (Q∗

n)−1∇n(G∗g))]

η

= iIm(i(G∗ f )|Σ − (Q∗

n)−1∇n(G∗ f ), i(G∗g)|Σ − (Q∗n)−1∇n(G∗g)

)Σ η

= −iIm(i(G∗ f )|Σ,(Q∗

n)−1∇n(G∗g))

Σ η − iIm((Q∗

n)−1∇n(G∗ f ), i(G∗g)|Σ)

Σ η

= i((G∗ f )|Σ,(Q∗

n)−1∇n(G∗g))

Σ ·η − i((Q∗

n)−1∇n(G∗ f ),(G∗g)|Σ)

Σ ·η

= i ·∫

Σ〈(G∗ f )|Σ,∇n(G∗g)〉 dV ·η − i ·

∫

Σ〈∇n(G∗ f ),(G∗g)|Σ〉 dV ·η .

Lemma 4.7.7 applied toP∗ concludes the proof.

Corollary 4.7.9. Let(M,E,P) be an object in the categoryGlobHyp with a twist structureQ. Choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. LetΦΣ be the quantum fieldfor P defined byΣ. If the supports of f and g∈ D(M,E∗) are causally independent, then

[ΦΣ( f ),ΦΣ(g)] = 0.

Proof. If the supports of supp( f ) and supp(g) are causally independent, thensupp(G∗ f ) ⊂ JM(supp( f )) and supp(g) are disjoint. Hence

[ΦΣ( f ),ΦΣ(g)] = i ·∫

M〈G∗ f ,g〉 dV = 0.

Proposition 4.7.10.Let(M,E,P) be an object in the categoryGlobHyp with a twist struc-ture Q. Choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. LetΦΣ be the quantumfield for P defined byΣ. LetΩ be the vacuum vector inF (HΣ).Then the linear span of the vectorsΦΣ( f1) · · ·ΦΣ( fn)Ω is dense inF (HΣ) where fj ∈D(M,E∗) and n∈ N.

Proof. By Lemma 4.6.9 (3) the span of vectors of the formθ (v1) · · ·θ (vn)Ω, v j ∈ HΣ,n ∈ N, is dense inF (HΣ). It therefore suffices to approximate vectors of the formθ (v1) · · ·θ (vn)Ω by vectors of the formΦΣ( f1) · · ·ΦΣ( fn)Ω. Any v j ∈ HΣ is of the formv j = wj + iz j with wj ,zj ∈ L2(Σ,E∗). SinceD(Σ,E∗) is dense inL2(Σ,E∗) we may as-sume without loss of generality thatwj ,zj ∈ D(Σ,E∗) by Proposition 4.7.5.By Theorem 3.2.11 there exists a solutionu j ∈C∞

sc(M,E∗) to the Cauchy problemPuj = 0with initial conditionsu j |Σ = zj and ∇nu j = −Q∗

nwj . By Theorem 3.4.7 there existsf j ∈ D(M,E∗) with G∗ f j = u j . ThenΦΣ( f j) = θ (−(Q∗

n)−1∇n(G∗ f j ) + i(G∗ f j)|Σ) =θ (−(Q∗

n)−1∇n(u j)+ iu j |Σ) = θ (wj + iz j) = θ (v j). This concludes the proof.

Remark 4.7.11. In the physics literature one usually also finds that the quantum fieldshould satisfy

ΦΣ( f ) = ΦΣ( f )∗. (4.14)


This simply expresses the fact that we are dealing with a realtheory and that the quantumfield takes its values in self-adjoint operators. Recall that we have assumedE to be arealvector bundle. Of course, one could complexifyE and extendΦΣ complex linearly suchthat (4.14) holds.

We relate the quantum field constructed in this section to theCCR-algebras studied earlier.

Proposition 4.7.12. Let (M,E,P) be an object in the categoryGlobHyp and let Q be atwist structure on E∗. Choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. Let ΦΣbe the quantum field for P defined byΣ.Then the map

WΣ : D(M,E∗) → L (F (HΣ)), WΣ( f ) = exp(i ΦΣ( f )),

yields a Weyl system of the symplectic vector spaceSYMPLSOLVE(M,E∗,P∗).

Proof. Recall that the symplectic vector space SYMPLSOLVE(M,E∗,P∗) is givenby V(M,E∗,G∗) = D(M,E∗)/ker(G∗) with symplectic form induced byω( f ,g) =∫

M〈G∗ f ,g〉 dV. By definitionWΣ( f ) = 1 holds for anyf ∈ ker(G∗), henceWΣ descendsto a mapV(M,E∗,G∗) → L (F (HΣ)).Let f ,g ∈ D(M,E∗). Set u := i(G∗ f )|Σ − (Q∗

n)−1∇n(G∗ f ) and v := i(G∗g)|Σ −(Q∗

n)−1∇n(G∗g)∈HΣ so thatΦΣ( f ) = θ (u) andΦΣ(g) = θ (v). Then by Lemma 4.6.9 (1)and by Proposition 4.7.8 we have

i Im(u,v)Σ · id = [θ (u),θ (v)] = [ΦΣ( f ),ΦΣ(g)] = i∫

M〈G∗ f ,g〉 dV · id,

hence

Im(u,v)Σ =

∫

M〈G∗ f ,g〉 dV = ω( f ,g).

Now the result follows from Proposition 4.6.10 (2).

Corollary 4.7.13. Let (M,E,P) be an object in the categoryGlobHyp and let Q be a twiststructure on E∗. Choose a spacelike smooth Cauchy hypersurfaceΣ ⊂ M. Let WΣ be theWeyl system defined byΦΣ.Then the CCR-algebra generated by the WΣ( f ), f ∈ D(M,E∗), is isomorphic toCCR(SYMPL(SOLVE(M,E∗,P∗))).

Proof. This is a direct consequence of Proposition 4.7.12 and of Theorem 4.2.9.

The construction of the quantum field on a globally hyperbolic Lorentzian manifold goesback to [Isham1978], [Hajicek1978], [Dimock1980], and others in the case of scalarfields, i. e., ifE is the trivial line bundle. See also the references in [Fulling1989] and[Wald1994]. In [Dimock1980] the formulaWΣ( f ) = exp(i ΦΣ( f )) in Proposition 4.7.12was used todefinethe CCR-algebra. It should be noted that this way one does notget atrue quantization functorGlobHyp → C ∗−Alg because one determines theC∗-algebraup toisomorphism only. This is caused by the fact that there is no canonical choice of Cauchyhypersurface. It seems that the approach based on algebras of observables as developed


in Sections 4.3 to 4.5 is more natural in the context of curvedspacetimes than the moretraditional approach via the Fock space.Wave equations for sections in nontrivial vector bundles also appear frequently. The ap-proach presented in this book works for linear wave equations in general but often extraproblems have to be taken care of. In [Dimock1992] the electromagnetic field is studied.Here one has to take the gauge freedom into account. For the Proca equation as studiede. g. in [Furlani1999] the extra constraintδA = 0 must be considered, compare Exam-ple 4.3.2. If one wants to study the Dirac equation itself rather than its square as we did inExample 4.3.3, then one has to use the canonical anticommutator relations (CAR) insteadof the CCR, see e. g. [Dimock1982].In the physics papers mentioned above the authors fix a wave equation, e. g. the Klein-Gordon equation, and then they set up a functorGlobHyp

naked→ C ∗−Alg . HereGlobHyp

nakedis the category whose objects are globally hyperbolic Lorentzian manifolds without anyfurther structure and the morphisms are the timeorientation preserving isometric embed-dings f : M1 → M2 such thatf (M1) is a causally compatible open subset ofM2. Therelation to our more universal functor CCRSYMPLSOLVE :GlobHyp → C ∗−Alg is asfollows:There is the forgetful functor FORGET :GlobHyp → GlobHyp naked given byFORGET(M,E,P) = M and FORGET( f ,F) = f . A geometric normally hyperbolic op-erator is a functor GOp :GlobHyp

naked→ GlobHyp such that FORGETGOp= id.

For example, the Klein-Gordon equation for fixed massmyields such a functor. One putsGOp(M) := (M,E,P) whereE is the trivial real line bundle overM with the canonicalinner product andP is the Klein-Gordon operatorP= +m2. On the level of morphisms,one sets GOp( f ) := ( f ,F) whereF is the embeddingM1 ×R → M2 ×R induced byf : M1 →M2. Similarly, the Yamabe operator, the wave equations for theelectromagneticfield and for the Proca field yield geometric normally hyperbolic operators.The square of the Dirac operator does not yield a geometric normally hyperbolic operatorbecause the construction of the spinor bundle depends on theadditional choice of a spinstructure. One can of course fix this by incorporating the spin structure into yet anothercategory, the category of globally hyperbolic Lorentzian manifolds equipped with a spinstructure, see [Verch2001, Sec. 3].In any case, given a geometric normally hyperbolic operatorGOp, thenCCRSYMPLSOLVEGOp :GlobHyp

naked→ C ∗−Alg is a locally covariant quantum

field theoryin the sense of [Brunetti-Fredenhagen-Verch2003, Def. 2.1].For introductions to quantum field theory on curved spacetimes from the physical pointof view the reader is referred to the books [Birrell-Davies1984], [Fulling1989], and[Wald1994].The passage from the abstract quantization procedure yielding quasi-localC∗-algebras tothe more familiar concept based on Fock space and quantum fields requires certain choices(Cauchy hypersurface) and additional structures (twist structure) and is therefore notcanonical. Furthermore, there are many more Hilbert space representations than the Fockspace representations constructed here and the question arises which ones are physicallyrelevant. A criterion in terms of micro-local analysis was found in [Radzikowski1996].As a matter of fact, the Fock space representations constructed here turn out not to satisfythis criterion and are therefore nowadays regarded as unphysical. A good geometric un-


derstanding of the physical Hilbert space representationson a general globally hyperbolicspacetime is still missing.Radzikowski’s work was developed further in [Brunetti-Fredenhagen-Kohler1996] andapplied in [Brunetti-Fredenhagen1997] to interacting fields. The theory of interactingquantum fields, in particular their renormalizability, currently forms an area of very activeresearch.

Appendix A

Background material

In Sections A.1 to A.4 the necessary terminology and basic facts from such diverse fieldsof mathematics as category theory, functional analysis, differential geometry, and differ-ential operators are presented. These sections are included for the convenience of thereader and are not meant to be a substitute for a thorough introduction to these topics.Section A.5 is of a different nature. Here we collect advanced material on Lorentziangeometry which is needed in the main text. In this section we give full proofs. Partly dueto the technical nature of many of these results they have notbeen included in the maintext in order not to distract the reader.

A.1 Categories

We start with basic definitions and examples from category theory (compare [Lang2002,Ch. 1,§ 11]). A nice introduction to further concepts related to categories can be foundin [MacLane1998].

Definition A.1.1. A categoryA consists of the following data:

• a class Obj(A ) whose members are calledobjects

• for any two objectsA,B∈ Obj(A ) there is a (possibly empty) set Mor(A,B) whoseelements are calledmorphisms,

• for any three objectsA,B,C ∈ Obj(A ) there is a map (called thecompositionofmorphisms)

Mor(B,C)×Mor(A,B) → Mor(A,C) , ( f ,g) 7→ f g,

such that the following axioms are fulfilled:

(1) If two pairs of objects(A,B) and(A′,B′) are not equal, then the sets Mor(A,B) andMor(A′,B′) are disjoint.

156 Chapter A. Background material

(2) For everyA∈ Obj(A ) there exists an element idA ∈ Mor(A,A) (called theidentitymorphismof A) such that for allB ∈ Obj(A ), for all f ∈ Mor(B,A) and allg ∈Mor(A,B) one has

idA f = f and g idA = g.

(3) The law of composition isassociative, i. e., for anyA,B,C,D ∈ Obj(A ) and for anyf ∈ Mor(A,B), g∈ Mor(B,C), h∈ Mor(C,D) we have

(hg) f = h (g f ).

