UV-Finite Gauge Theory and Quantum Gravity∗

C. F. Richardson

It is shown that UV-finite Lorentz covariant and gauge invariant quantum field theoriescan be constructed based on the premise that Lorentzian spacetime is emergent from astatistically averaged ensemble of discrete “pre-spacetimes” having Euclidean signature viaa local Wick rotation. A continuum approximation to the underlying discrete theory isdeveloped by expanding fields in terms of a finite number of eigenfunctions of an appropriategauge covariant Laplacian. The resulting Euclidean spacetime theory is gauge invariantand rotation covariant. It is shown that Wick rotation is uniquely defined locally withrespect to a frame at rest relative to a background gravitational field such that rotationcovariant results in Euclidean spacetime are mapped to Lorentz covariant results upon Wickrotation. Starting from Hamiltonian formulations, partition functions are derived for Yang-Mills theory and for quantum gravity that contain only a finite number of degrees of freedom.A perturbative expansion for the eigenfunctions is developed and it is shown that there areno UV divergences in the theory. For the case of Yang-Mills theory, quantities that aredivergent in the standard approach are replaced with finite quantities that depend on thelength scale, Ld, of the discrete pre-spacetimes. In the case of quantum gravity, it is shownthat a perturbation expansion in powers of LPl/Ld can be carried out if it is assumed thatthis is a small quantity. As concrete examples of the formalism, vacuum polarization andelectron self-energy at one loop order in QED are calculated and standard results are obtainedwith a finite physical cutoff of 2π/Ld.

I. INTRODUCTION

There are various indications that there may be only a finite number of degrees of freedomassociated with any finite volume of spacetime[1–5]. Traditional quantum field theory associates aninfinite number of degrees of freedom with each finite volume of spacetime and it has proven difficultto modify quantum field theory to have a finite number of degrees of freedom without spoilingLorentz covariance and/or gauge invariance[6]. Here we construct field theories having a finitenumber of degrees of freedom while maintaining both gauge invariance and Lorentz covariance. Wedo this by considering an emergent spacetime theory that includes a discrete structure and thendeveloping truncated continuous spacetime theories as continuous approximations to the discretetheory.

It is well known that one can Wick rotate from a description in Lorentzian spacetime to a de-scription in Euclidean spacetime. The generating functional for the Green’s functions in Lorentzianspacetime becomes a partition function in Euclidean spacetime. If we interpret the Euclidean space-time partition function as describing a statistical mechanical system, then it is straightforward toconstruct a rotationally covariant partition function for a system of extended 4-dimensional objectsin Euclidean spacetime - simply include in the definition of the partition function an average overall orientations of the arrangement of the extended objects. That spacetime or gravity may berelated to thermodynamics has been suggested previously; see for example, [7–11]. We considerthe extended objects to be components of a “pre-spacetime”. Lorentzian spacetime may be thenbe defined by averaging over all possible arrangements of the extended 4-d objects and then Wickrotating in a specific reference frame that we will describe below.

∗ This paper is available at http://archive.org/details/DiscreteEmergentSpacetime;and at http://uvfinite.weebly.com;Originally uploaded February 8, 2014; Revised April 30, 2015.

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In this “pre-spacetime” picture, quantum gravity can be described as assigning weights toconfigurations of the extended objects according to the geometry defined by each configuration.Quantum field theory can be described as the statistical mechanics of fields defined on the config-urations of the extended objects. Such an approach can be described as an “emergent spacetime”approach. That spacetime may be emergent has been previously suggested in various forms; seefor example, [12–21].

If we interpret the partition function as defining a thermodynamic average, this average shouldbe considered a local thermodynamic equilibrium (LTE) approximation. Locally, a frame canalways be found where the background spacetime metric is independent of time t up to termscubic in coordinate displacements and we will assume that LTE holds in this frame[22]. A localWick rotation can be unambiguously defined in such a frame. In other words, a background metricspecifies a frame in which LTE holds and in this frame a local Wick rotation is uniquely defined.As we will see in more detail below, this procedure results in a cutoff. However, since all observersagree on which frame this cutoff is specified, Lorentz covariance is maintained.

Since partition functions are only determined locally in this approach, we would have to piecelocal solutions together to determine global physics. Assuming scattering occurs locally, n-pointGreen’s functions could be extended to arbitrary regions of spacetime by determining the inter-acting n-point Green’s function in a local region, assuming that the Green’s functions are wellapproximated by combinations of 2-point Green’s functions on the boundary of the local region,and then extending the 2-point functions by solving the appropriate wave equation.

If we understood the details of the states and the interactions of the extended 4-d objects, wewould be able to define a partition function by summing over all such states. One approach to doingquantum field theory in this picture, would be to define a lattice gauge theory on an arbitrary lattice,perform standard calculations of correlation functions between fields defined at two coordinatevalues (defining a field at a coordinate point as the field on the 4-d object intersecting the coordinatepoint, for example) and average over all possible lattices. Since such an average includes an averageover all orientations, the resulting correlation functions will be SO(4) covariant and give Lorentzcovariant Green’s functions upon Wick rotation. However, such an approach is difficult to trackanalytically (beyond simply averaging simple cubic lattices over orientations and displacements)and difficult to extend to the case of quantum gravity without a better understanding of theunderlying states.

Instead of an approach involving averaging discrete degrees of freedom, we would like to developcontinuous approximations to the discrete structures that retain the essential feature of havingonly a finite number of degrees of freedom in a finite 4-volume of spacetime. To do so, we need todevelop a truncated theory in Euclidean spacetime and then Wick rotate the result to Lorentzianspacetime. It is straightforward to maintain Lorentz covariance in the sense discussed above in suchan approach - we simply require rotation (i.e., SO(4)) covariance in our truncated theory. It is alsonecessary to maintain gauge invariance in the truncated theory. We can achieve this by expanding ina truncated set of eigenfunctions of the appropriate gauge (and rotation) covariant Laplacian. Sucheigenfunctions are not generally known exactly and we will describe a perturbative approach fortheir evaluation. Denoting the length scale associated with the underlying discrete pre-spacetimeby Ld, the eigenfunctions can be labeled by a 4-vector with a cutoff of 2π/Ld. (This is, however,not a simple momentum cutoff which would not preserve gauge invariance). Note that Ld need notbe identified with the Planck length. It is possible that the discrete structure characterized by thelength scale Ld represents a fundamental building block of nature. Alternatively it is possible thatthe discrete structure arises as an effective property of some more fundamental theory. For thepurposes of the present paper where we are concerned with truncated continuous approximations,it does not matter whether we consider the discrete structure to be truly fundamental or merelyeffective.

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In the following, we develop truncated rotation and gauge covariant theories generally and deriveexpressions for the resulting partition function for Yang-Mills theory and for quantum gravity. Weargue that such theories are unitary since they can be derived from Hamiltonian formulations andare UV-finite since the fields are expanded in a finite basis. As examples of how to apply thetheory, we calculate the vacuum polarization in scalar and spinor QED and the electron self-energyfor momenta small compared to 2π/Ld in spinor QED. We obtain the standard results with noUV divergences encountered in the calculations. Rather, the cutoff appearing in the standardapproaches is replaced with the finite physical quantity 2π/Ld.

For Euclidean quantum gravity there is the well-known conformal mode problem. We willdescribe the approach of Schleich[23] and argue that this provides an entirely satisfactory solutionto the conformal mode problem in our approach. A perturbative expansion for quantum gravityin terms of eigenfunctions of the covariant Laplacian results in a power series in LPl/Ld. If weassume that this ratio of length scales is small compared to unity, then this approach provides asensible perturbative expansion for quantum gravity that is unitary (since the partition functionfollows from a Hamiltonian formulation) and that preserves general covariance. Interestingly, it isprecisely because quantum gravity is non-renormalizable that such an expansion scheme is possible.

This paper is organized as follows. In Section II, we describe a unique local Wick rotationand then in Section III we show how this ties in to Lorentz covariance. In Section IV, we showthat expanding in a finite number of eigenfunctions of a gauge covariant Laplacian preserves gaugeinvariance and we show how to determine the eigenfunctions. In Section V, we illustrate ourapproach for deriving a functional integral for a system with a finite number of degrees of freedomfor a scalar field and we extend this to the Yang-Mills case in Section VI and to quantum gravity inSection VII. We apply our approach to 1-loop calculations in QED in Section VIII and, finally, wediscuss some possible consequences of the theory in Section IX. We use natural units throughout.

II. WICK ROTATION

As explained in the introduction, it is natural to describe the partition function in quantumfield theory in Euclidean time. In the sections below, we will consider quantum gravity as well asYang-Mills theory and we will need to be able to define a Wick rotation in a curved spacetime.In considering quantum gravity, we would like to interpret the partition function as a statisticalaverage over discrete degrees of freedom. This implies a real metric description in Euclidean time.The Wick rotation to Lorentzian spacetime should also result in a real metric if it is to be physicallymeaningful. However, it is not possible for a general spacetime to have a real valued metric in theEuclidean time description that is Wick rotated to a real valued metric in Lorentzian time, unlessone modifies the definition of Wick rotation[24, 25]. This difficulty is due to the fact that anon-stationary spacetime generally does not represent a system that is in global thermodynamicequilibrium, but local thermodynamic equilibrium could be a reasonable approximation. Thatthermodynamics of spacetime should be understood locally has been emphasized in [26, 27]. Wewill now show that the notion of local thermodynamic equilibrium allows us to to define a uniquelocal Wick rotation in such a way that both the Lorentzian and Euclidean metrics are real.

The metric for an arbitrary stationary spacetime[28] can be written in the form

d2s = −N(~x)2(dt− wi(~x)dxi)2 + (δij + gij(~x))dxidxj (1)

where N(~x)2, wi(~x) and gij(~x)) are independent of the time t, and i and j represent spatialcoordinates. Wick rotation can be defined by t → −iτ and wi → iwi. Every spacetime is locallystationary. That is around any given point, any Lorentzian metric can be written in the form

d2s = −dt2(1 + n(~x)) + (δij + gij(~x))dxidxj + dtdxivi(~x) +O(δ3)dxαdxβ, (2)

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where n(~x), gij(~x),and vi(~x) are independent of the time coordinate t, δ is the coordinate distancefrom the reference point which is taken to be ~x = 0, t = 0. To order δ2, n, gij , and vi(~x) arequadratic in ~x. Note that equation (2) is of the form of equation (1) with N = 1 + n/2 andwi = vi/2. As we show in Appendix A, this result follows from starting with normal coordinates(gαβ = ηαβ; gαβ,γ = 0) at a point and considering coordinate transformations of the form

xα = xα +Aαβγδxβxγ xδ. (3)

The coefficientsAαβγδ can be chosen to eliminate all terms linear or quadratic in time. By eliminatingtime dependence from the metric, we obtain upon Wick rotation, a time-independent metric inEuclidean spacetime and so we can extend the Euclidean time coordinate τ so that it ranges from0 to β giving a natural thermodynamic interpretation. The Wick rotated metric is locally

d2s = dτ2(1 + n(~x)) + (δij + gij(~x))dxidxj + dτdxivi(~x) +O(δ3)dxαdxβ, (4)

Note that the term dtdxivi(~x) is the only term in equation (2) which gives a nonzero contributionto R0i which is proportional to T0i by the Einstein equation. Since T0i is proportional to a velocity,it is natural to define Wick rotation by t→ −iτ and simultaneously vi → ivi. This interpretationalso follows for Ricci-flat spacetime since the twist, which is wi in equation (1), is related to arotation in the spacetime geometry as is familiar from the Kerr black hole geometry.