Examples A.1.2. a) In the category of setsSet the class of objects Obj(Set ) consists ofall sets, and for any two setsA,B∈ Obj(Set ) the set Mor(A,B) consists of all maps fromA to B. Composition is the usual composition of maps.b) The objects of the categoryTop are the topological spaces, and the morphisms are thecontinuous maps.c) In the category of groupsGroups one considers the class Obj(Groups ) of all groups, andthe morphisms are the group homomorphisms.d) InAbelG r , the category of abelian groups, Obj(AbelG r ) is the class of all abelian groups,and again the morphisms are the group homomorphisms.

Definition A.1.3. Let A andB be two categories. ThenA is called afull subcategoryofB provided

(1) Obj(A ) ⊂ Obj(B ),

(2) for anyA,B∈Obj(A ) the set of morphisms ofA to B are the same in both categoriesA andB ,

(3) for all A,B,C∈ Obj(A ), any f ∈ Mor(A,B) and anyg∈ Mor(B,C) the compositesg f coincide inA andB ,

(4) for A∈ Obj(A ) the identity morphism idA is the same in bothA andB .

Examples A.1.4.a)Top is not a full subcategory ofSet because there are non-continuousmaps between topological spaces.b) AbelG r is a full subcategory ofGroups .

Definition A.1.5. LetA andB be categories. A(covariant) functorT fromA toB consistsof a mapT : Obj(A ) → Obj(B ) and mapsT : Mor(A,B) → Mor(TA,TB) for everyA,B∈Obj(A ) such that

(1) the composition is preserved, i. e., for allA,B,C∈ Obj(A ), for any f ∈ Mor(A,B)and for anyg∈ Mor(B,C) one has

T(g f ) = T(g)T( f ),

(2) T maps identities to identities, i. e., for anyA∈ Obj(A ) we get

T(idA) = idTA.

A.2. Functional analysis 157

In symbols one writesT : A → B .

Examples A.1.6.a) For every categoryA one has theidentity functorId : A → A whichis defined by Id(A) = A for all A ∈ Obj(A ) and Id( f ) = f for all f ∈ Mor(A,B) withA,B∈ Obj(A ).b) There is a functorF : Top → Set which maps each topological space to the underlyingset andF(g) = g for all A,B∈ Obj(Top ) and allg∈ Mor(A,B). This functorF is calledtheforgetful functorbecause it forgets the topological structure.c) Let A be a category. We fix an objectC ∈ Obj(A ). We defineT : A → Set by T(A) =Mor(C,A) for all A∈ Obj(A ) and by

Mor(A,B) → Mor(

Mor(C,A),Mor(C,B)),

f 7→(

g 7→ f g),

for all A,B∈ Obj(A ). It is easy to check thatT is a functor.

A.2 Functional analysis

In this section we give some background in functional analysis. More comprehensive ex-positions can be found e. g. in [Reed-Simon1980], [Reed-Simon1975], and [Rudin1973].

Definition A.2.1. A Banach spaceis a real or complex vector spaceX equipped with anorm‖ · ‖ such that every Cauchy sequence inX has a limit.

Examples A.2.2. a) ConsiderX = C0([0,1]), the space of continuous functions on theunit interval[0,1]. We pick thesupremum norm: For f ∈C0([0,1]) one puts

‖ f‖C0([0,1]) := supt∈[0,1]

∣∣ f (t)∣∣.

With this normX is Banach space. In this example the unit interval can be replaced byany compact topological space.b) More generally, letk ∈ N and letX = Ck([0,1]), the space ofk times continuouslydifferentiable functions on the unit interval[0,1]. TheCk-norm is defined by

‖ f‖Ck([0,1]) := maxℓ=0,...,k

‖ f (ℓ)‖C0([0,1])

where f (ℓ) denotes theℓth derivative of f ∈ X. ThenX = Ck([0,1]) together with theCk-norm is a Banach space.

Now let H be a complex vector space, and let(· , ·) be a (positive definite) Hermitianscalar product. The scalar product induces a norm onH,

‖x‖ :=√

(x,x) for all x∈ H.

Definition A.2.3. A complex vector spaceH endowed with Hermitian scalar product(· , ·)is called aHilbert spaceif H together with the norm induced by(· , ·) forms a Banachspace.


Example A.2.4. Consider the space ofsquare integrable functionson [0,1]:

L 2([0,1]) :=

f : [0,1]→ C

∣∣∣ f measurable and∫ 1

0| f (t)|2 dt < ∞

.

On L 2([0,1]) one gets a natural sesquilinear form(·, ·) by ( f ,g) :=∫ 1

0 f (t) ·g(t) dt forall f ,g∈ L 2([0,1]). ThenN :=

f ∈ L 2([0,1]) |( f , f ) = 0

is a linear subspace, and

one denotes the quotient vector space by

L2([0,1]) := L 2([0,1])/N .

The sesquilinear form(· , ·) induces a Hermitian scalar product onL2([0,1]). The Riesz-Fisher theorem [Reed-Simon1980, Example 2, p. 29] states that L2([0,1]) equipped withthis Hermitian scalar product is a Hilbert space.

Definition A.2.5. A semi-normon aK-vector spaceX, K = R or C, is a mapρ : X →[0,∞) such that

(1) ρ(x+y)≤ ρ(x)+ ρ(y) for anyx,y∈ X,

(2) ρ(α x) = |α|ρ(x) for anyx∈ X andα ∈ K.

A family of semi-normsρii∈I is said toseparate pointsif

(3) ρi(x) = 0 for all i ∈ I impliesx = 0.

Given a countable family of seminormsρkk∈N separating points one defines a metricdonX by setting forx,y∈ X:

d(x,y) :=∞

∑k=0

12k ·max

(1,ρk(x,y)

). (A.1)

Definition A.2.6. A Frechet spaceis aK-vector spaceX equipped with a countable fam-ily of semi-normsρkk∈N separating points such that the metricd given by (A.1) iscomplete. Thenatural topologyof a Frechet space is the one induced by this metricd.

Example A.2.7. Let C∞([0,1]) be the space of smooth functions on the interval[0,1]. Acountable family of semi-norms is given by theCk-norms as defined in Example A.2.2 b).In order to prove that this family of (semi-)norms turnsC∞([0,1]) into a Frechet space wewill show thatC∞([0,1]) equipped with the metricd given by (A.1) is complete.Let (gn)n be a Cauchy sequence inC∞([0,1]) with respect to the metricd. Then for anyk≥ 0 the sequence(gn)n is Cauchy with respect to theCk-norm. SinceCk([0,1]) togetherwith theCk-norm is a Banach space there exists a uniquehk ∈ Ck([0,1] such that(gn)n

converges tohk in theCk-norm. From the estimate‖ · ‖Ck([0,1]) ≤ ‖ · ‖Cℓ([0,1]) for k≤ ℓ weconclude thathk andhℓ coincide. Therefore, puttingh := h0 we obtainh∈C∞([0,1]) andd(h,gn) → 0 for n→ ∞. This shows the completeness ofC∞([0,1]).

If one wants to show that linear maps between Frechet spacesare homeomorphisms, thefollowing theorem is very helpful.

A.2. Functional analysis 159

Theorem A.2.8(Open Mapping Theorem). Let X and Y be Frechet spaces, and let f:X →Y be a continuous linear surjection. Then f is open, i. e., f isa homeomorphism.

Proof. See [Rudin1973, Cor. 2.12., p. 48] or [Reed-Simon1980, Thm.V.6, p. 132]

From now on we fix a Hilbert spaceH. A continuous linear mapH →H is calledboundedoperatoronH. But many operators occuring in analysis and mathematical physics are notcontinuous and not even defined on the whole Hilbert space. Therefore one introduces theconcept of unbounded operators.

Definition A.2.9. Let dom(A)⊂H be a linear subspace ofH. A linear mapA : dom(A)→H is called anunbounded operatorin H with domaindom(A). One says thatA is denselydefinedif dom(A) is a dense subspace ofH.

Example A.2.10. One can represent elements ofL2(R) by functions. The space ofsmooth functions with compact supportC∞

c (R) is regarded as a linear subspace,C∞c (R)⊂

L2(R). Then one can consider the differentiation operatorA := ddt as an unbounded oper-

ator inL2(R) with domain dom(A) = C∞c (R), andA is densely defined.

Definition A.2.11. Let A be an unbounded operator onH with domain dom(A). Thegraphof A is the set

Γ(A) :=(x,Ax)

∣∣x∈ dom(A)⊂ H ×H.

The operatorA is called aclosedoperator if its graphΓ(A) is a closed subset ofH ×H.

Definition A.2.12. Let A1 andA2 be operators onH. If dom(A1) ⊃ dom(A2) andA1x =A2x for all x∈ dom(A2) , thenA1 is said to be anextensionof A2. One then writesA1 ⊃A2.

Definition A.2.13. Let A be an operator onH. An operatorA is closableif it possessesa closed extension. In this case the closureΓ(A) of Γ(A) in H ×H is the graph of anoperator called theclosureof A.

Definition A.2.14. Let A be a densely defined operator onH. Then we put

dom(A∗) :=

x∈ H∣∣ there exists ay∈ H with (Az,x) = (z,y) for all z∈ dom(A)

.

For eachx∈ dom(A∗) we defineA∗x := y wherey is uniquely determined by the require-ment (Az,x) = (z,y) for all z∈ dom(A). Uniqueness ofy follows from dom(A) beingdense inH. We callA∗ theadjointof A.

Definition A.2.15. A densely defined operatorA on H is calledsymmetricif A∗ is anextension ofA, i. e., if dom(A)⊂ dom(A∗) andAx= A∗x for all x∈ dom(A). The operatorA is calledselfadjointif A = A∗, that is, ifA is symmetric and dom(A) = dom(A∗).

Any symmetric operator is closable with closureA = A∗∗.

Definition A.2.16. A symmetric operatorA is calledessentially selfadjointif its closureA is selfadjoint.


We conclude this section by stating a criterion for essential selfadjointness of a symmetricoperator.

Definition A.2.17. Let A be an operator on a Hilbert spaceH. Then one calls the setC∞(A) :=

⋂∞n=1dom(An) the set ofC∞-vectorsfor A. A vectorϕ ∈ C∞(A) is called an

analytic vectorfor A if∞

∑n=0

‖Anϕ‖n!

tn < ∞

for somet > 0.

Theorem A.2.18(Nelson’s Theorem). Let A be a symmetric operator on a Hilbert spaceH. If dom(A) contains a set of analytic vectors which is dense in H, then A is essentiallyselfadjoint.

Proof. See [Reed-Simon1975, Thm. X.39, p. 202].

If A is a selfadjoint operator andf : R → C is a bounded Borel-measurable function, thenone can define the bounded operatorf (A) in a natural manner. We use this to get theunitary operator exp(iA) in Section 4.6. Ifϕ is an analytic vector, then

exp(iA)ϕ =∞

∑n=0

in

n!Anϕ .

A.3 Differential geometry

In this section we introduce the basic geometrical objects such as manifolds and vectorbundles which are used throughout the text. A detailed introduction can be found e. g. in[Spivak1979] or in [Nicolaescu1996].

A.3.1 Differentiable manifolds

We start with the concept of a manifold. Loosely speaking, manifolds are spaces whichlook locally likeRn.

Definition A.3.1. Let n be an integer. A topological spaceM is called ann-dimensionaltopological manifoldif and only if

(1) its topology is Hausdorff and has a countable basis, and

(2) it is locally homeomorphic toRn, i. e., for everyp∈ M there exists an open neigh-borhoodU of p in M and a homeomorphismϕ : U → ϕ(U), whereϕ(U) is an opensubset ofRn.

Any such homeomorphismϕ : U → ϕ(U) ⊂ Rn is called a(local) chart of M. Thecoordinate functionsϕ j : U → R of ϕ = (ϕ1, . . . ,ϕn) are called thecoordinatesof thelocal chart. Anatlasof M is a family of local charts(Ui ,ϕi)i∈I of M which covers all ofM, i. e., ∪

i∈IUi = M.

A.3. Differential geometry 161

Definition A.3.2. Let M be a topological manifold. Asmooth atlasof M is an atlas(Ui ,ϕi)i∈I such that

ϕi ϕ−1j : ϕ j(Ui ∩U j) → ϕi(Ui ∩U j)

is a smooth map (as a map between open subsets ofRn) wheneverUi ∩U j 6= /0.

M

ϕi ϕ−1j

Rn

ϕ j (U j) ϕi(Ui)

ϕi

Ui

U j

ϕ j

Fig. 37: Smooth atlas

Not every topological manifold admits a smooth atlas. We shall only be interested inthose topological manifolds that do. Moreover, topological manifolds can have essentiallydifferent smooth atlases in the sense that they give rise to non-diffeomorphic smoothmanifolds. Hence the smooth atlas is an important additional piece of structure.

Definition A.3.3. A smooth manifoldis a topological manifoldM together with a maxi-mal smooth atlas.

Maximality means that there is no smooth atlas onM containing all local charts of thegiven atlas except for the given atlas itself. Every smooth atlas is contained in a uniquemaximal smooth atlas.In the following “manifold” will always mean “smooth manifold”. The smooth atlas willusually be suppressed in the notation.

Examples A.3.4. a) Every nonempty open subset ofRn is ann-dimensional manifold.More generally, any nonempty open subset of ann-dimensional manifold is itself ann-dimensional manifold.b) The product of anym-dimensional manifold with anyn-dimensional manifold is canon-ically an(m+n)-dimensional manifold.


c) Let n ≤ m. An n-dimensionalsubmanifoldN of an m-dimensional manifoldM is anonempty subsetN of M such that for everyp∈ N there exists a local chart(U,ϕ) of Maboutp with

ϕ(U ∩N) = ϕ(U)∩Rn,

where we identifyRn ∼= Rn×0 ⊂ Rm. Any submanifold is canonically a manifold. Inthe casen = m−1 the submanifoldN is calledhypersurfaceof M.

As in the case of open subsets ofRn, we have the concept ofdifferentiable mapbetweenmanifolds:

Definition A.3.5. Let M andN be manifolds and letp∈M. A continuous mapf : M →Nis said to bedifferentiableat the pointp if there exist local charts(U,ϕ) and(V,ψ) aboutp in M and aboutf (p) in N respectively, such thatf (U) ⊂V and

ψ f ϕ−1 : ϕ(U) → ψ(V)

is differentiable atϕ(p) ∈ ϕ(U). The map f is said to be differentiable onM if it isdifferentiable at every point ofM.Similarly, one definesCk-maps between smooth manifolds,k ∈ N∪∞. A C∞-map isalso called asmooth map.

MU b p

Rn

ϕ(U)

ϕ

f

Nbf (p)

V

ψ

ψ(V)

Rm

ψ f ϕ−1

Fig. 38: Differentiability of a map

Note that, ifψ f ϕ−1 is (Ck-)differentiable forsomelocal chartsϕ ,ψ as in Defini-tion A.3.5, then so isψ ′ f ϕ ′−1 for any other pair of local chartsϕ ′,ψ ′ obeying thesame conditions. This is a consequence of the fact that the atlases ofM andN have beenassumed to be smooth.In order to define the differential of a differentiable map between manifolds, we need theconcept oftangent space:


Definition A.3.6. Let M be a manifold andp∈ M. Consider the setTp of differentiablecurvesc : I → M with c(0) = p whereI is an open interval containing 0∈ R. Thetangentspaceof M at p is the quotient

TpM := Tp/ ∼,

where “∼” is the equivalence relation defined as follows: Two smooth curvesc1,c2 ∈ Tp

are equivalent if and only if there exists a local chart aboutp such that(ϕ c1)′(0) =

(ϕ c2)′(0).

M

b p

c

TpM

Fig. 39: Tangent spaceTpM

One checks that the definition of the equivalence relation does not depend on the choiceof local chart: If(ϕ c1)

′(0) = (ϕ c2)′(0) for one local chart(U,ϕ) with p ∈ U , then

(ψ c1)′(0) = (ψ c2)

′(0) for all local charts(V,ψ) with p∈V.Let n denote the dimension ofM. Denote the equivalence class ofc∈ Tp in TpM by [c].It can be easily shown that the map

Θϕ : TpM → Rn,

[c] 7→ (ϕ c)′(0),

is a well-defined bijection. Hence we can introduce a vector space structure onTpM bydeclaringΘϕ to be a linear isomorphism. This vector space structure is independent ofthe choice of local chart because for two local charts(U,ϕ) and(V,ψ) containingp themapΘψ Θ−1

ϕ = dϕ(p)(ψ ϕ−1) is linear.By definition, thetangent bundleof M is the disjoint union of all the tangent spaces ofM,

TM :=∪

p∈MTpM.

Definition A.3.7. Let f : M → N be a differentiable map between manifolds and letp∈M. Thedifferentialof f at p (also called thetangent mapof f at p) is the map

dp f : TpM → Tf (p)N, [c] 7→ [ f c].

Thedifferentialof f is the mapd f : TM → TN, d f|TpM:= dp f .


The mapdp f is well-defined and linear. The mapf is said to be animmersionor asubmersionif dp f is injective or surjective for everyp ∈ M respectively. Adiffeomor-phismbetween manifolds is a smooth bijective map whose inverse isalso smooth. Anembeddingis an immersionf : M → N such thatf (M) ⊂ N is a submanifold ofN andf : M → f (M) is a diffeomorphism.

Using local charts basically all local properties of differential calculus onRn can be trans-lated to manifolds. For example, we have thechain rule

dp(g f ) = df (p)gdp f ,

and theinverse function theoremwhich states that ifdp f : TpM → Tf (p)N is a linear iso-morphism, thenf maps a neighborhood ofp diffeomorphically onto a neighborhood off (p).

A.3.2 Vector bundles

We can think of the tangent bundle as a family of pairwise disjoint vector spacesparametrized by the points of the manifold. In a suitable sense these vector spaces de-pend smoothly on the base point. This is formalized by the concept of a vector bundle.

Definition A.3.8. Let K = R or C. Let E andM be manifolds of dimensionm+n andmrespectively. Letπ : E → M be a surjective smooth map. Let the fiberEp := π−1(p) carrya structure ofK-vector spaceVp for eachp ∈ M. The quadruple(E,π ,M,Vpp∈M) iscalled aK-vector bundleif for every p∈ M there exists an open neighborhoodU of p inM and a diffeomorphismΦ : π−1(U) →U ×Kn such that the following diagram

π−1(U)

π""FF

FFFF

FFF

Φ // U ×Kn

π1||xx

xxxx

xxx

U

(A.2)

commutes and for everyq∈U the mapπ2Φ|Eq : Eq →Kn is a vector space isomorphism.Hereπ1 : U×Kn →U denotes the projection onto the first factorU andπ2 :U×Kn →Kn

is the projection onto the second factorKn.


Mb

p

Ep

E

π

Fig. 40: Vector bundle

Such a mapΦ : π−1(U) → U ×Kn is called alocal trivialization of the vector bundle.The manifoldE is called thetotal space, M thebase, and the numbern the rank of thevector bundle. Often one simply speaks of the vector bundleE for brevity.A vector bundle is said to betrivial if it admits a global trivialization, that is, if there existsa diffeomorphism as in (A.2) withU = M.

Examples A.3.9.a) The tangent bundle of anyn-dimensional manifoldM is a real vectorbundle of rankn. The mapπ is given by the canonical mapπ(TpM) = p for all p∈ M.b) Most operations from linear algebra on vector spaces can be carried out fiberwise onvector bundles to give new vector bundles. For example, for agiven vector bundleEone can define thedual vector bundleE∗. Here one has by definition(E∗)p = (Ep)

∗.Similarly, one can define the exterior and the symmetric powers ofE. For givenK-vectorbundlesE andF one can form the direct sumE⊕F , the tensor productE⊗F, the bundleHomK(E,F) etc.c) The dual vector bundle of the tangent bundle is called thecotangent bundleand isdenoted byT∗M.d) Let n be the dimension ofM andk ∈ 0,1, . . . ,n. Thekth exterior power ofT∗M isthebundle of k-linear skew-symmetric formsonTM and is denoted byΛkT∗M. It is a realvector bundle of rank n!

k!(n−k)! . By conventionΛ0T∗M is the trivial real vector bundle ofrank 1.

Definition A.3.10. A sectionin a vector bundle(E,π ,M,Vpp∈M) is a maps : M → Esuch that

π s= idM.


Mb

p

s(M)

Ep

E

π

bs(p)

Fig. 41: Section in a vector bundle

SinceM andE are smooth manifolds we can speak aboutCk-sections,k∈ N∪∞. ThesetCk(M,E) of Ck-sections of a givenK-vector bundle forms aK-vector space, and amodule over the algebraCk(M,K) as well because multiplyingpointwiseanyCk-sectionwith anyCk-function one obtains a newCk-section.In each vector bundle there exists a canonical smooth section, namely thezero sectiondefined bys(x) := 0x ∈ Ex. However, there does not in general exist any smoothnowherevanishingsection. Moreover, the existence ofn everywhere linearly independent smoothsections in a vector bundle of rankn is equivalent to itstriviality .

Examples A.3.11.a) LetE = M ×Kn be the trivialK-vector bundle of rankn overM.

Then the sections ofE are essentially just theKn-valued functions onM.b) The sections inE = TM are called thevector fieldsonM. If (U,ϕ) is a local chart of then-dimensional manifoldM, then for eachj = 1, . . . ,n the curvec(t) = ϕ−1(ϕ(p)+ tej)

represents a tangent vector∂∂xj (p) wheree1, . . . ,en denote the standard basis ofRn. The

vector fields ∂∂x1 , . . . , ∂

∂xn are smooth onU and yield a basis ofTpM for everyp∈U ,

TpM = SpanR

(∂

∂x1 (p), . . . ,∂

∂xn (p)

).

c) The sections inE = T∗M are called the1-forms. Let (U,ϕ) be a local chart ofM. De-note the basis ofT∗

p M dual to ∂∂x1 (p), . . . , ∂

∂xn (p) by dx1(p), . . . ,dxn(p). Thendx1, . . . ,dxn

are smooth 1-forms onU .d) Fix k ∈ 0, . . . ,n. Sections inE = ΛkT∗M are calledk-forms. Given a local chart(U,ϕ) we get smoothk-forms which pointwise yield a basis ofΛkT∗M by

dxi1 ∧ . . .∧dxik, 1≤ i1 < .. . < ik ≤ n.


In particular, fork = n the bundleΛnT∗M has rank 1 and a local chart yields the smoothlocal sectiondx1∧ . . .∧dxn. Existence of a global smooth section inΛnT∗M is equivalentto M being orientable.e) For eachp ∈ M let |ΛM|p be the set of all functionsv : ΛnT∗

p M → R with v(λX) =|λ | · v(X) for all X ∈ ΛnT∗

p M and allλ ∈ R. Now |ΛM|p is a 1-dimensional real vectorspace and yields a vector bundle|ΛM| of rank 1 overM. Sections in|ΛM| are calleddensities.Given a local chart(U,ϕ) there is a smooth density|dx| defined onU and characterizedby

|dx|(dx1∧ . . .∧dxn) = 1.

The bundle|ΛM| is always trivial. Its importance lies in the fact that densities can beintegrated. There is a unique linear map

∫

M: D(M, |ΛM|) → R,

called theintegral, such that for any local chart(U,ϕ) and anyf ∈ D(U,R) we have∫

Mf |dx| =

∫

ϕ(U)( f ϕ−1)(x1, . . . ,xn)dx1 · · ·dxn

where the right hand side is the usual integral of functions on Rn andD(M,E) denotesthe set of smooth sections with compact support.f) Let E be a real vector bundle. Smooth sections inE∗⊗E∗ which are pointwisenonde-generate symmetricbilinear forms are calledsemi-Riemannian metricsor inner productson E. An inner product onE is calledRiemannian metricif it is pointwise positive def-inite. An inner product onE is called aLorentzian metricif it has pointwise signature(− + . . . +). In caseE = TM a Riemannian or Lorentzian metric onE is also called aRiemannian or Lorentzian metric onM respectively. ARiemannianor Lorentzian mani-fold is a manifoldM together with a Riemannian or Lorentzian metric onM respectively.Any semi-Riemannian metric onT∗M yields a nowhere vanishing smooth density dV onM. In local coordinates, write the semi-Riemannian metric as

n

∑i, j=1

gi j dxi ⊗dxj .