Our proof (see Appendix A)of equation (2) shows the the coordinate system is uniquely specifiedup to spatial transformations and up to coordinate transformations of the form

t = t+Aijkxixjxk. (5)

with Aijk = A(ijk). Spatial coordinate transformations do not affect the definition of the timecoordinate and thus do not affect the Wick rotation. The local Wick rotation as we have defined itis a map from a real Lorentzian geometry to a real Euclidean geometry. Transformations of the formof equation (5) shift vi as vi → v′i = vi + 6Aijkx

jxk. Under Wick rotation, v′i → iv′i. This defines aWick rotated metric given by equation (4) with vi replaced by v′i. The modified Euclidean metric isequivalent (i.e., defines the same geometry) to the Wick rotated metric determined without makingthe transformation (5). This can be seen by considering the transformation

τ = τ −Aijkxixjxk. (6)

which results in the original Wick rotated metric. It follows that our prescription for local Wickrotation in uniquely defined. This is an intuitively reasonable result: the gravitational field (i.e.,the metric) defines a local Lorentz frame where the metric is locally stationary. In this frame,the degrees of freedom underlying spacetime are in local thermodynamic equilibrium which can bedescribed in terms of the Wick rotated metric. The prescription for Wick rotation is unique sinceany boost to the local frame would give a time dependent metric which cannot be directly Wickrotated. This is reasonable since degrees of freedom which are in a thermal distribution in oneframe will not have a thermal distribution in a boosted frame.

The upper limit of the τ integral in defining a partition function is typically denoted β. Inorder to unambiguously interpret β as an inverse temperature, we need it to be a physical distancenot just a coordinate distance, because the temperature should not depend on an arbitrary choiceof coordinates. If we write the Euclidean metric in the form of equation (4) then the coordinatedistance is the physical distance along ~x = 0 and we can unambiguously interpret this distance asthe inverse temperature that would be measured by observers at ~x = 0. The inverse temperaturefor ~x 6= 0 is then the physical length of a period for ~x 6= 0 and this is given by

1

T (~x)= β

(1 + n(~x)

)1/2=|g00(~x)|1/2

T (~x = 0). (7)

This implies that T (~x)|g00(~x)|1/2 is constant which is the Tolman-Ehrenfest effect[29–31].

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III. LORENTZ COVARIANCE

The theories that we discuss below require a cutoff. Since we have a unique frame defined bythe metric, all observers agree on which frame this cutoff is specified and Lorentz covariance ismaintained. The Green’s functions in our approach will include a manifestly covariant term timesa factor like Θ((2π/Ld)

2 − (k2 + w2)) where k is a wavevector as measured in the unique localframe and w, which on-shell is (k2 +m2)1/2, is the frequency in the same frame. Although at firstsight such a term might appear to violate Lorentz covariance, it is actually a Lorentz scalar. Thisis because k2 is not the square of a wavevector determined in the frame of an arbitrary observer,but instead is defined in a way in which all observers agree - it is therefore simply a scalar that doesnot depend on the reference frame. To illustrate Lorentz covariance in more detail consider a non-interacting scalar field in an almost-flat spacetime background. That is we consider the backgroundspacetime to establish the local frame for Wick rotation but otherwise negligibly differs from flatspacetime. The Euclidean Green’s function can be written

G(k) =Θ((2π/Ld)

2 − k2)

k2(8)

where 2π/Ld is a momentum cutoff. We will, in general, not be able to truncate the theory with asimple momentum cut-off, but this works for a non-interacting scalar field since there is no gaugeinvariance to preserve. Wick rotating gives

G(k) =Θ((2π/Ld)

2 − k2 − w2)

k2(9)

where the k2 in the denominator is now −w2 + k2. The k2 appearing in the step function simplyrecords which states prior to Wick rotation gives a non-zero result and as such the w2 term does notget an additional minus sign upon Wick rotation. Now consider a boosted frame where particleswith momenta k in the un-boosted frame have momenta k in the boosted frame where kµ = Λµν kν .The Green’s function is

G(k) =Θ((2π/Ld)

2 −∑µ=4

µ=1(Λµν kν)2)

k2=

Θ((2π/Ld)

2 − k2 − w2)

k2= G(k) (10)

where the argument of the step function is determined by the requirement that the modes are cutoff in the unique frame where LTE holds and the Wick rotation is valid. Since all observers agreeon this frame, the step function does not spoil Lorentz covariance.

IV. EXPANSION IN A FINITE BASIS

A. Gauge Invariance

As mentioned in the introduction, since we must average over discrete configurations in Eu-clidean spacetime, a possible approach would be to perform a lattice gauge theory calculation foran arbitrary lattice and average the result over all possible lattices. The result of the average isrotation invariant since all possible lattice orientations are averaged over. We therefore obtain aLorentz covariant result after Wick rotation. While such an approach is faithful to the physicalprinciples that we have described, it is not easily tractable analytically and it is not easily ex-tendable to the case of quantum gravity without additional information on the discrete structureswe would be averaging over. We will therefore take an alternative approach were we approximatethe discrete structure by a finite number of continuous, spherically symmetric, gauge covariant

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functions. Doing so allows analytical results to be obtained, but since we are smearing out thediscrete structures underlying spacetime, we cannot expect the resulting models to accurately cap-ture physics on an energy scale comparable to the inverse length scale of the discrete structures.Nevertheless, we will be able to obtain the correct low energy physics without UV-divergencesappearing since only a finite number of degrees of freedom are included.

Our approach will be to expand all fields in terms of eigenfunctions of a gauge covariant Lapla-cian and write functional integrals in terms of the coefficients, ak, of this expansion. Truncating theexpansion at any given order preserves gauge invariance. To see this, consider Yang-Mills theory.Under a gauge transformation[32]

Aµ → w−1Aµw + i(∂µw−1)w, (11)

the Lie-algebra valued eigenfunctions, Φiµ, of the gauge covariant Laplacian DµDµ transform as[32]

Φiµ → w−1Φiµw (12)

where the covariant derivative acting on Lie-algebra valued functions is given by[32]

Dµ = ∂µ + i[Aµ, ] (13)

and

w = eiγata (14)

for gauge parameters γa and Lie algebra generators ta. (We are using a normalization for the gaugefields such that the coupling constant appears in the trF 2 part of the action rather than in thecovariant derivative.) We also expand matter fields in terms of the gauge covariant Laplacian−iγµDµ where the covariant derivative acting on matter fields is given by Dµ = ∂µ + iAµ. Undera gauge transformation the eigenfunctions, φi, of this Laplacian transform as

φi → w−1φi (15)

The covariant transformation properties of equations (12) and (15) allow us to preserve gaugeinvariance when fields are expanded in a finite number of eigenfunctions. To see this, let us expandthe vector potential in a finite number of terms as A0

µ =∑

i aiΦ0iµ and expand Ψ in a finite number

of terms as Ψ =∑

i ciφ0i . The superscript 0 indicates that the functions are determined in the

absence of a pure gauge term. It is convenient to perform a gauge transformation in order tointroduce a pure gauge term that will help us keep track of further gauge transformations. Wewrite

Aµ =∑i

aiΦiµ + i(∂µv−1)v (16)

where Φiµ = v−1Φ0iµv and

Ψ =∑i

ciφi (17)

where φi = v−1φ0i . Under a gauge transformation we have

i(∂µv−1)v → iw−1(∂µv

−1)vw + i(∂µw−1)w (18)

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It is readily verified that if equations (16) and (17) are substituted into the Yang-Mills action, theaction is unchanged under the transformations of equations (12), (15) and (18). It follows thatgauge invariance is preserved when the fields are expanded in any finite number of eigenfunctionsof the gauge covariant Laplacians. The same conclusion holds in the case of general relativitywhere we expand the metric in terms of eigenfunctions of � = ∇a∇a. Under diffeomorphisms,tensor eigenfunctions transform as tensors, and similarly for vector and scalar eigenfunctions, andthe Ricci scalar constructed from such eigenfunctions transforms as a scalar.

However, we generally don’t know the eigenfunctions exactly and so they must be determinedperturbatively. This generates additional interactions that must be included. We will label theeigenfunctions by a wavevector k and include all eigenfunctions labeled by k ≤ 2π/Ld with Ld thephysical length scale associated with the discrete spacetime structure.

B. Perturbation Theory

We wish to expand all fields in terms of eigenfunctions of an appropriate gauge covariant Lapla-cian. Since in general we do not know how to determine the eigenfunctions of the full Laplacian,we will need a perturbative expansion. Suppose, for example, that the gauge covariant Laplaciancan be written as � = �(0) + V and we wish to determine the eigenfunctions as a perturbationexpansion in V . In this section, we take the background spacetime to be flat and we considera finite 4-box with width L and apply degenerate perturbation theory to the discrete spectrum.Divide momentum space into thin shells and denote the set of states that diagonalize V in a givenshell by |p, a〉, where p denotes the momentum magnitude and a denotes the additional parametersneeded to specify the eigenfunction (e.g., angular variables). The width of the shells should be largeenough to contain many momentum states so that in the large L limit, the sum over momentumstates within a shell can be written in terms of an integral over the momentum in the shell. Wewill take the width of the shells to go to zero after taking the large L limit. Degenerate stateperturbation theory gives

Φk,a = |k, a〉+∑′

p,b|p, b〉〈p, b|V |k, a〉

p2 − k2+∑′

p,q,b,c|p, b〉〈p, b|V |q, c〉〈q, c|V |k, a〉

(p2 − k2)(q2 − k2)

+∑′

p,q1,q2,b,c1,c2|p, b〉〈p, b|V |q1, c1〉〈q1, c1|V |q2, c2〉〈q2, c2|V |k, a〉

(p2 − k2)(q21 − k2)(q2

2 − k2)+ ... (19)

where the primes in the sums indicate that terms with any two momenta in the same shell shouldnot be included. (We will generally use primes to indicate that a sum over momenta is restrictedin some way. Here, the sums are restricted to exclude poles in the numerator. In other contexts, aprime may be used to indicate a truncated sum.) Standard degenerate state perturbation theorygenerates terms with two or more momenta equal. However, these terms do not contribute in thethermodynamic limit since such terms contain the same factors of 1/V (4) as the other terms ofthe same order but contains fewer sums to contribute to limits such as 1/V (4)

∑k →

∫d4k/(2π)4

[33]. In equation (19), Φk,a is not normalized. The normalized eigenfunction can be written asΦk,a = Φk,a/Nk,a where to leading order the normalization factor is given by

N2k,a ≡

∫d4xΦ∗k,a(x)Φk,a(x) = 1 +

∑′

p,b

|〈p, b|V |k, a〉|2

(p2 − k2)2, (20)

which has a well-defined L→∞ limit as can be seen by expanding the numerator about p = k inpowers of p− k. The p = k term does not contribute in the large L limit because |〈k, b|V |k, a〉|2 ∝δa,bL4 contains a factor of 1/L4 and the minimum value of |p2 − k2| included in the sum is on the

order of (k + p)/L.

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An expansion such as that given in equation (19) may or may not provide a convergent expressionfor the exact eigenfunction. But this not particularly relevant for our approach. Rather, regardlessof how good an approximation equation (19) provides for the exact eigenfunction, at any finiteorder of perturbation theory, it provides a suitable basis of functions satisfying the symmetryrequirements of Section IV A.

Expanding in this way in principle provides a perturbative expansion of the eigenfunctions and,as we have noted, expanding fields in a finite number of such eigenfunctions would provide a finitebasis for quantum field theory that preserves gauge invariance. In particular, a suitable finite basisis Φk,a with k ≤ 2π/Ld.[34] However, in practice, we will usually not know how to diagonalize Vto obtain |p, a〉 which is needed to construct Φk,a. We would therefore like to bypass the problemof diagonalizing V and expand the eigenfunctions in a plane wave basis. We can write

|p, a〉 =∑

kα|k2=p2

Cp,a;kα |kα〉 (21)

where Cp,a;kα are expansion coefficients between the two basis sets and the sum is over states in ashell in momentum space. Substituting equation (21) into equation (19) and making used of thecompleteness of states, we see that

Φp,a =∑

kα|k2=p2

Cp,a;kαΦkα (22)

where

Φkα = |kα〉+∑′

p|p〉〈p|V |k〉

p2 − k2+∑′

p,q|p〉 〈p|V |q〉〈q|V |k〉

(p2 − k2)(q2 − k2)

+∑′

p,q1,q2|p〉 〈p|V |q1〉〈q1|V |q2〉〈q2|V |k〉

(p2 − k2)(q21 − k2)(q2

2 − k2)+ ... (23)

For simplicity of notation we have defined |p〉 ≡ |pα〉. The sums in the expansion for Φkα becomewell defined principal parts integrals in the large L limit.

The basis Φkα is generally not orthogonal since each Φkα is a linear combination of the orthogonaleigenfunctions Φk,a and so Φkα is not necessarily orthogonal to Φk′α with k′α different from kα butwith the same magnitude. We can form an orthonormal basis by taking combinations of Φkα statesas

Φkα = Φkα +∑p

Nk,pΦpα (24)

and then determining the orthonormalization coefficients Nk,p by requiring∫d4xΦ∗pα(x)Φkα(x) =

δpα,kα . It is straightforward to determine Nk,p perturbatively. To leading order, we have (forHermitian V )

Nk,p = −1

2

∑′

q

〈p|V |q〉〈q|V |k〉(q2 − p2)(q2 − k2)

δ|k|,|p|, (25)

where δ|k|,|p| is unity if k and p are in the same shell and is otherwise zero. In calculations involvingintegrals of Nk,p over k and p, we should take the limit that L → ∞ before taking the limit thatthe widths of the shells go to zero. This typically results in a zero contribution because in thelarge L limit, the sums become well-defined, finite principal-parts integrals and in taking the limitthat the widths of the shells go to zero, the δ|k|,|p| term reduces the integral to zero. This does not

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occur, however, if a momentum conservation delta function forces kα = pα, for example, since thesum over q could then not be written as a principle parts integral around poles at p and at k. Inthis case, the diagonal part, Nk,k, of the orthonormalization coefficients, Nk,p, can enter into thecalculation. This is given by

Nk,k = −1

2

∑′

q

〈k|V |q〉〈q|V |k〉(q2 − k2)2

. (26)

Since Φkα spans the same set of states as Φp,a we can use Φkα ≡ Φk as a basis. This allows usto avoid diagonalizing V , but leads to a normalization coefficient Nk,k that does not have a finitelarge L limit. Since physical results cannot depend on the basis chosen for the expansion and sucha divergence would not occur if the Φp,a basis had been used, physical quantities must either notdepend on Nk,k or the divergence in Nk,k must cancel with a related divergence. We will explicitlydemonstrate below that Nk,k drops out of the calculation of vacuum polarization in QED at oneloop order and that the divergence in Nk,k cancels with a similar divergence arising from a wavefunction overlap integral in determining the electron self-energy at one loop order.

Since Φkα(x) has an analogous expansion and the same gauge covariance symmetry as theeigenfunctions Φp,a(x), we will loosely refer to Φkα(x) (or Φkα(x)) as eigenfunctions and

λk ≡∫d4xΦ∗kα(x)�Φkα(x) (27)

= −k2 + 〈k|V |k〉+∑′

p

|〈p|V |k〉|2

p2 − k2+ ... (28)

as eigenvalues even though it would be more accurate to reserve those terms for Φp,a(x) and thecorresponding eigenvalues

∫d4xΦ∗p,a(x)�Φp,a(x). Note that∫

d4xΦ∗kα(x)�Φpα(x) ∝ δ|k|,|p|. (29)

For similar reasons as described above for Nk,p, such δ|k|,|p| terms typically only contribute in thethermodynamic (large L) limit when a momentum conservation delta function forces |k| = |p|.

V. RESTRICTION TO PHYSICAL STATES

To illustrate how the functional integral for quantum field theory with a finite number of degreesof freedom can be derived, we will first consider a scalar field Φ(x). By discretizing in M spacetimepoints, where M is finite but much larger than the number of discrete physical degrees of freedomN , our system can be described in terms of an M -dimensional configuration space M with ann-dimensional subspace of physical states N . We can write the partition function as a functionalintegral over the physical space N by writing the measure for functional integral over M as

d[Φ]Nd[δΦ]M−N , (30)

where d[δΦ]M−N is the measure for infinitesimal variations about N in M−N , and then identi-fying d[Φ]N as the measure on N .

We may define a physical subspace by restricting Φ(x) to the functions

ΦP (x) ≡∑k

akΦk(x; [ak]), (31)

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where the sum is over N terms and we allow the basis functions to depend on the field since wewill need to do this when we discuss gauge theories. The partition function can be constructedby inserting a set of states that are complete in the physical subspace at Mt time slices. We maydiscretize in the spatial coordinates as well for a total of M discrete points in the spacetime. Wewill work in Euclidean time, but it is straight forward to work in Lorentzian time by simply Wickrotating each basis function. Since we need to integrate only over physical states, we can restrictthe complete set of states to a set of states that are complete in the physical subspace. Thisimplies that we should integrate only over the physical N dimensional subspace, N , of the full Mdimensional space, M. These consideration yield

tre−Hβ =

∫dµP (Φ)〈Φ(xi, β)|e−Hε|Φ(xi, τj)〉〈Φ(xi, τj)|e−Hε...e−Hε|Φ(xi, 0)〉 (32)

where dµP is the measure on the physical subspace defined by equation (31) induced by the measureΠαdΦα on the full M-dimensional space. We will see shortly how to relate dµP (Φ) to Πkak. Notethat we do not introduce any additional time dependence by restricting the integration to N and soall the time dependence in correlation functions arise from the e−Hτ factors which give unitary timeevolution upon Wick rotation. As pointed out elsewhere[35], we can obtain a covariant measureif we discretize uniformly in physical distances rather than coordinate distances. So in a curvedspacetime, we chose the discretization such that the time slices are separated by constant Nlδτwhere Nl is the lapse and each time slice contains a point in each

√hδx region where h is the

determinant of the metric on the constant time slice. We introduce a total of M such points in thespacetime.

We now introduce a complete set of momentum eigenstates at each of the M points. One coulduse a restricted set of momentum states, but it is easier to include complete states at each pointsince we will exactly integrate out the momenta. A restricted set of momentum states give thesame result as the unrestricted set as long as the space spanned by the restricted set includes theclassical momentum for each field configuration (i.e., as long as ∂

∂ΦL can be decomposed in the

restricted basis) so that the shift necessary to perform the functional integral can be performedexactly. After integrating over momenta we obtain the standard partition function but integratedover the subspace N of physical states.

Let us introduce a basis Ψk(xi; [ak]) for M−N and expand Φ(x) as

Φ(xi) =∑k

akΦk(xi; [ak]) +∑k

ckΨk(xi; [ak]) (33)

where the sum over ak terms is restricted to k ≤ 2π/Ld and the sum over ck terms is restricted tok > 2π/Ld. The functional integral over the M Φ(xi) variables can be replaced by a integrals overthe N ak variables and the M − N ck variables by introducing an appropriate Jacobian factor.The Jacobian matrix is indexed by M k variables and M xi variables. It is convenient to considerthe Jacobian matrix times its transpose so that the matrix is indexed by M by M k indices. Thesquare of the determinant of the Jacobian matrix can be written as

J2 = det(∑

i

∆∂ΦP (xi)

∂a−k

∂ΦP (xi)

∂ak′

)(34)

where ak = ak for k ≤ 2π/Ld and ak = ck for k > 2π/Ld and the determinant is of an M by Mmatrix in k, k′. ∆ is an arbitrary constant that may be selected to be the 4-volume for each of theM points so that the sum over i may be written as an integral over volume in the large M limitfor fixed k, k′. The functional measure can be written as

ΠidΦ(xi) =((Πkdak)J/D

)((Πkdck)D

)(35)

11

where

D2 = det(∑

i

∆∂ΦP (xi)

∂c−k

∂ΦP (xi)

∂ck′

)(36)

and the determinant is of an M − N by M − N matrix in k, k′. The factor (Πkdck)D can beidentified as the measure d[δΦ]M−N onM−N and (Πkdak)J/D can be identified as the measured[Φ]N on N . To see that this identification is correct, note that if Ψk(xi; [ak]) is an orthonormalbasis forM−N , then D = 1 and the measure induced on N can be obtained simply by droppingthe Πkdck factor. The factor of D in (Πkdck)D is precisely what is needed to make the measured[δΦ]M−N independent of the basis chosen for M−N .

An alternate way to obtain this result is to introduce a factor into the partition function thatprojects out unphysical paths. A suitable factor is simply

exp(− 1

2εp

∑i

∆(δΦ)2)

(37)

where εp is to be taken to zero. Integrating out d[δΦ] then directly produces the factor of 1/N ind[Φ]N .

As we will see, in the case of QED, an orthonormal basis can be chosen so that J = D = 1 upto irrelevant constants. This is not the case for non-abelian gauge theories however. In this case,the factor J/D gives a finite contribution for any given finite order perturbation theory. To Seethis, note that J2 can be written

J2 = det(A) det(∆− ΓA−1B) = det(∆) det(A−B∆−1Γ) (38)

where A,B,Γ, and ∆ are blocks in the matrix∑i

∆∂ΦP (xi)

∂a−k

∂ΦP (xi)

∂ak′(39)

for k, k′ ≤ 2π/Ld; k ≤ 2π/Ld and k′ > 2π/Ld; k > 2π/Ld and k′ ≤ 2π/Ld; and for k, k′ > 2π/Ld,respectively. J/D can the be written in terms of a determinant of an N by N dimensional matrixas

J/D = det1/2(A−B∆−1Γ). (40)

Using basis functions determined perturbatively, the matrices B and Γ have only a finite number ofentries for an any finite order perturbation theory in the large M limit, and so the sums involved informing B∆−1Γ have only a finite number of nonzero terms. The quantity J/D is therefore finite.