Then the induced density is given by

dV =√|det(gi j )| |dx|.

Therefore there is a canonical way to form the integral∫

M f dV of any functionf ∈D(M)on a Riemannian or Lorentzian manifold.g) If E is a complex vector bundle aHermitian metricon E is by definition a smoothsection ofE∗⊗RE∗ (thereal tensor product ofE∗ with itself) which is a Hermitian scalarproduct on each fiber.

Definition A.3.12. Let (E,π ,M,Vpp∈M) and(E′,π ′,M′,V ′p′p′∈M′) beK-vector bun-

dles. Avector-bundle-homomorphismfrom E to E′ is a pair( f ,F) where


(1) f : M → M′ is a smooth map,

(2) F : E → E′ is a smooth map such that the diagram

E

π

F // E′

π ′

M

f // M′

commutes and such thatF|Ep: Ep → E′

f (p) is K-linear for everyp∈ M.

If M = M′ and f = idM a vector-bundle-homomorphism is simply a smooth section inHomK(E,E′) → M.

Remark A.3.13. Let E be a real vector bundle with inner product〈· , ·〉. Then we get avector-bundle-isomorphismZ : E → E∗,Z(X) = 〈X, ·〉.In particular, on a Riemannian or Lorentzian manifoldM with E = TM one can definethegradientof a differentiable functionf : M → R by gradf :=Z−1(d f) = (d f)♯ . Thedifferentiald f is a 1-form defined independently of the metric while the gradient gradfis a vector field whose definition does depend on the semi-Riemannian metric.

A.3.3 Connections on vector bundles

For a differentiable functionf : M → R on a (smooth) manifoldM its derivative in direc-tion X ∈C∞(M,TM) is defined by

∂X f := d f(X).

We have defined the concept of differentiability of a sections in a vector bundle. What isthe derivative ofs?Without further structure there is no canonical way of defining this. A rule for differenti-ation of sections in a vector bundle is called a connection.

Definition A.3.14. Let (E,π ,M,Vpp∈M) be aK-vector bundle,K := R or C. A con-nection(or covariant derivative) onE is aR-bilinear map

∇ : C∞(M,T M)×C∞(M,E) → C∞(M,E),

(X,s) 7→ ∇Xs,

with the following properties:

(1) The map∇ is C∞(M)-linear in the first argument, i. e.,

∇ f Xs= f ∇Xs

holds for all f ∈C∞(M), X ∈C∞(M,TM) ands∈C∞(M,E).


(2) The map∇ is aderivationwith respect to the second argument, i. e., it isK-bilinearand

∇X( f ·s) = ∂X f ·s+ f ·∇Xs

holds for all f ∈C∞(M), X ∈C∞(M,TM) ands∈C∞(M,E).

The properties of a connection imply that the value of∇Xsat a given pointp∈M dependsonly onX(p) and on the values ofson a curve representingX(p).Let ∇ be a connection on a vector bundleE over M. Let c : [a,b] → M be a smoothcurve. Givens0 ∈ Ec(a) there is a unique smooth solutions : [a,b] → E, t 7→ s(t) ∈ Ec(t),satisfyings(a) = s0 and

∇cs= 0. (A.3)

This follows from the fact that in local coordinates (A.3) isa linear ordinary differentialequation of first order. The map

Πc : Ec(a) → Ec(b), s0 7→ s(b),

is calledparallel transport. It is easy to see thatΠc is a linear isomorphism. This showsthat a connection allows us via its parallel transport to “connect” different fibers of thevector bundle. This is the origin of the term “connection”. Be aware that in general theparallel transportΠc does depend on the choice of curvec connecting its endpoints.Any connection∇ on a vector bundleE induces a connection, also denoted by∇, on thedual vector bundleE∗ by

(∇Xθ )(s) := ∂X (θ (s))−θ (∇Xs)

for all X ∈ C∞(M,TM), θ ∈ C∞(M,E∗) ands∈ C∞(M,E). Hereθ (s) ∈ C∞(M) is thefunction onM obtained by pointwise evaluation ofθ (p) ∈ E∗

p ons(p) ∈ Ep.Similarly, tensor products, exterior and symmetric products, and direct sums inherit con-nections from the connections on the vector bundles out of which they are built. Forexample, two connections∇ and∇′ on E andE′ respectively induce a connectionD onE⊗E′ by

DX(s⊗s′) := (∇Xs)⊗s′ +s⊗ (∇′Xs′)

and a connectionD onE⊕E′ by

DX(s⊕s′) := (∇Xs)⊕ (∇′Xs′)

for all X ∈C∞(M,T M), s∈C∞(M,E) ands′ ∈C∞(M,E′).If a vector bundleE carries a semi-Riemannian or Hermitian metric〈· , ·〉, then a connec-tion ∇ onE is calledmetric if the following Leibniz rule holds:

∂X(〈s,s′〉) = 〈∇Xs,s′〉+ 〈s,∇Xs′〉

for all X ∈C∞(M,T M) ands,s′ ∈C∞(M,E).Given two vector fieldsX,Y ∈ C∞(M,TM) there is a unique vector field[X,Y] ∈C∞(M,T M) characterized by

∂[X,Y] f = ∂X∂Y f − ∂Y∂X f


for all f ∈C∞(M). The map[·, ·] : C∞(M,TM)×C∞(M,T M)→C∞(M,TM) is called theLie bracket. It is R-bilinear, skew-symmetric and satisfies theJacobi identity

[[X,Y],Z]+ [[Y,Z],X]+ [[Z,X],Y] = 0.

Definition A.3.15. Let ∇ be a connection on a vector bundleE. Thecurvature tensorof∇ is the map

R : C∞(M,T M)×C∞(M,T M)×C∞(M,E) →C∞(M,E),

(X,Y,s) 7→ R(X,Y)s := ∇X(∇Ys)−∇Y(∇Xs)−∇[X,Y]s.

One can check that the value ofR(X,Y)sat any pointp∈ M depends only onX(p), Y(p),ands(p). Thus the curvature tensor can be regarded as a section,R∈ C∞(M,Λ2T∗M ⊗HomK(E,E)).Now let M be ann-dimensional manifold with semi-Riemannian metricg on TM. It canbe shown that there exists a unique metric connection∇ onTM satisfying

∇XY−∇YX = [X,Y]

for all vector fieldsX andY onM. This connection is called theLevi-Civitaconnection ofthe semi-Riemannian manifold(M,g). Its curvature tensorR is theRiemannian curvaturetensorof (M,g). TheRicci curvatureric ∈C∞(M,T∗M⊗T∗M) is defined by

ric(X,Y) :=n

∑j=1

ε j g(R(X,ej)ej ,Y)

wheree1, . . . ,en are smooth locally defined vector fields which are pointwise orthonormalwith respect tog andε j = g(ej ,ej) = ±1. It can easily be checked that this definition isindependent of the choice of the vector fieldse1, . . . ,en. Similarly, thescalar curvatureisthe function scal∈C∞(M,R) defined by

scal :=n

∑j=1

ε j ric(ej ,ej).

A.4 Differential operators

In this section we explain the concept of linear differential operators and we definethe principal symbol. A detailed introduction to the topic can be found e. g. in[Nicolaescu1996, Ch. 9]. As before we writeK = R or C.

Definition A.4.1. Let E andF beK-vector bundles of rankn andm respectively over ad-dimensional manifoldM. A linear differential operator of order at most kfrom E to Fis aK-linear map

L : C∞(M,E) →C∞(M,F)

A.4. Differential operators 171

which can locally be described as follows: For everyp ∈ M there exists an opencoordinate-neighborhoodU of p in M on whichE andF are trivialized and there aresmooth mapsAα : U → HomK(Kn,Km) such that onU

Ls= ∑|α |≤k

Aα∂ |α |s∂xα .

Here summation is taken over all multiindicesα = (α1, . . . ,αd) ∈ Nd with |α| :=

∑dr=1 αr ≤ k. Moreover, ∂ |α|

∂xα := ∂α1+···+αd

∂xα11 ···∂x

αdd

. In this definition we have used the local

trivializations to identify sections inE with Kn-valued functions and sections inF with

Km-valued functions onU . If L is a linear differential operator of order at mostk, but notof order at mostk−1, then we say thatL is of order k.

Note that zero-order differential operators are nothing but sections of HomK(E,F), i. e.,they are vector-bundle-homomorphisms fromE to F.

Definition A.4.2. Let L be a linear differential operator of orderk from E to F . Theprincipal symbolof L is the map

σL : T∗M → HomK(E,F)

defined locally as follows: For a givenp ∈ M write L = ∑|α |≤k Aα∂ |α|∂xα in a coordinate

neighborhood ofp with respect to local trivializations ofE andF as in Definition A.4.1.For everyξ = ∑d

r=1 ξr ·dxr ∈ T∗p M we have with respect to these trivializations,

σL(ξ ) := ∑|α |=k

ξ αAα(p)

whereξ α := ξ α11 · · ·ξ αd

d . Here we have used the local trivializations ofE andF to identifyHomK(E,F) with HomK(Kn,Km).

One can show that the principal symbol of a differential operator is well-defined, that is,it is independent of the choice of the local coordinates and trivializations. Moreover, theprincipal symbol of a differential operator of orderk is, by definition, a homogeneouspolynomial of degreek onT∗M.

Example A.4.3. The gradient is a linear differential operator of first order

grad :C∞(M,R) →C∞(M,T M)

with principal symbolσgrad(ξ ) f = f ·ξ ♯.

Example A.4.4. The divergence yields a first order linear differential operator

div : C∞(M,TM) →C∞(M,R)

with principal symbolσdiv(ξ )X = ξ (X).


Example A.4.5. For eachk∈ N there is a unique linear first order differential operator

d : C∞(M,ΛkT∗M) →C∞(M,Λk+1T∗M),

calledexterior differential, such that

(1) for k = 0 the exterior differential coincides with the differential defined in Defini-tion A.3.7, after the canonical identificationTyR = R,

(2) d2 = 0 :C∞(M,ΛkT∗M) →C∞(M,Λk+2T∗M) for all k,

(3) d(ω ∧ η) = (dω) ∧ η + (−1)kω ∧ dη for all ω ∈ C∞(M,ΛkT∗M) and η ∈C∞(M,Λl T∗M).

Its principal symbol is given by

σd(ξ )ω = ξ ∧ω .

Example A.4.6. A connection∇ on a vector bundleE can be considered as a first orderlinear differential operator

∇ : C∞(M,E) →C∞(M,T∗M⊗E).

Its principal symbol is easily be seen to be

σ∇(ξ )e= ξ ⊗e.

Example A.4.7. If L is a linear differential operator of order 0, i. e.,L ∈C∞(M,Hom(E,F)), then

σL(ξ ) = L.

Remark A.4.8. If L1 : C∞(M,E) →C∞(M,F) is a linear differential operator of orderkandL2 : C∞(M,F) →C∞(M,G) is a linear differential operator of orderl , thenL2 L1 isa linear differential operator of orderk+ l . The principal symbols satisfy

σL2L1(ξ ) = σL2(ξ )σL1(ξ ).

A.5 More on Lorentzian geometry

This section is a rather heterogeneous collection of results on Lorentzian manifolds. Wegive full proofs. This material has been collected in this appendix in order not to overloadSection 1.3 with technical statements.Throughout this sectionM denotes a Lorentzian manifold.

Lemma A.5.1. Let the causal relation≤ on M be closed, i. e., for all convergent se-quences pn→p and qn→q in M with pn ≤ qn we have p≤ q.Then for every compact subset K of M the subsets JM

+ (K) and JM− (K) are closed.

A.5. More on Lorentzian geometry 173

Proof. Let (qn)n∈N be any sequence inJM+ (K) converging inM andq ∈ M be its limit.