The resulting partition function can be written as

Z = Πkdak(J/D)|ck=0 e−S (41)

with J given by equation (34) and D given by equation (36). This is covariant since we have dis-cretized in physical distances rather than coordinate distances. If we had quantized by discretizingin constant coordinate lengths rather than physical lengths, we would not obtain a covariant con-figuration space measure after integrating over momenta. This is because as discussed in [35], themeasure would contain spurious factors of the lapse that result from treating degrees of freedom asif they were distributed uniformly in coordinate lengths rather than physical distances. As shownin [35], a spurious factor of the lapse arises for each fermionic degree of freedom and a spuriousfactor of the inverse lapse arises for each bosonic degree of freedom when the time discretization

12

is uniform in coordinate time rather than physical time. An alternate approach to dealing withthese spurious lapse factors is to introduce additional ghost fields to cancel the lapse factors. As anexample consider QED or quantum gravity where there are two physical degrees of freedom. Thecanonical measure determined with uniform coordinate discretization will contain two spuriousfactors of the lapse. We can eliminate these factors by adding ghost fields having two fermionicdegrees of freedom and then integrating out these fields. We will illustrate this approach in sec-tion VII. This is a convenient technique for quantum gravity where, as discussed in [35], a directderivation of the canonical measure by discretizing uniformly in physical time is problematic.

The derivation of the partition function starts with a Hamiltonian formulation where unitarityis manifest and the truncation to physical states is done in a way which preserves unitarity. Thisis because the truncation is done at a path level. To see this consider the quantity (Wick rotatedto Lorentzian spacetime)∫

dΦ(xi, t2)〈Φ(xi, t3)|e−iH(t3−t2)|Φ(xi, t2)〉〈Φ(xi, t2)|e−iH(t2−t1)|Φ(xi, t1)〉 (42)

where each of the factors

〈Φ(xi, tj)|e−iH(tj−ti)|Φ(xi, ti)〉 (43)

are determined by integrating over only physical paths. The only |Φ(xi, t2)〉 states that give a non-zero contribution in the truncated theory are those on a physical path. This means that |Φ(xi, t2)〉contributes only if there is some spacetime region around (xi, t2) where Φ(x, t) can be written inthe form of equation (31). Summing over only physical |Φ(xi, t2)〉 states, equation (42) gives

〈Φ(xi, t3)|e−iH(t3−t1)|Φ(xi, t1)〉 (44)

which implies unitarity. For the abelien case, unitarity of the S-matrix follows from the unitarityof the untruncated theory and the optical theorem. This is because in the truncated theory eachinternal propagator is cut off in the same way as external lines and so the optical theorem issatisfied for the truncated theory if it is satisfied for the untruncated theory. For the non-abelieancase, the the analysis is more complicated, but we show elsewhere[36] that the theory possessesBRST symmetries which ensure unitary S-matrices between physical states.

VI. YANG-MILLS THEORY

A. Derivation of Partition Function

Recall that the familiar Faddeev expression for the partition function for a theory with firstclass constraints, such as Yang-Mills theory, can be obtained by writing the partition function interms of physical variables q∗, p∗, introducing auxiliary fields Q,Π as functional integrals over deltafunctions, and then performing a canonical transformation to return to q, p coordinates[37, 38].This leaves a delta function for the gauge conditions, a delta function for the constraints, and aPoisson bracket term. The delta function for the constraints and the delta function for the gaugecondition can be written in terms of functional integrals over Lagrange multiplier fields λc and λg,respectively. We may consider q, λc and λg together to define extended configuration variables. Aswe will see, we can choose gauge conditions and basis functions such that the Fadeev determinant isunity. The resulting partition function is similar to the scalar field partition function but with morefield variables. We may then truncate the extended variables in much the same way as for the scalarfield case provided that we truncate the extended variables in a way that preserves gauge invariance

13

so that the unphysical auxiliary fields do not contribute to the partition function. To implementthis procedure, we define a “physical” subspace for the gauge field in terms of a finite set of basisfunctions that include both gauge fixed parts and variations about the gauge fixed parts. It isnecessary to truncate the gauge variation part as well as the gauge fixed part since these variationsare components of the extended configuration variables that we wish to truncate. We use a gaugecovariant truncation by expanding in gauge covariant eigenfunctions as described previously andwe preserve unitarity since the truncation preserves gauge invariance. In the approach taken here,the gauge fixed part of the gauge field is explicitly isolated and, as we will see in Section VII,this approach extends directly to the quantum gravity case where we need to explicitly isolate theconformal mode. In addition, our approach allows QED-like Ward identities to be derived for thenon-abelian case as we show in the next section.

In order to define a functional integral over the physical subspace, we need to be able to splitthe various components of the functional integral into contributions from the physical subspaceand contributions from the orthogonal subspace. We can do this by dividing the expansion of boththe physical and unphysical parts of the gauge fields into components in a gauge-fixed subspaceand components representing variations about this subspace. For the physical part of the gauge

field we denote the basis for the gauge fixed part as A(gf)ikµ . We have

Aiµ =∑

k≤2π/Ld

(aikA(gf)ikµ + dikDµα

ik) +

∑k>2π/Ld

(aikA(gf)ikµ + dikDµα

ik) (45)

where Dµ is the gauge covariant derivative and we denote the expansion coefficients for the physicalsubspace N by aik and dik for the gauge fixed parts and the infinitesimal gauge transformation parts,respectively, and the corresponding expansion coefficients for the orthogonal subspace M−N byaik and dik. α

ik can be taken to be a complete orthonormal set of scalar eigenfunctions of the gauge

covariant Laplacian. The quantity i labels the basis. For simplicity of notation we use the genericlabel i repeatedly for such a label. The range of i will vary depending on what basis it labels; thisshould be clear from the context.

In order to expand in this way, we need to choose our basis functions such that they containlongitudinal functions of the form Dµφk. We can do this by taking φk(x) to be scalar eigenfunctionsof � = DµDµ and defining a basis for our physical subspace to be functions of the form Dµφk(x)plus a basis of transverse functions. In general, the vector eigenfunctions of � will not be purelytransverse or purely longitudinal since Dν does not commute with �. However, we can form asuitable basis of transverse modes by taking linear combinations of vector eigenfunctions of � withfunctions of the form Dµφk(x) to construct functions satisfying DµAµ = 0. We must then take

our gauge fixing condition such that the set of functions A(gf)ikµ for k ≤ 2π/Ld are in the physical

subspace defined by the basis of transverse and longitudinal modes. A natural choice is to require

A(gf)ikµ to be purely transverse, i.e., DµA

(gf)ikµ = 0.

We take the gauge condition to be

0 = F (x; [Aµ]) = α(x) (46)

so that the determinant of δF (x′)/δα(x) is unity. The measure is then determined by changingvariables to aik, d

ik, a

ik, d

ik and then projecting out the aik, d

ik variables in the same way as described

for the scalar field. We then implement the gauge fixing condition which eliminates the dik variables.Recall that canonical quantization gives a matching number of gauge fixing delta functions andconstraints which are implemented with the Ai0 field as a Lagrange multiplier field. Since we projectout the Ai0k for k > 2π/Ld along with dik for k > 2π/Ld, we are left with one gauge fixing delta

14

function for each dik for k ≤ 2π/Ld. The resulting measure on the physical subspace is then

dµP (A) =(Πj,k≤2π/Ldda

jk

)J/D (47)

where J is the determinant of the Jacobian matrix from Aiµ(x) variable to aik, dik, a

ik, d

ik variables, D

(as in the scalar field case) is the square root of the determinant of the block of squared Jacobianmatrix with k, k′ > 2π/Ld, and aik, d

ik are set to zero after taking the derivatives to form the

Jacobian matrix. By choosing the gauge fixing function in this way, we do not obtain additionalghost terms from evaluating the gauge fixing determinant. However, the usual Faddeev-Popovghosts appear in the Jacobian associated with changing variables to aik, d

ik.

The square of the Jacobian J can be written analogously to equation (34) and the square rootcan be taken by introducing a set of Grassmann ghosts and a set of real scalar ghosts. Another setof real scalar ghosts can be introduced to produce the factor of 1/D. The partition function canthen be written as

Z = Πkdajkdf

ikdf

ikdb

ikdb

ike−SYM−Sgh (48)

where SYM is the Yang-Mills action and the term implementing the Jacobian factors can be written

Sgh =∑

k,k′,i,i′

(f ikfi′k′ + bi−kb

i′k′ + bi−k b

i′k′)(∑m

∆∂Ajµ(xm)

∂ai−k

∂Aµj (xm)

∂ai′k′

)=

∑k,k′,i,i′

(f ikfi′k′ + bi−kb

i′k′ + bi−k b

i′k′)(∫

d4x√g∂Ajµ(x)

∂ai−k

∂Aµj (x)

∂ai′k′

)(49)

where aik stands for (aik, dik) for k ≤ 2π/Ld and (aik, d

ik) for k > 2π/Ld. The j label contains a

gauge label as well a basis function label. k and k′ are unrestricted for f ik, fik, and dbik, and k and

k′ are restricted for dbik to be greater than 2π/Ld. Only a finite number of terms contribute, sincethe contribution from dbik cancels the contribution from the other ghost fields when both momentaare greater than 2π/Ld (this can also be seen from considering the ratio J/D - see the discussionaround equation (40)). Since only a finite number of terms contribute, the second form follows inthe limit that M → ∞. It is readily verified that equation (48) is gauge invariant. In particular,

we can change the gauge by replacing the gauge fixed field A(gf)ikµ with δγA

(gf)ikµ where the notation

δγ indicates a gauge transformation for some gauge parameter γ that is independent of the gaugefield. Such transformations do not alter equation (48). We will make use of this invariance in thenext section to derive Ward identities.

The partition function of equation (48) can be explicitly evaluated at quadratic order. Foreach gauge boson, we obtain three factors of

∏k≤2π/Ld

k−1 from the ajk fields and one factor of∏k≤2π/Ld

k from the ghost fields, so the partition function is

Z =

′∏k−2Ng (50)

where Ng is the number of gauge bosons. This gives

lnZ = −Ng

∑k

∑n

ln(k2 + w2n)Θ((2π/Ld)

2 − (k2 + w2n)) (51)

which is the expected form for 2Ng bosonic degrees of freedom. This is a well-defined partitionfunction that can be evaluated without further regularization. In Appendix B, we show how to

15

exactly evaluate the frequency sum in equation (51) and we show that the standard results for thelow energy physics are obtained.

At one loop order, calculations based on equation (48) become algebraically much more involved,especially for non-abelian fields. In the present paper, we will content ourselves with evaluatingvarious quantities for QED at one loop order. We will verify that standard results are obtained forQED in Section VIII.