By definition, there exists a sequence(pn)n∈N in K with pn ≤ qn for everyn. SinceKis compact we may assume, after to passing to a subsequence, that(pn)n∈N converges tosomep∈ K. Since≤ is closed we getp≤ q , henceq∈ JM

+ (K). This shows thatJM+ (K)

is closed. The proof forJM− (K) is the same.

Remark A.5.2. If K is only assumed to beclosedin Lemma A.5.1, thenJM± (K) need not

be closed. The following picture shows a curveK, closed as a subset and asymptotic to alightlike line in 2-dimensional Minkowski space. Its causal futureJM

+ (K) is the open halfplane bounded by this lightlike line.

JM+ (K)

x0 (time direction)

K

Fig. 42: Causal futureJM+ (K) is open

Lemma A.5.3. Let M be a timeoriented Lorentzian manifold. Let K⊂ M be a compactsubset. Let A⊂ M be a subset such that, for every x∈ M, the intersection A∩ JM

− (x) isrelatively compact in M.Then A∩ JM

− (K) is a relatively compact subset of M. Similarly, if A∩ JM+ (x) is relatively

compact for every x∈ M, then A∩JM+ (K) is relatively compact.

A

K

JM− (K)∩A

Fig. 43:A∩JM− (K) is relatively compact


Proof. It suffices to consider the first case. The family of open setsIM− (x), x ∈ M, is an

open covering ofM. SinceK is compact it is covered by a finite number of such sets,

K ⊂ IM− (x1)∪ . . .∪ IM

− (xl ).

We conclude

JM− (K) ⊂ J−

(IM− (x1)∪ . . .∪ IM

− (xl ))⊂ JM

− (x1)∪ . . .∪JM− (xl ).

Since eachA∩JM− (x j) is relatively compact, we have thatA∩JM

− (K)⊂⋃ lj=1

(A∩JM

− (x j ))

is contained in a compact set.

Corollary A.5.4. Let S be a Cauchy hypersurface in a globally hyperbolic Lorentzianmanifold M and let K⊂M be compact. Then JM

± (K)∩S and JM± (K)∩JM∓ (S) are compact.

Proof. The causal future of every Cauchy hypersurface is past-compact. This followse. g. from [O’Neill1983, Chap. 14, Lemma 40]. Applying LemmaA.5.3 toA := JM

+ (S)we conclude thatJM

− (K)∩JM+ (S) is relatively compact inM. By [O’Neill1983, Chap. 14,

Lemma 22] the subsetsJM± (S) and the causal relation “≤” are closed. By Lemma A.5.1

JM− (K)∩JM

+ (S) is closed, hence compact.SinceS is a closed subset ofJM

+ (S) we also have thatJM− (K)∩S is compact.

The statements onJM+ (K)∩JM

− (S) and onJM+ (K)∩Sare analogous.

Lemma A.5.5. Let M be a timeoriented convex domain. Then the causal relation “≤”is closed. In particular, the causal future and the causal past of each point are closedsubsets of M.

Proof. Let p, pi ,q,qi ∈ M with lim i→∞ pi = p, limi→∞ qi = q, and pi ≤ qi for all i. Wehave to showp≤ q.Let x∈ TpM be the unique vector such thatq = expp(x) and, similarly, for eachi let xi ∈Tpi M be such that exppi

(xi) = qi. Sincepi ≤ qi and since exppimapsJ+(0)∩exp−1

pi(M)

in Tpi M diffeomorphically ontoJM+ (pi), we havexi ∈ J+(0), hence〈xi ,xi〉 ≤ 0. From

lim i→∞ pi = p and limi→∞ qi = q we conclude limi→∞ xi = x and therefore〈x,x〉 ≤ 0. Thusx∈ J+(0)∪J−(0) ⊂ TpM.Now letT be a smooth vector field onM representing the timeorientation. In other words,T is timelike and future directed. Then〈T,xi〉 ≤ 0 becausexi is future directed and so〈T,x〉 ≤ 0 as well. Thusx∈ J+(0) ⊂ TpM and hencep≤ q.

Lemma A.5.6. Let M be a timeoriented Lorentzian manifold and let S⊂M be a spacelikehypersurface. Then for every point p in S, there exists a basis of open neighborhoodsΩof p in M such that S∩Ω is a Cauchy hypersurface inΩ.

Proof. Let p ∈ S. Since every spacelike hypersurface is locally acausal there exists anopen neighborhoodU of p in M such thatS∩U is an acausal spacelike hypersurface ofU . Let Ω be the Cauchy development ofS∩U in U .


S

U

ΩS∩U

Fig. 44: Cauchy development ofS∩U in U

SinceΩ is the Cauchy development of an acausal hypersurface containing p, it is an openneighborhood ofp in U and hence also inM. It follows from the definition of the Cauchydevelopment thatΩ∩S= S∩U and thatS∩Ω is a Cauchy hypersurface ofΩ.Given any neighborhoodV of p the neighborhoodU from above can be chosen to be con-tained inV. HenceΩ is also contained inV. Therefore we get a basis of neighborhoodsΩ with the required properties.

On globally hyperbolic manifolds the relation≤ is always closed [O’Neill1983, Chap. 14,Lemma 22]. The statement that the setsJM

+ (p)∩JM− (q) are compact can be strengthened

as follows:

Lemma A.5.7. Let K,K′ ⊂ M two compact subsets of a globally hyperbolic Lorentzianmanifold M. Then JM+ (K)∩JM

− (K′) is compact.

Proof. Let p in M. By the definition of global-hyperbolicity, the subsetJM+ (p) is past

compact inM. It follows from Lemma A.5.3 thatJM+ (p)∩ JM

− (K′) is relatively compactin M. Since the relation≤ is closed onM, the setsJM

+ (p) and JM− (K′) are closed by

Lemma A.5.1. HenceJM+ (p)∩ JM

− (K′) is actually compact. This holds for everyp∈ M,i. e., JM

− (K′) is future compact inM. It follows again from Lemma A.5.3 thatJM+ (K)∩

JM− (K′) is relatively compact inM, hence compact by Lemma A.5.1.

Lemma A.5.8. LetΩ ⊂ M be a nonempty open subset of a timeoriented Lorentzian man-ifold M. Let JM

+ (p)∩ JM− (q) be contained inΩ for all p,q ∈ Ω. ThenΩ is causally

compatible.If furthermore M is globally hyperbolic, thenΩ is globally hyperbolic as well.

Proof. We first show thatJM± (p)∩Ω = JΩ

± (p) for all p ∈ Ω. Let p ∈ Ω be fixed. TheinclusionJΩ

±(p) ⊂ JM± (p)∩Ω is obvious. To show the opposite inclusion letq∈ JM

+ (p)∩Ω. Then there exists a future directed causal curvec : [0,1] → M with c(0) = p andc(1) = q. For everyz∈ c([0,1]) we havez∈ JM

+ (p)∩ JM− (q) ⊂ Ω, i. e., c([0,1]) ⊂ Ω.

Thereforeq ∈ JΩ+(p). HenceJM

+ (p)∩Ω ⊂ JΩ+(p) andJM

− (p)∩Ω ⊂ JΩ−(p) can be seen

similarly.


We have shownJM± (p)∩Ω = JΩ

±(p), i. e., Ω is a causally compatible subset ofM. Letnow M be globally hyperbolic. Then since for any two pointsp,q ∈ Ω the intersectionJM+ (p)∩JM

− (q) is contained inΩ the subset

JΩ+(p)∩JΩ

−(q) = JM+ (p)∩JM

− (q)∩Ω = JM+ (p)∩JM

− (q)

is compact. Remark 1.3.9 concludes the proof.

Lemma A.5.9. For any acausal hypersurface S in a timeoriented Lorentzianmanifold theCauchy development D(S) is a causally compatible and globally hyperbolic open subsetof M.

Proof. Let S be an acausal hypersurface in a timeoriented Lorentzian manifold M. By[O’Neill1983, Chap. 14, Lemma 43]D(S) is an open and globally hyperbolic subset ofM. Let p,q∈ D(S). Let z∈ JM

+ (p)∩ JM− (q). We choose a future directed causal curvec

from p throughz to q. Extendc to an inextendible causal curve inM, again denoted byc. Sincep∈ D(S) the curvec must meetS. SinceS is acausal this intersection point isunique.Now let c be any inextendible causal curve throughz. If c intersectsSbeforez, then lookat the inextendible curve obtained by first following ˜c until zand then followingc. This isan inextendible causal curve throughq. Sinceq∈ D(S) this curve must intersectS. Thisintersection point must come beforez, hence lie on ˜c. Similarly, if c intersectsS at orafterz, then look at the inextendible curve obtained by first following c until z and thenfollowing c. Again, this curve is inextendible causal and goes throughp∈ D(S). Hence itmust hitSand this intersection point must come before or atz, thus it must again lie on ˜c.In any case ˜c intersectsS. This showsz∈ D(S). We have provedJM

+ (p)∩JM− (q) ⊂ D(S).

By Lemma A.5.8D(S) is causally compatible inM.

Note furthermore that, by the definition ofD(S), the acausal subsetS is a Cauchy hyper-surface ofD(S).

Lemma A.5.10. Let M be a globally hyperbolic Lorentzian manifold. LetΩ ⊂ M bea causally compatible and globally hyperbolic open subset.Assume that there exists aCauchy hypersurfaceΣ of Ω which is also a Cauchy hypersurface of M.Then every Cauchy hypersurface ofΩ is also a Cauchy hypersurface of M.

Proof. Let S be any Cauchy hypersurface ofΩ. SinceΩ is causally compatible inM,achronality ofS in Ω implies achronality ofS in M.Let c : I → M be any inextendible timelike curve inM. SinceΣ is a Cauchy hypersurfaceof M there exists somet0 ∈ I with c(t0) ∈ Σ ⊂ Ω. Let I ′ ⊂ I be the connected componentof c−1(Ω) containingt0. ThenI ′ is an open interval andc|I ′ is an inextendible timelikecurve inΩ. Therefore it must hitS. ThusS is a Cauchy hypersurface inM.

Given any compact subsetK of a globally hyperbolic manifoldM, one can construct acausally compatible globally hyperbolic open subset ofM which is “causally indepen-dent” ofK:


Lemma A.5.11. Let K be a compact subset of a globally hyperbolic Lorentzianmani-fold M. Then the subset M\ JM(K) is, when nonempty, a causally compatible globallyhyperbolic open subset of M.

Proof. SinceM is globally hyperbolic andK is compact it follows from Lemma A.5.1thatJM(K) is closed inM, henceM \JM(K) is open. Next we show thatJM

+ (x)∩JM− (y)⊂

M \JM(K) for any two pointsx,y∈ M \J(K). It will then follow from Lemma A.5.8 thatM \ JM(K) is causally compatible and globally hyperbolic.Let x,y ∈ M \ JM(K) and pickz∈ JM

+ (x)∩ JM− (y). If z /∈ M \ JM(K), thenz∈ JM

+ (K)∪JM− (K). If z∈ JM

+ (K), theny ∈ JM+ (z) ⊂ JM

+ (JM+ (K)) = JM

+ (K) in contradiction toy 6∈JM(K). Similarly, if z ∈ JM

− (K) we getx ∈ JM− (K), again a contradiction. Therefore

z∈ M \ JM(K). This showsJM+ (x)∩JM

− (y) ⊂ M \ JM(K).

Next we prove the existence of a causally compatible globally hyperbolic neighborhoodof any compact subset in any globally hyperbolic manifold. First we need a technicallemma.

Lemma A.5.12. Let A and B two nonempty subsets of a globally hyperbolic Lorentzianmanifold M. ThenΩ := IM

+ (A)∩ IM− (B) is a globally hyperbolic causally compatible open

subset of M.Furthermore, if A and B are relatively compact in M, then so isΩ.

Proof. Since the chronological future and past of any subset ofM are open, so isΩ.For anyx,y ∈ Ω we haveJM

+ (x)∩ JM− (y) ⊂ Ω becauseJM

+ (x) ⊂ JM+ (IM

+ (A)) = IM+ (A) and

JM− (y) ⊂ JM

− (IM− (B)) = IM

− (B). Lemma A.5.8 implies thatΩ is globally hyperbolic andcausally compatible.If furthermoreA andB are relatively compact, then

Ω ⊂ JM+ (A)∩JM

− (B) ⊂ JM+ (A)∩JM

− (B)

andJM+ (A)∩JM

− (B) is compact by Lemma A.5.7. HenceΩ is relatively compact inM.

Proposition A.5.13. Let K be a compact subset of a globally hyperbolic Lorentzianman-ifold M. Then there exists a relatively compact causally compatible globally hyperbolicopen subset O of M containing K.

Proof. Let h : M → R be a Cauchy time-function as in Corollary 1.3.12. The level setsSt := h−1(t) are Cauchy hypersurfaces for eacht ∈ h(M). SinceK is compact so ish(K). Hence there exist numberst+,t− ∈ h(M) such that

St± ∩JM∓ (K) = /0,

that is, such thatK lies in the past ofSt+ and in the future ofSt− . We consider the open set

O := IM−(JM+ (K)∩St+

)∩ IM

+

(JM− (K)∩St−

).


K

St+

St−

JM+ (K)∩St+

JM− (K)∩St−

O

Fig. 45: Construction of globally hyperbolic neighborhoodO of K

We showK ⊂ O. Let p ∈ K. By the choice oft± we haveK ⊂ JM∓ (St±). Choose any

inextendible future directed timelike curve starting atp. SinceSt+ is a Cauchy hypersur-face, it is hit exactly once by this curve at a pointq. Thereforeq ∈ IM

+ (K)∩St+ hencep ∈ IM

− (q) ⊂ IM− (IM

+ (K)∩St+) ⊂ IM− (JM

+ (K)∩St+). Analogouslyp ∈ IM+

(JM− (K)∩St−

).

Thereforep∈ O.It follows from Lemma A.5.12 thatO is a causally compatible globally hyperbolic opensubset ofM. Since every Cauchy hypersurface is future and past compact, the subsetsJM− (K)∩St− and JM

+ (K)∩St+ are relatively compact by Lemma A.5.3. According toLemma A.5.12 the subsetO is also relatively compact. This finishes the proof.

Lemma A.5.14. Let (S,g0) be a connected Riemannian manifold. Let I⊂ R be an openinterval and let f: I → R be a smooth positive function. Let M= I ×S and g= − dt2 +f (t)2g0. We give M the timeorientation with respect to which the vector field ∂

∂ t is futuredirected.Then(M,g) is globally hyperbolic if and only if(S,g0) is complete.

Proof. Let (S,g0) be complete. Each slicet0×S in M is certainly achronal,t0 ∈ I . Weshow that they are Cauchy hypersurfaces by proving that eachinextendible causal curvemeets all the slices.Let c(s) = (t(s),x(s)) be a causal curve inM = I ×S. Without loss of generality we mayassume thatc is future directed, i. e.,t ′(s) > 0. We can reparametrizec and uset as thecurve parameter, i. e.,c is of the formc(t) = (t,x(t)).Suppose thatc is inextendible. We have to show thatc is defined on all ofI . Assumethatc is defined only on a proper subinterval(α,β ) ⊂ I with, say,α ∈ I . Pickε > 0 with[α −ε,α +ε]⊂ I . Then there exist constantsC2 >C1 > 0 such thatC1 ≤ f (t)≤C2 for allt ∈ [α −ε,α +ε]. The curvec being causal means 0≥ g(c′(t),c′(t)) =−1+ f (t)2‖x′(t)‖2

where‖ · ‖ is the norm induced byg0. Hence‖x′(t)‖ ≤ 1f (t) ≤

1C1

for all t ∈ (α,α + ε).

Now let (ti)i be a sequence withti ց α. For sufficiently largei we haveti ∈ (α,α + ε).For j > i ≫ 0 the length of the part ofx from ti to t j is bounded from above by

t j−tiC1

. Thus


we have for the Riemannian distance

dist(x(t j ),x(ti)) ≤t j − ti

C1.

Hence(x(ti))i is a Cauchy sequence and since(S,g0) is complete it converges to a pointp ∈ S. This limit point p does not depend on the choice of Cauchy sequence becausethe union of two such Cauchy sequences is again a Cauchy sequence with a unique limitpoint. This shows that the curvex can be extended continuously by puttingx(α) := p.We extendx in an arbitrary fashion beyondα to a piecewiseC1-curve with‖x′(t)‖ ≤ 1

C2

for all t ∈ (α − ε,α). This yields an extension ofc with

g(c′(t),c′(t)) = −1+ f (t)2‖x′(t)‖2 ≤−1+f (t)2

C22

≤ 0.

Thus this extension is causal in contradiction to the inextendibility of c.Conversely, assume that(M,g) is globally hyperbolic. We fixt0 ∈ I and chooseε > 0 suchthat[t0−ε, t0+ε]⊂ I . There is a constantη > 0 such that 1

f (t) ≥η for all t ∈ [t0−ε,t0+ε].

Fix p ∈ S. For anyq ∈ S with dist(p,q) ≤ εη2 there is a smooth curvex in S of length

at mostεη joining p andq. We may parametrizex on [t0,t0 + ε] such thatx(t0) = q,x(t0 + ε) = p and‖x′‖ ≤ η . Now the curvec(t) := (t,x(t)) is causal because

g(c′,c′) = −1+ f 2‖x′‖2 ≤−1+ f 2η2 ≤ 0.

Moreover,c(t0)= (t0,q) andc(t0+ε) = (t0+ε, p). Thus(t0,q)∈ JM− (t0+ε, p). Similarly,

one sees(t0,q) ∈ JM+ (t0 − ε, p). Hence the closed ballBr(p) in S is contained in the

compact setJM+ (t0− ε, p)∩JM

− (t0 + ε, p) and therefore compact itself, wherer = ηε2 . We

have shown that all closed balls of the fixed radiusr > 0 in Sare compact.Every metric space with this property is complete. Namely, let (pi)i be a Cauchy se-quence. Then there existsi0 > 0 such that dist(pi , p j) ≤ r wheneveri, j ≥ i0. Thusp j ∈ Br(pi0) for all j ≥ i0. Since any Cauchy sequence in the compact ballBr(pi0) mustconverge we have shown completeness.

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[Wald1994] R. M. WALD : Quantum field theory in curved spacetime and black holethermodynamics. University of Chicago Press, Chicago, 1994

[Warner1983] F. W. WARNER: Foundations of Differentiable Manifolds and Lie Groups.Springer-Verlag, New York-Berlin-Heidelberg, 1983

Figures 185

Figures

A∩JM− (K) is relatively compact . . . 174

Ω is geodesically starshaped w. r. t.x 20‖(z1−a)−1‖ is bounded . . . . . . . . . . 110

Advanced fundamental solution atx isnot unique onM . . . . . . . . . .72

Advanced fundamental solution inx onthe (open) upper half-cylinderis not unique . . . . . . . . . . . . 101

Cauchy development. . . . . . . . . . . . . . .22Cauchy developmentD(O) is not rela-

tively compact inM . . . . . 136Cauchy development ofS∩U in U . 175Cauchy hypersurfaceS met by a time-

like curve . . . . . . . . . . . . . . . .21Causal and chronological future and

past of subsetA of Minkowskispace with one point removed18

Causal futureJM+ (K) is open . . . . . . .173

Causally compatible subset ofMinkowski space . . . . . . . . . 19

Construction ofΩ andε . . . . . . . . . . . .85Construction ofρ . . . . . . . . . . . . . . . . 138Construction of globally hyperbolic

neighborhoodO of K . . . . 178Construction of the neighborhoodV of

p . . . . . . . . . . . . . . . . . . . . . . . 82Construction of the sequence(xi)i . . . 84Convexity versus causality . . . . . . . . . 21

Definition ofθr(q) . . . . . . . . . . . . . . . . .83Differentiability of a map . . . . . . . . . 162Domain which is not causally compati-

ble in Minkowski space . . . 19

Functors which yield nets of algebras135

Global fundamental solution; construc-tion of y, y′ andy′′ . . . . . . . . 88

Graphs of Riesz distributionsR+(α) intwo dimensions . . . . . . . . . . 13

Light cone in Minkowski space . . . . . 10

Pseudohyperbolicspace; construction ofΦ(p,y) . . . . . . . . . . . . . . . . . . 97

Quantization functors . . . . . . . . . . . . . 130

Section in a vector bundle . . . . . . . . . 166Section ofSj for fixedx . . . . . . . . . . . . 49Smooth atlas . . . . . . . . . . . . . . . . . . . . . 161Strong causality condition . . . . . . . . . . 23Support ofϕ . . . . . . . . . . . . . . . . . . . . . . 14

Tangent spaceTpM . . . . . . . . . . . . . . . 163The spacelike helixM1 is not causally

compatible inM2 . . . . . . . .127The subsetA is past compact . . . . . . . 19The time separation function is not finite

on anti-deSitter spacetime100

Uniqueness of fundamental solution;JM+ (pi)∩A⊂ Ω for i ≫ 0 . 71

Uniqueness of fundamental solution;construction ofy . . . . . . . . . 69

Uniqueness of fundamental solution;construction ofz . . . . . . . . . 70

Uniqueness of fundamental solution; se-quencepii converging toz70

Uniqueness of solution to Cauchy prob-lem; u vanishes onJM

− (p) ∩JΩ+(St0 ∩Ω) . . . . . . . . . . . . . . 78

Uniqueness of solution to Cauchy prob-lem; domain whereu vanishes77

Vector bundle . . . . . . . . . . . . . . . . . . . . 165

186 Symbols

Symbols

, d’Alembert operator . . . . . . . . . . . . 26

∇, connection-d’Alembert operator 34g, d’Alembert operator for metricg98‖ϕ‖Ck(A), Ck-norm of sectionϕ over a

setA . . . . . . . . . . . . . . . . . . . . . 2♯, isomorphismT∗M → TM induced by

Lorentzian metric . . . . . . . . 25

a, annihilation operator . . . . . . . . . . . 142A, Aα , C∗-algebra . . . . . . . . . . . . . . . . 131a∗, creation operator . . . . . . . . . . . . . . 142(A,‖ · ‖,∗), C∗-algebra . . . . . . . . . . . . 104A∗, adjoint of the operatorA . . . . . . . 159A×, invertible elements inA . . . . . . . 106A , B , categories . . . . . . . . . . . . . . . . . . 155|dx|, density provided by a local chart

167|ΛM|, bundle of densities overM . . 167αO2,O1, morphism

CCRSYMPL(ιO2,O1) inC ∗−Alg . . . . . . . . . . . . . . . . . 132

, isomorphismTM → T∗M induced byLorentzian metric . . . . . . . . 25

B, BO, C∗-algebra . . . . . . . . . . . . . . . 134Br(p), open Riemannian ball of radius

r aboutp in Cauchy hypersur-face . . . . . . . . . . . . . . . . . . . . . 81

C(α,n), coefficients involved in defini-tion of Riesz distributions . 10

C+(0), future light cone in Minkowskispace. . . . . . . . . . . . . . . . . . . .10

C−(0), past light cone in Minkowskispace. . . . . . . . . . . . . . . . . . . .10

C∞sc(M,E), smooth sections with space-

like compact support . . . . . .90C∞(A), set ofC∞-vectors forA . . . . . 160C∞(X), space of smooth functions onX

158C∞

0 (X), smooth functions vanishing atinfinity . . . . . . . . . . . . . . . . . 105

C0(X), continuous functions vanishingat infinity . . . . . . . . . . . . . . .105

C ∗−Alg , category ofC∗-algebras and in-jective∗-morphisms . . . . . 122

CCR, functorSymplVec → C ∗−Alg . . 122CCR(S), ∗-morphism induced by a sym-

plectic linear map . . . . . . . 122CCR(V,ω), CCR-algebra of a symplec-

tic vector space . . . . . . . . . 117Ck(M,E), space ofCk-sections inE 166[Aα ,Aβ

], commutator ofAα and Aβ131

, composition of morphisms in a cate-gory . . . . . . . . . . . . . . . . . . . 155

C0(X), space of continuous functions onX . . . . . . . . . . . . . . . . . . . . . . 157