Before exploring Ward identities in the next section, let us summarize some properties of thepartition function that is evident from our approach. First, perturbation theory will give finiteresults order by order, since each order of perturbation theory involves a finite number of integrals,each over a finite range of momenta. Second, physical quantities calculated from equation (48)will be gauge invariant since a gauge covariant expansion was used. Third, Lorentz covarianceholds as previously discussed. Finally, unitarity holds since the partition function follows from aHamiltonian formulation where unitarity is manifest.

B. Ward Identities

In this section, we show that non-abelian Ward identities can be derived for equation (48) thatresemble standard Ward identities for QED. First note that matter fields can be introduced in ourformalism by adding the appropriate term to the action and expanding the matter fields in gaugecovariant eigenfunctions. Here we consider a Dirac fermion field ψ =

∑ckψk belonging to some

representation of the gauge group and we do not include chiral fermions[39]. In our formalism weend up integrating only over a gauge fixed subspace rather than an unrestricted integration withan action modified by a gauge fixing term. We can add a gauge fixing term proportional to 1

ξ to theaction and an additional functional integration and recover the results from the partition functionwithout the gauge fixing term by taking ξ to be infinitesimal. We will take the basis functions for

the gauge fixed field to satisfy DµA(gf)ikµ = 0 where Dµ is the covariant derivative defined using

the gauge fixed field A(gf)aµ so that DµA

(gf)aµ = ∂µA

(gf)aµ = 0. A general non-gauge fixed gauge

field can be written as a transverse part satisfying DµAaµ = 0 plus a longitudinal part of the formDµλ

a for some Lie algebra valued scalar field λa. We can add a Dµλa term to the gauge field and

integrate over the λa field if an appropriate gauge fixing term is included. For infinitesimal ξ wecan identify λa with 1

gγa where γa is a gauge transformation parameter and g is the Yang-Mills

coupling constant. The longitudinal part of the gauge field can then be written as AaLµ = 1gDµγ

a

for infinitesimal ξ. We can expand the gauge function γ(x) =∑

k gakγ

ak(x) in some basis γak(x)

and perform the functional integral over gak rather than over λa. Note that since matter fields areexpanded in eigenfunctions of the appropriate gauge covariant Laplacian, the matter fields alsoundergo a gauge transformation with gauge transformation parameter γa as AaLµ is varied.

To define the gauge fixing term note that∫[dγa] det(∂µDµ) exp[−(∂µDµγ)2/(2ξ)] = 1 (52)

(up to irrelevant constants) where the functional measure [dγ] can be expressed as Πdgak for or-thonormal γak(x) basis functions. Inserting equation(52) into equation (48) leads to the generatingfunctional

Z[J, η, η] = Πkdajkdf

ikdf

ikdb

ikdckdck[dALµ]e−STOT (53)

where

STOT = SYM + Sgh +

∫d4x

((∂µAaLµ)2/(2ξ) + JaµAaµ + ηψ + ψη

)(54)

16

and the functional measure [dALµ] is shorthand for (Πgak) det(∂αDα).

Although it is not required for deriving the Ward identities, SYM and Sgh may be consideredto depend on the gauge transformed field

Aaµ = δγ(x)A(gf)aµ = A(gf)a

µ +1

gDµγ

a, (55)

rather than on the gauge fixed field, where the last equality holds for infinitesimal ξ. SYM [Aaµ] isindependent of γa(x) since it is gauge invariant. If we choose the basis γak(x) to be independent ofaik, d

ik, ck, and the ghost fields, then Sgh[Aaµ] will also be independent of γa(x) and so equation (53)

with SYM and Sgh taken to be functionals of the gauge transformed field gives the same partitionfunction in the absence of source terms as the partition function defined in equation (48) whereSYM and Sgh are functionals of the gauge fixed field. To see that Sgh[Aaµ] is independent of γa(x)when the γak are independent of the other fields, note that a gauge transformation for arbitrary γa

can be written as

δγ(x)Aµ = e−iγAµeiγ − i

ge−iγ∂µe

iγ . (56)

In the ghost action, the derivative term gives zero after differentiating with aik to form Sgh. Thederivative only acts on Aµ in the first term and the factors eiγ do not contribute to Sgh. Thisfollows because the sum over j in equation (49) includes a sum over gauge field index and for gaugefields Aaµ and Ba

ν we have the following transformation if we drop the derivative term

∑a

AaµBaν =

1

Ctr(AµBν

)→ 1

Ctr(e−iγAµe

iγe−iγBνeiγ)=

1

Ctr(AµBν

)=∑a

AaµBaν (57)

where C is a constant.

To derive the Ward identities, change variables in equation (53) from gak to gak + δak for constantinfinitesimal shifts δak so that γa → γa + δa and AaLµ → AaLµ + 1

gDµδa. The measure does not

change and the action changes by

δSTOT =

∫d4x

((∂µAaLµ)(∂νDνδ

a)/ξ + JaµδAaµ + ηδψ + δψη

)(58)

where δAaµ represents the change in Aaµ when gak is shifted to gak + δak , and similarly for ψ and ψ.Since the functional integral must be invariant for arbitrary δak , we have

0 =

⟨(∂µAaLµ)(∂νDνδ

a)/ξ + JaµδAaµ + ηδψ + δψη

⟩(59)

where 〈...〉 denotes an average under the functional integral. Since we are taking the limit ξ → 0,δAaµ is simply 1

gDµδa = 1

gDµ

(∑k δ

akγ

ak

)and the matter fields also follow from the infinitesimal form

of the gauge transformation. Integrating by parts we have

0 =⟨Dν∂

ν∂µAaLµ/ξ −1

gDµJ

aµ + i(ηtaψ − ψtaη)⟩

(60)

Equation (60) can be written

0 = − 1

gξDν∂

ν∂µPL

(δW

δJaµ

)A=− δW

δJaµ

− 1

gDµJ

aµ|A=− δWδJaµ

− i(ηta δWδη

+δW

δηtaη) (61)

17

where W [J, η, η] = lnZ[J, η, η] and the sign change in the last term comes from commuting Grass-mann variables to take the derivative. PL() is the projection operator onto the longitudinal partof the gauge field and can be written as PL(Aaµ) = Dµ(DαDα)−1DβAaβ

Following Ryder[40], but using differing sign conventions since we are working in Euclideanspacetime, we can Legendre transform to form the vertex function Γ, which in our sign conventionmay be defined as

Γ[A,ψ, ψ] = −W [J, η, η]−∫d4x(JaµAaµ + ηψ + ψη

)(62)

We have

0 = − 1

gξDν∂

ν∂µPL(Aaµ)− 1

gDµ

δΓ

δAaµ− i( δΓδψtaψ + ψta

δΓ

δψ

)(63)

For deriving relations involving gauge field propagators and gauge-field matter vertex functions,the first term can be replaced with the first order term in Aaµ, − 1

gξ�∂µAaµ, but relations among

higher order Green’s functions require that higher order terms be kept except in the case of QEDwhere the first order term is exact. Functionally differentiating equation (63) with respect to ψand then ψ and setting remaining fields to zero gives

∂µx1δ3Γ

δψ(x3)δψ(x2)δAaµ(x1)= ig

(δ2Γ

δψ(x3)δψ(x2)taδ(x2 − x1)− ta δ2Γ

δψ(x3)δψ(x2)δ(x3 − x1)

)(64)

which can be recognized as a Ward-Takahashi identity relating the derivative of a vertex functionto the difference in inverse propagators.

The Ward identity for the polarization can be obtained as follows. First note that evaluatingequation (63) with all fields equal to zero gives ∂µ

δΓδAaµ|A=0 = 0. The only solution to this equation

consistent with Lorentz covariance (since there are no 4-vectors around in the absence of fields) issimply δΓ

δAaµ|A=0 = 0. Next, functionally differentiate equation (63) with respect to Aaµ and then set

all fields to zero. In momentum space this gives

0 =1

ξk2kµ − kν

(Dνµ

)−1(65)

where Dνµ is the gauge field propagator (both the context and the number of indices distinguish

this from the covariant derivative). Denoting the non-interacting propagator by D(0)abµν = δabD

(0)µν

we have

D(0)µν =

δµν − kµkνk2

+ ξkµkνk2

(66)

which implies

(D(0)µν

)−1= k2(δµν − kµkν) +

k2

ξkµkν (67)

Defining the polarization Πµν as the difference between the non-interacting inverse propagator andthe interacting inverse propagator, equation (65) then implies

kµΠµν = 0. (68)

18

VII. QUANTUM GRAVITY

Quantum gravity can be developed in much the same way as Yang-Mills theory. There are how-ever two significant differences. First, for quantum gravity in Euclidean spacetime, the conformalmode gives a divergent contribution to the functional integral if the action is assumed to be thenaively Wick rotated version of the Einstein-Hilbert action. However, the solution to the conformalmode problem suggested by Schleich[23] (similar results were also obtained by others[41, 42]) is acompletely satisfactory solution to the conformal mode problem from the perspective of the presenttheory. We will outline Schleich’s argument below. The second major difference with Yang-Millstheory relates to renormalizability. In the case of Yang-Mills theory, perturbation theory givesexpansions in powers of a renormalized coupling constant. In contrast to this, quantum gravity isnon-renormalizable. As we show in some detail below, the non-renormalizability allows a perturba-tive expansion to be developed. The quantity LPl/Ld, which we assume to be small, becomes theexpansion parameter and the non-renormalizable nature of the theory ensures that higher orderloop diagrams gives contributions with higher powers of LPl/Ld so that we have a well definedperturbative expansion.

The essential premise of our approach is that there are only a finite number of physical degreesof freedom associated with a given fixed finite 4-volume in Euclidean spacetime. If we allow themetric tensor to vary arbitrarily, we would have fluctuations with arbitrarily large 4-volume. Wewould then not be able to restrict the partition function to a finite number of field integrations.It is therefore natural to work in the canonical ensemble where the 4-volume is held fixed and themetric is expanded in a finite number of modes. We will discuss the grand canonical ensembleelsewhere[43].

A. General Formulation

We could expand the metric as

gαβ = g(0)αβ + hαβ (69)

where g(0)αβ is a background metric. We can decompose hαβ into a transverse, a traceless part, a

scalar and a vector part. We may define transverse and traceless modes relative to the full metricand write

hαβ = φTTαβ +1

4gαβφ+∇(αξβ) (70)

where φTTαβ is transverse and traceless, φ is a scalar, ξβ is a vector and ∇α is the covariant derivativecompatible with the full metric. However, we need to isolate the conformal mode factor. Analternative approach is to first parameterize a gauge fixed metric with ξβ = 0 and then parameterizegauge transformations induced by ξβ. Writing 1− 1

4φ = e2σ we have

g(gf)αβ = e2σ

(g

(0)αβ + φTTαβ

)(71)

and

gαβ = Φ∗ξ

[g

(gf)αβ

]. (72)

where Φ∗ξ is the pullback by the diffeomorphism generated by the vector field ξα. We may expand

the conformal factor in some set of modes φi as

σ = σ0 +∑k

ckφk. (73)

19

Since we are working in the canonical ensemble, we chose σ0 so that the 4-volume inside some fixedregion[44] is held constant. This makes σ0 a function of the other fields. The remaining fields maybe expanded as

φTTαβ =∑i,k

aikφTT (i)kαβ (74)

and

ξα =∑i,k

dikξikα. (75)

In equations (74) and (75), the index i labels the modes (e.g., transverse modes).The appropriate action in Euclidean spacetime is S[g] = S[ck, a

ik] = SEH [ick, a

ik]. As discussed

in detail by Schleich[23], the Einstein-Hilbert action with an additional factor of i multiplyingthe conformal factor in Euclidean spacetime gives perturbatively identical results to the Einstein-Hilbert action in Lorentzian spacetime. In Euclidean spacetime this action gives a well definedfunctional integral in contrast to the naively Wick rotated Einstein-Hilbert action which does notdue to the conformal mode problem.