C0(X), space ofk times continuouslydifferentiable functions onX157

d, exterior differentiation . . . . . . . . . .172D(S), Cauchy development of a subsetS

22D(M,E), compactly supported smooth

sections in vector bundleEoverM . . . . . . . . . . . . . . . . . . . 2

D ′(M,E,W), W-valued distributions inE . . . . . . . . . . . . . . . . . . . . . . . . 2

Dm(M,E∗), compactly supportedCm-sections inE∗ . . . . . . . . . . . . . 3

δ , codifferential . . . . . . . . . . . . . . . . . . 124δx, delta-distribution . . . . . . . . . . . . . . . . 3∂X f := d f(X) . . . . . . . . . . . . . . . . . . . 168

∂∂xj , local vector field provided by a

chart . . . . . . . . . . . . . . . . . . . 166d f , differential of f . . . . . . . . . . . . . . 163dp f , differential of f at p . . . . . . . . . 163divX, divergence of vector fieldX . . .25dom(A), domain of a linear operatorA

159dV, semi-Riemannian volume density

167

Symbols 187

dxj , local 1-form provided by a chart166

E⊠F∗, exterior tensor product of vectorbundlesE andF∗ . . . . . . . . . .4

E∗, dual vector bundle . . . . . . . . . . . . 165Ep, fiber ofE abovep . . . . . . . . . . . . 164ε j := g(ej ,ej ) = ±1 . . . . . . . . . . . . . . 170E⊗F, tensor product of vector bundles

165E⊗R F , real tensor product of complex

vector bundles . . . . . . . . . . 167ext, extension of section. . . . . . . . . . .126

F±(x), global fundamental solution . .87FΩ± (·), fundamental solution for domain

Ω . . . . . . . . . . . . . . . . . . . . . . . 55Falg(H), algebraic symmetric Fock

space ofH . . . . . . . . . . . . . .141F (H), symmetric Fock space ofH 141Z, isomorphismE → E∗ induced by an

inner product . . . . . . . . . . . 123f (A), function f applied to selfadjoint

operatorA . . . . . . . . . . . . . . 160

γ, quadratic form associated toMinkowski product . . . . . . . 10

G = G+ −G− . . . . . . . . . . . . . . . . . . . . . 90G+, advanced Green’s operator . . . . . 89G−, retarded Green’s operator . . . . . . 89Γx = γ exp−1

x . . . . . . . . . . . . . . . . . . . . 28Γ(x,y) = Γx(y) . . . . . . . . . . . . . . . . . . . . 45Γ(A), graph ofA . . . . . . . . . . . . . . . . . 159GOp, geometric normally hyperbolic

operator . . . . . . . . . . . . . . . . 153GlobHyp , category of globally hyper-

bolic manifolds equipped withformally selfadjoint normallyhyperbolic operators . . . . . 125

GlobHyp naked , category of globally hy-perbolic manifolds withoutfurther structure . . . . . . . . . 153

gradf , gradient of functionf . . . . . . 168

Hn1 , pseudohyperbolic space . . . . . . . . 95

Hess( f )|x, Hessian of function f atpointx . . . . . . . . . . . . . . . . . . 26

Hn1 , anti-deSitter spacetime . . . . . . . . . 98

(·, ·), Hermitian scalar product . . . . . 140HomK(E,F), bundle of homomor-

phisms between two bundles165

I+(0), chronological future inMinkowski space . . . . . . . . . 10

IM+ (x), chronological future of pointx in

M . . . . . . . . . . . . . . . . . . . . . . .18I−(0), chronological past in Minkowski

space. . . . . . . . . . . . . . . . . . . .10IM− (x), chronological past of pointx in

M . . . . . . . . . . . . . . . . . . . . . . .18IM+ (A), chronological future of subsetA

of M . . . . . . . . . . . . . . . . . . . . 18IM− (A), chronological past of subsetA of

M . . . . . . . . . . . . . . . . . . . . . . .18idA, identity morphism ofA . . . . . . . 156Im, imaginary part . . . . . . . . . . . . . . . 145〈·, ·〉, (nondegenerate) inner product on

a vector bundle. . . . . . . . . .123ιO2,O1, morphism inLorFund induced by

inclusionO1 ⊂ O2 . . . . . . .132

J+(0) causal future in Minkowski space10

JM+ (x), causal future of pointx in M . 18

J−(0) causal past in Minkowski space10JM− (x), causal past of pointx in M . . . 18

JM+ (A), causal future of subsetA of M18

JM− (A), causal past of subsetA of M . 18

JM(A) := JM− (A)∪JM

+ (A) . . . . . . . . . . .18

K±(x,y), error term for approximatefundamental solution . . . . . 47

K, field R or C . . . . . . . . . . . . . . . . . . . . . 2K±, integral operator with kernelK±53

L (H), bounded operators on HilbertspaceH . . . . . . . . . . . . . . . . 104

L[c], length of curvec. . . . . . . . . . . . . . 24L1

loc(M,E), locally integrable sections invector bundleE . . . . . . . . . . . 3

L 2(X), space of square integrable func-tions onX . . . . . . . . . . . . . . 158

188 Symbols

L2(X), space of classes of square inte-grable functions onX . . . . 158

L2(Σ,E∗), Hilbert space of square inte-grable sections inE∗ over Σ149

ΛkT∗M, bundle ofk-forms onM . . . 165[X,Y], Lie bracket ofX with Y . . . . . 170LorFund , category of timeoriented

Lorentzian manifoldsequipped with formallyselfadjoint normally hyper-bolic operators and Green’soperators . . . . . . . . . . . . . . . 128

m, mass of a spin-1 particle . . . . . . . .124µx, local density function . . . . . . . . . . .20Mor(A,B), set of morphisms fromA to

B . . . . . . . . . . . . . . . . . . . . . . 155

n, unit normal field . . . . . . . . . . . . . . . . 25∇, connection on a vector bundle . . 123‖ · ‖1, norm on〈W(V)〉 . . . . . . . . . . . .119‖ · ‖max, C∗-norm on〈W(V)〉 . . . . . . 120

Obj(A ), class of objects in a categoryA155

ω , symplectic form . . . . . . . . . . . . . . . 129ω , skew-symmetric bilinear form induc-

ing the symplectic formω129Ω, vacuum vector . . . . . . . . . . . . . . . . 141⊥, orthogonality relation in a set . . . 130

P(2), operatorP applied w. r. t. secondvariable . . . . . . . . . . . . . . . . . 43

Φ, morphism CCRSYMPL( f ,F) inC ∗−Alg . . . . . . . . . . . . . . . . . 134

ΦΣ, quantum field defined byΣ . . . . 149Φ(y,s) = expx(s·exp−1

x (y)) . . . . . . . . 40π , projection from the total space of a

vector bundle onto its base164Πc, parallel transport along the curvec

169Πx

y : Ex → Ey, parallel translation alongthe geodesic fromx to y . . . 40

Q, twist structure . . . . . . . . . . . . . . . . . 148

QuasiLocA lg , category of quasi-localC∗-algebras . . . . . . . . . . . . . . . . 131

QuasiLocA lgweak

, category of weakquasi-localC∗-algebras . . 131

R, curvature tensor of a connection .170RΩ

+(α,x), advanced Riesz distributionon domainΩ at pointx. . . .29

RΩ−(α,x), retarded Riesz distribution on

domainΩ at pointx . . . . . . 29R+(x), formal advanced fundamental

solution . . . . . . . . . . . . . . . . . 40R−(x), formal retarded fundamental so-

lution . . . . . . . . . . . . . . . . . . . 40RN+k

± (x), truncated formal fundamentalsolution . . . . . . . . . . . . . . . . . 57

R+(x), approximate advanced funda-mental solution . . . . . . . . . . 53

R−(x), approximate retarded funda-mental solution . . . . . . . . . . 53

rA(a), resolvent set ofa∈ A . . . . . . . 107res, restriction of a section . . . . . . . . 128ρA(a), spectral radius ofa∈ A . . . . . 107ric, Ricci curvature . . . . . . . . . . . . . . . 170

scal, scalar curvature . . . . . . . . . . . . . 170σA(a), spectrum ofa∈ A. . . . . . . . . .107σL, principal symbol ofL . . . . . . . . . 171SOLVE, functor GlobHyp → LorFund

128singsupp(T), singular support of distri-

butionT . . . . . . . . . . . . . . . . . . 7supp(T), support of distributionT . . . .6⊙n

algH, algebraicnth symmetric tensorproduct ofH . . . . . . . . . . . .140⊙nH, nth symmetric tensor product ofH . . . . . . . . . . . . . . . . . . . . . .140

SymplVec , category of symplectic vectorspaces and symplectic linearmaps . . . . . . . . . . . . . . . . . . .122

SYMPL, functor LorFund → SymplVec129

τ(p,q), time-separation of pointsp,q24T∗M, cotangent bundle ofM . . . . . . 165θ , Segal field . . . . . . . . . . . . . . . . . . . . 143

Symbols 189

Θϕ , bijective mapTpM → Rn inducedby a chartϕ . . . . . . . . . . . . 163

θr , auxiliary function in proof of globalexistence of fundamental so-lutions . . . . . . . . . . . . . . . . . . 82

TM, tangent bundle ofM . . . . . . . . . 163TpM, tangent space ofM at p . . . . . . 163

u±, solution for inhomogeneous waveequation . . . . . . . . . . . . . . . . . 62

Vkx , Hadamard coefficient at pointx . 40

Vk(x,y) = Vkx (y) . . . . . . . . . . . . . . . . . . . 42

V(M,E,G) = D(M,E)/ker(G) . . . . 129

W(ϕ), Weyl-system . . . . . . . . . . . . . . 116〈W(V)〉, linear span of theW(ϕ), ϕ ∈V

119

x≤ y, causality relation . . . . . . . . . . . . 18x<y, (strict) causality relation. . . . . . .18

190 Index

IndexSymbols

∗–automorphism . . . . . . . . . . . . . . . . .114∗–morphism . . . . . . . . . . . . . . . . . . . . .114

Aacausal subset . . . . . . . . . . . . . . . . . . . . .21achronal subset . . . . . . . . . . . . . . . . . . . .21adjoint operator . . . . . . . . . . . . . . . . . .159advanced fundamental solution . .38, 55advanced Green’s operator . . . . . .89, 92advanced Riesz distributions . . . . 12,29algebra of continuous functions vanish-

ing at infinity . . . . . . . . . . .105algebra of smooth functions vanishing at

infinity . . . . . . . . . . . . . . . . .105algebraic symmetric Fock space . . .141algebraic symmetric tensor product140analytic vector . . . . . . . . . . . . . . . . . . .160annihilation operator . . . . . . . . . . . . . .143anti-deSitter spacetime . . . . . 95,98, 136approximate fundamental solution . .53,

57associativity law . . . . . . . . . . . . . . . . .156asymptotic expansion of fundamental

solution . . . . . . . . . . . . . . . . .59atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . .160

BBanach space . . . . . . . . . . . . . . . . . . . .157base of a vector bundle. . . . . . . . . . . .165Bogoliubov transformation . . . . . . . .122bosonic quasi-localC∗-algebra . . . .131bounded operator . . . . . . . . . . . . . . . . .159bundle ofk-forms onM . . . . . . . . . . .165

CC∗-algebra . . . . . . . . . . . . . . . . . . . . . . .104

– quasi-local . . . . . . . . . . . . . . . .131– weak quasi-local . . . . . . . . . . .131

C∗-norm. . . . . . . . . . . . . . . . . . . . . . . . .104C∗-property . . . . . . . . . . . . . . . . . . . . . .104C∗-subalgebra. . . . . . . . . . . . . . . . . . . .105C∗-subalgebra generated by a set . . .105

C∞-vector . . . . . . . . . . . . . . . . . . . . . . .160C0-norm. . . . . . . . . . . . . . . . . . . . . . . . .157canonical anticommutator relations153canonical commutator relations . . .116,

143, 145, 152category . . . . . . . . . . . . . . . . . . . . . . . . .155Cauchy development of a subset22, 81,

176Cauchy hypersurface . . . . . . . . . . . . . .21Cauchy problem . . . . . 67, 73,76, 79, 85Cauchy time-function . . . . . . . . . . . . . .24causal curve . . . . . . . . . . . . . . . . . . . . . .17causal domain . . . . . . . . . . . . . . . . . . . . .20causal future . . . . . . . . . . . . . . . . . . . . . .18causal past . . . . . . . . . . . . . . . . . . . . . . . .18causality condition . . . . . . . . . . . . .22, 68causally compatible subset .19, 92, 126,

175, 177causally independent subsets . .132, 176CCR-algebra. . . . . . . . . . . . . . . . . . . . .121CCR-representation . . . . . . . . . . . . . .121chain rule. . . . . . . . . . . . . . . . . . . . . . . .164chart . . . . . . . . . . . . . . . . . . . . . . . . . . . .160chronological future . . . . . . . . . . . . . . .18chronological past . . . . . . . . . . . . . . . . .18Ck-norm . . . . . . . . . . . . . . . . . . . . . .2, 157closable operator . . . . . . . . . . . . . . . . .159closed operator . . . . . . . . . . . . . . . . . . .159closure of an operator . . . . . . . . . . . . .159composition of morphisms in a category

155conformal change of metrics . . . . . . . .93connection induced on powers of vector

bundles . . . . . . . . . . . . . . . .169connection on a vector bundle .168, 172connection-d’Alembert operator . . . .34convex domain . . . . . . . . . . . . . . .20, 174coordinates in a manifold . . . . . . . . .160cotangent bundle . . . . . . . . . . . . . . . . .165covariant derivative . . . . . . . . . . . . . . .168covariant functor . . . . . . . . . . . . . . . . .156covariant Klein-Gordon operator . . .124

Index 191

creation operator . . . . . . . . . . . . . . . . .143current density in electrodynamics .125curvature tensor . . . . . . . . . . . . . . . . . .170

Dd’Alembert operator . . .11,26, 135, 148delta-distribution . . . . . . . . . . . . . . . . . . .3densely defined operator . . . . . . . . . .159density on a manifold . . . . . . . . . . . . .167derivation . . . . . . . . . . . . . . . . . . . . . . .169diffeomorphism . . . . . . . . . . . . . . . . . .164differentiable manifold . . . . . . . . . . . .160differentiable map . . . . . . . . . . . . . . . .162differential form. . . . . . . . . . . . . . . . . .166differential of a map . . . . . . . . . . . . . .163differential operator . . . . . . . . . . . . 5,170dimension of a manifold . . . . . . . . . .160Dirac operator . . 34,125, 135, 148, 153directed set . . . . . . . . . . . . . . . . . . . . . .130

– with orthogonality relation . .130– with weak orthogonality relation

131distribution . . . . . . . . . . . . . . . . . . . . . . . .2divergence of a vector field . . . .25, 171domain of a linear operator . . . . . . . .159dual vector bundle . . . . . . . . . . . . . . . .165

EEinstein cylinder . . . . . . . . . . . . . . . . . .93electrodynamics . . . . . . . . . . . . .125, 149electromagnetic field . . . . . . . . . 153,153embedding . . . . . . . . . . . . . . . . . . . . . .164essentially selfadjoint operator . . . . .159extension of an operator . . . . . . . . . . .159exterior differential . . . . . . . . . . 124,172

Ffield operator . . . . . . . . . . . . . . . . . . . .149field strength in electrodynamics . .125,

135Fock space . . . . . . . . . . . . . 139,141, 149forgetful functor . . . . . . . . . . . . . . . . .157formal adjoint of differential operator .5formal advanced fundamental solution

40, 44, 57

formal retarded fundamental solution40,44, 57

formally selfadjoint differential operator123, 128

Frechet space . . . . . . . . . . . . . . . . . . . .158full subcategory . . . . . . . . . . . . . . . . . .156functor . . . . . . . . . . . . . . . . . . . . . 123,156fundamental solution .38, 56, 68, 75,87future compact subset . . . . . . . . . . . . . .18future compact support . . . . . . . . . . . . .68future-directed vector . . . . . . . . . . . . . .10future-stretched subset . . . . . . . . . . . . .52

GGauss’ divergence theorem . . . . . . . . .25geodesically convex domain . . . . . . . .20geodesically starshaped domain . . . . .20geometric normally hyperbolic operator

153globally hyperbolic manifold . . .23, 72,

175, 177gradient of a function . . . . . 24,168, 171graph of an operator . . . . . . . . . . . . . .159Green’s operator . . . . . . . . .88, 127, 150

HHaag-Kastler axioms . . . . . . . . . . . . .136Hadamard coefficients . . . . . . . . . .40, 42Hermitian metric . . . . . . . . . . . . . . . . .167Hermitian scalar product . . . . . . . . . .140Hessian of a function . . . . . . . . . . . . . .26Hilbert space . . . . . . . . . . . . . . . . . . . .157hypersurface . . . . . . . . . . . . . . . . . . . . .162

Iidentity morphism . . . . . . . . . . . . . . . .156immersion . . . . . . . . . . . . . . . . . . . . . . .164inextendible curve . . . . . . . . . . . . . . . . .18inhomogeneous wave equation . . . . . .62inner product on a vector bundle . . .167integration on a manifold. . . . . . . . . .167inverse function theorem . . . . . . . . . .164isometric element of aC∗–algebra .111

JJacobi identity . . . . . . . . . . . . . . . . . . .170

192 Index

Kk-form . . . . . . . . . . . . . . . . . . . . . . . . . .166Klein-Gordon operator . .124, 135, 148,

153

LLeibniz rule . . . . . . . . . . . . . . . . . . . . .169length of a curve . . . . . . . . . . . . . . . . . .24Levi-Civita connection . . . . . . . 124,170Lie bracket . . . . . . . . . . . . . . . . . . . . . .170lightlike curve. . . . . . . . . . . . . . . . . . . . .17lightlike vector . . . . . . . . . . . . . . . . . . . .10local trivialization . . . . . . . . . . . . . . . .165locally covariant quantum field theory

153Lorentz gauge . . . . . . . . . . . . . . . . . . .125Lorentzian metric . . . . . . . . . . . . . . . .167Lorentzian orthonormal basis . . . . . . .11

Mmanifold

– Lorentzian . . . . . . . . . . . .167, 172– Riemannian . . . . . . . . . . . . . . .167– smooth . . . . . . . . . . . . . . . . . . . .161– topological . . . . . . . . . . . . . . . .160

map between manifolds . . . . . . . . . . .162Maxwell equations . . . . . . . . . . . . . . .125metric connection on a vector bundle

123,169morphism between (weak) quasi-local

C∗-algebras . . . . . . . . . . . . .131morphism in a category . . . . . . . . . . .155

NNelson’s theorem. . . . . . . . . . . . . . . . .160normal coordinates . . . . . . . . . . . . . . . .26normal element of aC∗-algebra . . . .111normally hyperbolic operator . . . . . . .33

Oobject in a category . . . . . . . . . . . . . . .1551-form . . . . . . . . . . . . . . . . . . . . . . . . . .166open mapping theorem. . . . . . . . . . . .159operator norm. . . . . . . . . . . . . . . . . . . .104order of a differential operator . . . . .171order of a distribution . . . . . . . . . . . . . . .3

orientable manifold . . . . . . . . . . . . . . .167orientable vector bundle. . . . . . . . . . .167orthogonality relation . . . . . . . . . . . . .130

Pparallel transport . . . . . . . . . . . . . . . . .169past compact subset . . . . . . . . . . . . . . .18past compact support . . . . . . . . . . . . . .68past-stretched subset . . . . . . . . . . . . . . .52P-compatible connection . . . . . . . . . . .36primitiveC∗–algebra . . . . . . . . . . . . .137principal symbol of a differential opera-

tor . . . . . . . . . . . . . . . . . . . . .171Proca equation . . . . .124, 135, 149, 153product manifold . . . . . . . . . . . . . . . . .161pseudohyperbolic space . . . . . . . . . . . .95

Qquantum field . . . . . . . . . . . . . . . 147,149quantum mechanical wave function124

Rrank of a vector bundle. . . . . . . . . . . .165RCCSV-domain . . . . . . . . . . . . . . . . . . .79resolvent . . . . . . . . . . . . . . . . . . . . . . . .107resolvent set . . . . . . . . . . . . . . . . . . . . .107retarded fundamental solution . . .38, 55retarded Green’s operator . . . . . . .89, 92retarded Riesz distributions . . . . . 12,29Ricci curvature . . . . . . . . . . . . . . . . . . .170Riemannian curvature tensor . . . . . .170Riemannian metric . . . . . . . . . . . . . . .167Riesz distributions on a domain . . . . .29Riesz distributions on Minkowski space

12Riesz-Fisher theorem . . . . . . . . . . . . .158

Sscalar curvature . . . . . . . . . . . . . . . . . .170second quantization . . . . . . . . . . . . . .140section in a vector bundle . . . . . . . . .165Segal field . . . . . . . . . . . . . . . . . . . . . . .143selfadjoint element of aC∗-algebra .105selfadjoint operator . . . . . . . . . . . . . . .159semi-norm . . . . . . . . . . . . . . . . . . . . . . .158semi-Riemannian metric . . . . . . . . . .167

Index 193

separating points (semi-norms) . . . .158simpleC∗–algebra . . . . . . . . . . .120, 122singular support of a distribution . . . . .7smooth atlas . . . . . . . . . . . . . . . . . . . . .161smooth map . . . . . . . . . . . . . . . . . . . . .162spacelike compact support . . . . . . . . . .90spacelike curve . . . . . . . . . . . . . . . . . . . .17spacelike vector . . . . . . . . . . . . . . . . . . .10spectral radius . . . . . . . . . . . . . . . . . . .107spectrum . . . . . . . . . . . . . . . . . . . . . . . .107spinor bundle . . . . . . . . . . . . . . . .125, 148square integrable function . . . . . . . . .158strictly future-stretched subset . . . . . .52strictly past-stretched subset . . . . . . . .52strong causality condition . . . . . . . . . .22submanifold . . . . . . . . . . . . . . . . . . . . .162submersion . . . . . . . . . . . . . . . . . . . . . .164support of a distribution . . . . . . . . . . . . .6supremum norm. . . . . . . . . . . . . . . . . .157symmetric operator . . . . . . . . . . . . . . .159symmetric tensor product . . . . . . . . .140symplectic linear map . . . . . . . . . . . .122symplectic vector space . . . . . . . . . . .116

Ttangent bundle of a manifold . .163, 165tangent map . . . . . . . . . . . . . . . . . . . . .163tangent space . . . . . . . . . . . . . . . . . . . .163test sections . . . . . . . . . . . . . . . . . . . . . . . .2time-separation . . . . . . . . . . . . . . . .24, 68time-slice axiom . . . . . . . . . . . . . . . . .136timelike curve . . . . . . . . . . . . . . . . . . . . .17timelike vector . . . . . . . . . . . . . . . . . . . .10timeorientation . . . . . . . . . . . . . . . . . . . .10topology of a Frechet space . . . . . . .158total space of a vector bundle . . . . . .165trivial vector bundle . . . . . . . . . .165, 166trivialization . . . . . . . . . . . . . . . . . . . . .165truncated formal fundamental solution

57twist structure of spink/2 . . . . . . . . .148

Uunbounded operator . . . . . . . . . . . . . .159unitary element of aC∗-algebra . . . .111

universal property of the symmetric ten-sor product . . . . . . . . . . . . .140

Vvacuum vector . . . . . . . . . . . . . . . . . . .141vector bundle . . . . . . . . . . . . . . . . . . . .164vector field . . . . . . . . . . . . . . . . . . . . . .166vector potential in electrodynamics.125vector-bundle-homomorphism167, 171

Wwave equation. . . . . . . . . . . . . . . . . . . . .37weak topology for distributions . . . . . .7Weitzenbock formula . . . . . . . .124, 125Weyl system . . . . . . . . . . . .116, 145, 152

YYamabe operator . . . . .98, 124, 135, 148

Zzero section . . . . . . . . . . . . . . . . . . . . . .166

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