In our action, σ or ck represent the conformal mode for a fixed gauge slice, which differs from thedefinition of Schleich. However, Schleich’s analysis applies directly to σ. We may write correlationfunctions with quadratic terms only in the exponent and expand in algebraic powers of the fieldsoutside of the exponent. Then note that if the sign of the part of the Lorentzian action quadraticin ck is reversed and if all factors of ck appearing in algebraic powers outside the exponent arereplaced with ick, the expression does not change. Then Wick rotation results in an extra minussign in the quadratic term in the exponent giving a well behaved conformal mode in Euclideanspacetime. The net result is equivalent to multiplying each ck by a factor of i in the Euclideanaction. Note that this “conformal rotation” must be applied to all factors of ck, including thoseassociated with ghost terms. We may view Wick rotation as mapping a real Lorentzian metric toa complex metric on Euclidean spacetime of the form

gαβ = Φ∗ξ [ick, aik][e2σ0[ick,a

ik]+2i

∑k ciφk

(g

(0)αβ + φTTαβ

)](76)

which can be mapped to the real Euclidean metric of equations (71) and (72) by conformal rotation.From the point of view of a discrete spacetime approach, the discrete structure defines, at least atlow energies, a real Euclidean metric, but the effective action describing the distribution of metricsis more easily described in terms of the complex field gαβ. As we will see, this approach leads to

a partition function that can be written as an integral of e−Seff [ick,aik,ghosts,J

αβ ] over aik, ck andghost fields, where Jαβ is a source coupled to the metric gαβ for producing correlation functions.Although the action here is in a complex form, only real terms contribute to correlation functions.We would obtain equivalent results by defining

e−Seff [ck,aik,ghosts,J

αβ ] = Re(e−Seff [ick,a

ik,ghosts,J

αβ ])

(77)

where Seff [ck, aik, ghosts, J

αβ] is real, at least perturbatively.As pointed out by Schleich, the action defined in this way is not local in the metric g. One

might object that a fundamental theory of gravity might be expected to be local in g. However,in our approach g is not a fundamental field but rather a low energy approximation to discretedegrees of freedom of geometries defined by the arrangement of the discrete 4-dimensional buildingblocks of spacetime. So it is not reasonable to require that the action be local in g. Note, however,that although the action is not local in g, it is separately local in σ and in φTT .

20

To derive the partition function from canonical quantization, we can work directly in Lorentzianspacetime with functions defined by Wick rotation of basis functions defined in Euclidean spacetimeor we can work in Euclidean spacetime with the complex metric gαβ. Alternatively, we can workwith the Euclidean metric gαβ and take ck → ick in the resulting effective action. We will take thisapproach here. As shown in [35], canonical quantization gives the measure

µ[g] =∏xp

1

g3

∏µ≥ν

dgµνδ(Φα) det(LξβΦα) =

∏xc

1

g3/2

∏µ≥ν

dgµνδ(Φα) det(LξβΦα) (78)

where the notation∏xp

indicates that the discrete points used in defining the functional integral are

uniformly distributed in physical distances while∏xc

indicates uniform discretization in coordinatedistances. The equality in equation (78) is to be interpreted as follows. As discussed in [35], themetric can be decomposed in a suitable basis and both the first and second forms of the measurecan be written in terms of the expansion coefficients. The equality in equation (78) means thatone obtains the same measure in terms of the expansion coefficients starting from either the firstor the second forms. The second form, which allows us to derive the truncated theory in a waythat closely parallels Yang-Mills theory, can be derived by introducing ghost fields to cancel thespurious factors of the lapse that would otherwise appear in a uniform coordinate discretization aswe now show.

Consider the action

L =√g(gαβ∂αΦ∂βΦ +m2ΦΦ

)(79)

where Φ is a fermionic scalar field. This would violate the spin-statistics theorem, but we willconsider this to be a ghost field with no physical consequences. We take m >> 2π/Ld so thereare no propagating modes. By discretizing uniformly in physical lengths as discussed in [35], thepartition function for Φ can be written

Z =

∫ ∏xp

dΦ(x)dΦ(x) exp

(−∫d4xL

)(80)

Expanding Φ in a basis as Φ =∑

k akΦk with Φk satisfying∫d4x√gΦ∗kΦk′ = δk,k′ and similarly

for Φ, equation (80) becomes

Z =

∫ ∏k

dakdak exp

(−∫d4xL

)≈∫ ∏

k

dakdak exp

(−m2

∑k

akak

)(81)

where the approximation on the right hand side becomes exact in the limit that m → ∞. Sincethere are a finite number of terms, this partition function is just a finite (for finite m) constantthat does not contribute to any physics. Now consider quantizing the same action and quantize bydiscretizing uniformly in coordinates. Instead of equation (81) we obtain

Z =

∫ (∏k

dakdak

)(∏xc

N2)

exp

(−∫d4xL

)(82)

As discussed in [35], the extra factor of N2 appearing in the measure is an artifact of not discretizinguniformly in physical distances. However, if we consider the ghost fields Φ and Φ simultaneouslywith quantum gravity and discretize uniformly in coordinate distances rather than physical dis-tances, the erroneous factors of N cancel. The standard result for the measure in quantum gravityis[45]

µ[g] =∏xc

1

N2g3/2

(∏µ≥ν

dgµν

)δ(Φα) det(LξβΦα) (83)

21

The total measure when gravity and the ghost fields are quantized using uniform coordinate dis-tances and the ghost fields subsequently integrated out is given by the right hand side of equation(78).

We would like to replace the integral over gαβ with an integral over Ai,k ≡ (aik, ck, dik). To do

this, first define an inner product on the space of metrics:

(gαβ, gγδ) =

∫d4x√ggαβG

αβγδgγδ (84)

with Gαβγδ given by the DeWitt supermetric (i.e., metric on the space of metrics)

Gαβγδ =1

2

(gαγgβδ + gαδgβγ + Cgαβgγδ

)(85)

where C is any constant greater than −1/2. Since we wish to construct a well defined ghost actionto implement the Jacobian from gαβ variables to Ai variables, we need to require C > −1/2 so thatthe supermetric is positive definite. We may simply take C = 0. Up to irrelevant constant factors,we can write the Jacobian from g(xm) variables (with xm distributed uniformly in coordinatedistances) to Ai variables as

det∂gαβ(xn)

∂Aik= det1/2

[∫d4x√g∂gαβ(x)

∂Ai(−k′)Gαβγδ

∂gγδ(x)

∂Ajk

](86)

since in 4-d det(√gG) is a constant. If we were using xm distributed in physical distances rather

than coordinate distances, we would have an additional factor of detG ∝ g−5/2 in this term butwe would end up at the same final result.

We will now apply the the steps leading from equation (45) to equation (49). The main dif-ferences here are the additional tensor index on various quantities and how factors of g entervarious equations. We will work with uniform coordinate discretization using the right hand sideof equation (78). We write

gµν = g(gf)µν +

∑k≤2π/Ld

dik∇(µξiν)k +

∑k>2π/Ld

dik∇(µξiν)k (87)

where dik and dik are expansion coefficients for the infinitesimal gauge transformation and g(gf)µν is

given by equation (71) with modes coefficients ck, aik for k ≤ 2π/Ld and ck, a

ik for k > 2π/Ld. In

order to expand in this way, we need to choose our basis functions such that they contain longitu-dinal functions of the form ∇(µφν)k. We can do this by taking φµk(x) to be vector eigenfunctions of� = ∇µ∇µ and defining a basis for our physical subspace to be functions of the form ∇(µφν)k plusa basis of conformal modes and transverse-traceless functions. In general, the eigenfunctions of �will not be purely transverse or purely longitudinal since ∇ν does not commute with �. However,we can form a suitable basis of transverse modes by taking linear combinations of tensor eigen-functions of � with functions of the form ∇(µφν)k to construct functions satisfying ∇µΦTT

µν = 0.Instead of using equation (71) for the gauge fixed metric, we may take any metric of the form ofequation (72) as our gauge fixed metric with ξµ a fixed element of our basis and with σ and ΦTT

modes restricted to our basis set. A natural choice for the gauge condition is to simply take thegauge fixed metric to be given by equation (71) and to require that the longitudinal modes vanish.

We take the gauge condition to be

0 = Φµ(x; [gµν ]) = ξµ(x) (88)

22

so that the determinant of LξβΦα is unity, where

ξν(x) =∑

k≤2π/Ld

dikξiνk +

∑k>2π/Ld

dikξiνk (89)

The gauge fixing delta function can be replaced with a gauge fixing term in the usual way toobtain

Πxcδ(Φµ)→ exp

(− 1

2εg

∫d4x√gΦµΦµ

)(Πxcg

3/2) (90)

Taking εg → 0 recovers the product of delta functions and we will eventually take this limit. Theprojection onto the physical subspace can be implemented by introducing a factor

exp(− 1

2εp

∫d4x√g(δgαβ)Gαβγδ(δgγδ)

)(91)

into the partition function and taking the limit εp → 0. Here δgαβ is the full metric given byequation (87) minus the metric given by equation (87) with all mode coefficients having k > 2π/Ldset to zero. In order to correctly balance gauge conditions and constraints we must take the limitεp → 0 and then take the limit εg → 0. With this limiting procedure the term from equation (91)projects out fields with k > 2π/Ld including dik fields and the term from equation (90) projectsout dik terms with k ≤ 2π/Ld. This results in the measure

µP [g] =(Πi,k≤2π/Ldda

ikdck

)J/D (92)

where J is the square root of the determinant of the matrix

M =[∫

d4x√g∂gαβ(x)

∂Ai(−k′)Gαβγδ

∂gγδ(x)

∂Ajk

](93)

and D is the square root of the determinant of the block of M having k, k′ > 2π/Ld.The partition function can be written

Q =

∫ ∏daikdckdf

jkdf

jkdb

jkdb

jke−SEH [a,ic]−Sgh[a,ic,f,b] (94)

where

Sgh =∑ijkk′

(f ik′fjk + bi−k′b

jk + bi−k′ b

jk)

∫d4x√g∂gαβ(x)

∂Ai(−k′)Gαβγδ

∂gγδ(x)

∂Ajk, (95)

and f ik and f ik are fermionic ghost fields labeled by arbitrary k, bik is a real scalar ghost field labeledby arbitrary k and bik is real scalar ghost for k > 2π/Ld. The index i labels transverse-tracelessmodes for i = 1...5, the conformal mode for i = 6 and longitudinal modes for i = 7...10. As inthe Yang-Mills case, only a finite number of ghost terms contribute, since the contribution from bikcancels the contribution from the other ghost fields when both momenta are greater than 2π/Ldand so momenta integrals for the ghost loops are effectively cut off (see discussion around equation(40)). Equation (94) is our key result for quantum gravity. We denote the partition function forquantum gravity as Q since it can be regarded as a canonical ensemble partition function havinga fixed number of gravitational degrees of freedom. We will discuss a grand canonical ensemble in[43].

Many properties of equation (94) follow as in the Yang-Mills case. Ward identities can be derivedas in Section VI B and in particular it is easy to see that the polarization for the graviton propagator

23

must be transverse. Diffeomorphism covariance follows from the covariant expansion that was usedand local Lorentz covariance follows from the discussion of Section III. Unitarity follows, as in theYang-Mills case, from the fact that the starting point was a Hermitian Hamiltonian, the reduction tothe physical subspace is done at the path level in a way that preserves unitarity, and no dependenceon unphysical auxiliary fields was introduced since gauge invariance was preserved. As we will seein the following sections, equation (94) is not only finite, but gives a well defined perturbativeexpansion in powers of LPl/Ld if this is assumed to be small enough to act as an expansionparameter. Finally, note that even though we have not introduced a cosmological constant intothe action, the stationary phase approximation to equation (94) yields Einstein’s equation with acosmological constant. This is because if we consider the action to be a functional of the metric anddetermine the stationary phase approximation by extremizing the action with respect to variationsof the metric, we must introduce a Lagrange multiplier to keep the 4-volume fixed. This Lagrangemultiplier plays the role of the cosmological constant. We elaborate on the cosmological constantin our formalism elsewhere[43].

B. Quadratic Fluctuations in Quantum Gravity

We consider quadratic fluctuations in the metric about flat spacetime. Neglecting the cosmo-logical constant, the Einstein-Hilbert action to quadratic order in hµν = gµν − δµν is given by

16πSEH = −1

4hµν�hµν +

1

8h�h− 1

2(∂µhµν −

1

2∂νh)2 (96)

where h = δµνhµν . It is sufficient to consider the metric to first order in σ and ξ. We have

gµν = δµν + 2σδµν + ΦTTµν +∇(µξν) (97)

where the last term is an infinitesimal gauge term that does not contribute to the Einstein-Hilbertaction but does contribute to the ghost action. h is given by 8σ in 4 spacetime dimensions. Wechose the basis functions for ΦTT

µν so that the normalization condition∫d4xΦ

∗TT (i)kµν ΦTT (j) µν

p = δijδkp (98)

holds, where i, j label the 5 independent transverse traceless basis functions for each momentum.Similarly we take the basis for σ to be orthonormal. Equation (96) then becomes

16πSEH [a, c] = −6∑k

k2ckc−k +1

4

∑i,k

k2aikai−k. (99)

So we see that e−SEH [a,ic] can be integrated over aik and ck giving six factors of∏k−1.

At quadratic level, the ghost part of the action corresponding to ai fields are diagonal and withthe normalization condition of equation (98) we have for the transverse traceless contribution tothe ghost action

STTgh =

i=5∑i=1

∑k

(f ikfik + bikb

i−k + bik b

i−k). (100)

Integrating these ghost terms fields gives a constant factor.

24

Let us divide ξµ into a longitudinal part and a transverse part. The transverse part has threecomponents which we label ξiµ for i = 8, 9, 10. (We label these as 8, 9, 10 to match the last threeindices in f i and bi as described in the previous section.) These terms give a contribution to theghost action of

STgh =1

2

i=10∑i=8

∑k

k2(f ikfik + bikb

i−k + bik b

i−k) (101)

Integrating these ghost fields give three factors of∏k≤2π/Ld

k.

The remaining components, labeled 6 and 7 and corresponding to the conformal mode and thelongitudinal gauge mode, respectively, contribute to a non-diagonal part of the ghost action. Therelevant part of the metric can be written as

g(6−7)αβk = dkk(αkβ)e

ikx + 2ckδαβeikx (102)

Denote b6,k the bosonic ghost field associated with ck and b7,k the bosonic ghost field associatedwith dk. Taking C = 0 in equation (85), the ghost action contains

(b6,k b7,k

)(16 2k2k k2

)(b6,−kb7,−k

)(103)

This can be diagonalized if desired, but this is not necessary for our purposes. Note that theeigenvalues are both positive and so the integrals over the ghost fields are well defined. Integratingover the bosonic ghosts and the similar term for the fermionic ghosts gives one factor of

∏k.

We have obtained a total of four factors of∏k from the ghost fields and six factors of

∏k−1

from the six ak and ck fields, so the partition function is

Q =′∏k−2 (104)

or

lnQ = −∑k

∑n

ln(k2 + w2n)Θ((2π/Ld)

2 − (k2 + w2n)) (105)

which is the expected form for 2 bosonic degrees of freedom. This is the same result that weobtained for the Yang-Mills case and it is evaluated in Appendix B.

C. Perturbative Expansion in Quantum Gravity

The functions σ, ΦTTαβ and ξα can be expanded in eigenfunctions of an appropriate covariant

Laplacian as described generally in section IV B. An appropriate Laplacian is � = gab∇a∇b. (Thisis, up to a sign, the Bochner Laplacian. Other Laplacians could be considered, but this is thesimplest choice.) σ0 is fixed by requiring that the 4-volume remain fixed. This means that σ0

should be expanded in a power series of the other fields. As previously mentioned, a technicaldifficulty arises in constructing basis functions for ΦTT : eigenfunctions of � will not be purelytransverse since ∇ν does not commute with �. However, we can form a suitable basis of transversemodes by taking linear combinations of tensor eigenfunctions of � with functions of the form∇(µξν)k where ξνk(x) is a vector eigenfunction of �. This provides a well defined finite set of gaugecovariant basis functions for expanding the metric.

25

In contrast to Yang-Mills theory, quantum gravity is non-renormalizable. Diagrams that areincreasingly divergent in the conventional approach, are proportional to increasingly large powersof the cutoff in our approach. Since the cutoff is proportional to LPl/Ld, increasingly high orderdiagrams contains factors LPl/Ld to increasingly high powers. If we assume that LPl/Ld is small,then this quantity becomes an expansion parameter that allows a well defined perturbative expan-sion to be carried out for quantum gravity. This can be seen by scaling every length scale in theproblem by Ld. This leaves a factor of L2

d/L2Pl in front of the action. A further rescaling of the

fields aik, ck by Ld/LPl shows that expansions in powers of the rescaled aik, ck give an extra factorof LPl/Ld with each factor of aik, ck beyond quadratic order. This shows that LPl/Ld plays therole of a coupling constant.

We can explicitly demonstrate such a rescaling as follows. The typical quadratic term in theaction can be written in the form

1

L2Pl

∑k

k2aka−k → V (4) 1

L2Pl

∫d4k/(2π)4k2aka−k = V

∫d4k/(2π)4k2aka−k (106)

where k = Ld/(2π)k, ak = ak2π

LdLPl, V = V (4)

(2π/Ld

)4and ak represents any of the aik or ck terms

in our expansion of the metric. A term containing n ak terms and no ghost terms can be writtenin the general form(

2πLPlLd

)n−2

V n/2

∫d4k1

(2π)4

∫d4k2

(2π)4...

∫d4kn−1

(2π)4f(ki)ak1ak2...akn (107)

where kn is minus the sum of the remaining n−1 momenta and f(ki) is a function of the momentathat depends on the term in question. This factor can simply be a dot product of two momenta or itcan contain integrable singular terms arising from the perturbative expansion of the eigenfunctions.Ghost fields not in the ∇(αξβ) sector can be scaled by a factor of k so that the ghost action isquadratic in k. The ghost fields then scale in the same was as the ak fields. From the scalingdescribed here, we conclude that

(2πLPlLd

)serves as an expansion parameter.

If Ld is large, we therefore obtain Einstein’s equation (with a cosmological constant) as astationary phase approximation to the partition function with perturbations around the stationaryphase solutions giving small contributions. If Ld were not large, it is not clear that Einstein’sequation would be meaningful, since fluctuations could not be neglected. On the other hand, if Ldwere large compared to 103LPl, it would likely cause difficulties with any reasonable GUT model.It would seem that a reasonable guess might be that Ld is somewhere in the range of about 10−103

Planck lengths. An interesting possibility is that Ld might be on the GUT scale and that gravitycould be included with the other forces in a GUT scheme.

VIII. QUANTUM ELECTRODYNAMICS

A. Scalar QED

We expand the scalar field in eigenfunctions of the gauge covariant Laplacian as

Φ(x) =∑k

ckΦk(x) (108)

where Φk(x) is related to the unnormalized functions Φk(x) as in equation(24) and to second order

√V (4)Φk = eikx +

∑′

peipx〈p|V |k〉p2 − k2

+∑′

p,qeipx

〈p|V |q〉〈q|V |k〉(p2 − k2)(q2 − k2)

(109)

26

where

V = 2ieAµ∂µ + ie(∂µAµ)− e2AµAµ. (110)

The gauge covariant Laplacian for the vector potential is just the ordinary Laplacian and so thevector potential can be expanded as

Aµ(x) =1√V (4)

∑k

abkvbµke

ikx (111)

where vbµk are transverse unit vectors for b=1,2,3, and for b = 4, v4µk is the unit vector parallel

or anti-parallel with k. It is convenient to define vbµk such that vbµk = vbµ(−k) so that the reality

condition on Aµ gives (abk)∗ = ab−k.

We can then apply the techniques of Section VI A except that we need to add matter fieldsto the partition function. To do this we need the Jacobian of the transformation from f(x) =

(Φ(x), Φ(x), Aµ(x)) to bk = (ck, c†k, a

ik). Realizing that the complex field notation is a shorthand

for separately integrating over the real and the imaginary parts of the field and that the Jacobianmatrix between Re[Φ], Im[Φ] and Φ, Φ is just a constant, the Jacobian can be written, up toirrelevant constant factors, as

J2 =( ∂f∂bk

,∂f

∂bk

)(112)

where the inner product is defined by

(f1, f2) =

∫d4xf1(x) · f2(x) =

∫d4x(Φ1(x)Φ2(x) + Φ1(x)Φ2(x) +A1µ(x)Aµ2 (x)). (113)

Arranging the indicies of the Jacobian matrix with Φ and Φ entries ahead of all Aµ entries, we seethat the matrix is upper tridiagonal so that it’s determinant is the product of the diagonal entries.This product is independent of the fields and so is a constant that can be dropped. The functionalmeasure of equation (47) is then Πkda

1kda

2kda

3kk where the factor of k arises from the difference

in the Jacobian in equation (47), which is the Jacobian for transforming from Aµ(x) variables toa1k, a

2k, a

3k, dk = a4

k/k variables, and the constant Jacobian of equation (112).We can introduce a gauge parameter by inserting the factor

1 =∏k

∫da4

kδ(ka4k −Xk)k (114)

into the functional integral where Xk is arbitrary. Then following the common procedure, wemultiply by e−(Xk)2/2ξ and integrate over Xk for a fixed gauge parameter ξ. After introducing thegauge parameter, the functional measure becomes∏

dckdc∗kda

αkk

2e−(ka4k)2/2ξ (115)

where α ranges over the three transverse modes and the longitudinal mode and the factor of k2 isneeded only to keep track of the appropriate number of degrees of freedom in the partition function(i.e., it cancels 2 of the 4 factors of k−1 arising from integrating the 4 aα fields). For calculatingcorrelation functions the factor of k2 can be dropped since it is independent of the fields.

The scalar field part of the action contains a term proportional to∑k,k′

c∗kck′

∫d4xΦ∗k(x)�Φk′(x). (116)

27

Each term in this sum is proportional to δ|k|,|k′| and will therefore contribute to the polarizationin the thermodynamic limit (the large L limit discussed in Section IV B) only if a momentumconservation delta function forces |k| = |k′| which occurs only in a loop diagram having k = k′.Dropping the off-diagonal terms that don’t contribute to the polarization in the thermodynamiclimit, the scalar field part of the action can be written

S = −∑k

c∗kck(λk[A]−m2) (117)

where the sign convention that the partition function contains the factor e−S is used, λk[A] is theeigenvalue of the gauge covariant Laplacian determined as in equation (27), and the sum shouldbe understood to be restricted to k no more than 2π/Ld. The eigenvalue can be written

λk[A] = −k2 −∑′

pe2Ap ·A−p + e2

∑′

p

((k + p) ·Ap−k)((k + p) ·Ak−p)p2 − k2

. (118)

The polarization is then given by

Πabq =

1

V (4)

∑′

k

2

k2 +m2

[−e2δµν + 4e2 (kµ + qµ/2)(kν + qν/2)

(k + q)2 − k2

]vaµq vbν−q (119)

where a, b label the basis as in equation (111). By symmetry Πabq is proportional to δab for a, b =

1, 2, 3 and Π4bq = Πa4

q = 0 for all a and b. Using i, j to denote the transverse components, we have

Πijq =

1

3δij

1

V (4)

∑′

k

2

k2 +m2

[−e2δµν + 4e2 kµkν

(k + q)2 − k2

](δµν − qµqν) (120)

or

Πijq =

1

3δij

1

V (4)

∑′

k

2

k2 +m2

[−3e2 + 4e2k2 1− x2

q2 + 2kqx

](121)

where x is the cosine of the angle between k and q. Performing a 4-d spherical average over x, wecan write

Πijq =

1

3δij

1

V (4)

∑′

k

2

k2 +m2

2

π

∫ 1

−1dx(1− x2)1/2

[−3e2 + 4e2k2 1− x2

q2 + 2kqx

]. (122)

Using the principle parts integral

2

π

∫ 1

−1dx

(1− x2)3/2

(q/2k) + x=( q

2k

)(3− 2(q/2k)2) + 2[(q/2k)2 − 1]3/2Θ((q/2k)− 1), (123)

we have

Πijq =

4e2

3δij∫k<2π/Ld

d4k

(2π)4

1

k2 +m2

[−( q

2k

)2+

2k

q

(( q2k

)2 − 1)3/2

Θ(q − 2k)]. (124)

Using Πµνq =

∑Πabq v

aµq vbν−q =

∑Πijq v

iµq v

jν−q, we have

Πµνq =

4e2

3(qµqν − q2δµν)

∫k<2π/Ld

d4k

(2π)4

1

k2 +m2

[1

4k2− 2k

q3

(( q2k

)2 − 1)3/2

Θ(q − 2k)

]≡ (q2δµν − qµqν)Π(q2) (125)

28

which is the form required by gauge invariance and Lorentz covariance. So we have

Π(q2 = 0) = −e2 4

3

∫k<2π/Ld

d4k

(2π)4

1

k2 +m2

1

4k2= − e2

48π2ln

(1 + (2π/(mLd))

2

)(126)

and

Π(q2)−Π(q2 = 0) = e2 4

3

∫k<2π/Ld

d4k

(2π)4

1

k2 +m2

2k

q3

(( q2k

)2 − 1)3/2

Θ(q − 2k). (127)

Substituting z = 2k/q this gives

Π(q2)−Π(q2 = 0) =e2q2

96π2

∫ 1

0dzz4(z−2 − 1)3/2 1

q2z2/4 +m2. (128)

Define x± = 12 [1± (1− z2)1/2] then z2 = 4x±(1− x±) and [z−2 − 1]1/2dz = ∓2[z−2 − 1]dx±. Using

the lower sign converts the integral over z into an integral over x− from 0 to 1/2, while usingthe upper sign converts the integral over z into an integral over x+ from 1/2 to 1. Adding bothexpressions and dividing by 2, we have

Π(q2)−Π(q2 = 0) =e2q2

96π2

∫ 1

0dx

(1− 2x)4

q2x(1− x) +m2. (129)

Using

d

dxln(m2 + q2x(1− x)

)=

q2(1− 2x)

q2x(1− x) +m2(130)

and integrating by parts, we have

Π(q2)−Π(q2 = 0) = − e2

4π2

∫ 1

0dx (x− 1/2)2 ln

(m2

m2 + q2x(1− x)

)(131)

which agrees with the result obtained in the standard way[46]. We now turn to the more physicallyinteresting case of spinor QED.

B. Spinor QED

The Euclidean action is

S =

∫d4xΨ(γµ(∂µ + ieAµ) +m)Ψ +

1

4FµνFµν (132)

We will refer to the operator −iγµ∂µ as the non-interacting Euclidean Dirac operator. (The factorof i is needed for Hermicity. The factor of −1 is an arbitrary factor that we include to makethe ups eigenvectors, which are defined below, have a positive eigenvalue.) The Euclidean Diracoperator is Hermitian and therefore has real eigenvalues and its eigenvectors form a complete set.To determine the eigenstate, we write the eigenvalue problem as(

−i∂0 −σi∂iσi∂i i∂0

)(ΨA

ΨB

)= λ

(ΨA

ΨB

)(133)

where we have used the gamma matrix representation from Sakurai[47]. We substitute upseipx

with 4-spinor u to find the eigenvalue. This implies ΨA = iσ·pp0−λΨB and ΨB = iσ·p

p0+λΨA. Combining

29

these we have λ = ±(pαpα)1/2. Note that we could have immediately determined the eigenvaluesby applying the Dirac operator twice and using the algebra of the gamma matrices.

Denoting the eigenvectors with positive eigenvalue as upseipx and the eigenvectors with negative

eigenvalues as vpseipx, it is straightforward to verify that

ups = N

(ξs

iσ·pp0+(pαpα)1/2

ξs

)(134)

and

vps = N

(iσ·p

p0+(pαpα)1/2ξs

ξs

)(135)

where N2 = p0+(pαpα)1/2

2(pαpα)1/2. From this we obtain

∑s

upsu†ps =

γµpµ + (pαpα)1/2

2(pαpα)1/2(136)

and ∑s

vpsv†ps =

−γµpµ + (pαpα)1/2

2(pαpα)1/2. (137)

Note that these equations follow from rotation invariance after being verified in a frame wherep is in the x0 direction. Note also that u†u and v†v are rotation invariant. There is no need for afactor of γ0 to obtain a rotation invariant quantity. This difference in the Euclidean case from theLorentz case arises because the generator of rotations Sµν = i

4 [γµ, γν ] is Hermitian in the Euclideancase, but the corresponding generator of Lorentz transformations is not Hermitian because thereis no basis for Lorentzian gamma matrices where all gamma matrices are Hermitian.

We will drop V0 terms since these can only contribute to tadpole diagrams which vanish bysymmetry. Let λksw[A] be the eigenvalue (determined as in equation (27)) of the interactingEuclidean Dirac operator −iγµ(∂µ + ieAµ), where k is a momentum index, s is a spin index, andw = ±1 denotes the positive or negative eigenvalue. Then to second order

λksw[A] = wk|k|+∑′

p,wp+k

(〈kswk|e /A−p(wp+k(/p+ /k) + |p+ k|)e /Ap|kswk〉

2|k + p|(wk|k| − wp+k|p+ k|)

)(138)

or

λksw[A] = wk|k| − e2∑′

p

(〈kswk| /A−p((/p+ /k) + wk|k|) /Ap|kswk〉

(p+ k)2 − k2

). (139)

Dropping off-diagonal terms that do not contribute to the polarization in the thermodynamic limit,the fermion and interaction part of the action becomes

S =∑

c∗kswcksw(iλksw +m) (140)

and we find

Πabq = −e2 1

V (4)

∑′

k,s,wk

2i

iwkk +m

(〈kswk|/vb−q((/q + /k) + wk|k|)/vaq |kswk〉

(q + k)2 − k2

). (141)

30

Using equations (136) and (137), this can be written

Πabq = −e2 1

V (4)

∑′

k,wk

i

(iwkk +m)k

(tr[/vb−q((/q + /k) + wk|k|)/vaq(wk/k + k)]

(q + k)2 − k2

). (142)

Using the fact that the trace of an odd number of gamma matrices vanish and using

tr[/vb−q(/q + /k)/vaq/k] = 4((k + q) · vaq )(k · vb−q) + 4(k · vaq )((k + q) · vb−q)− 4(vaq · vb−q)(k2 + q · k) (143)

and

tr[/vb−q/vaq ] = 4(vaq · vb−q), (144)

we have Πabq = 0 if either a or b are longitudinal and for the transverse components we have

Πijq = −e2 1

V (4)

∑′

k,wk

4iwk(iwkk +m)k

(2(k · viq)(k · v

j−q)− (viq · v

j−q)(q · k)

(q + k)2 − k2

)(145)

or

Πijq = −e2 1

V (4)

∑′

k

8

k2 +m2

(2(k · viq)(k · v

j−q)− (viq · v

j−q)(q · k)

(q + k)2 − k2

)(146)

which can be written

Πijq = −e2 δ

ij

3

1

V (4)

∑′

k

8

k2 +m2

(2kµkν − δµν(q · k)

(q + k)2 − k2

)(δµν − qµqν). (147)

Letting x be the cosine of the angle between k and q we have

Πijq = −e2 δ

ij

3

1

V (4)

∑′

k

8

k2 +m2

(2k2(1− x2)− 3qkx

q2 + 2kqx

)(148)

or

Πijq = −e2 δ

ij

3

∫k<2π/Ld

d4k

(2π)4

4

k2 +m2

2

π

∫ 1

−1dx(1− x2)1/2

((2k/q)(1− x2)− 3x

(q/2k) + x

). (149)

Using the principal part integral

2

π

∫ 1

−1dxx(1− x2)1/2

(q/2k) + x= (1− 2(q/2k)2) +

( qk

)[(q/2k)2 − 1]1/2Θ((q/2k)− 1) (150)

and equation (123), we have

Πijq = −δij e

2

3

∫k<2π/Ld

d4k

(2π)4

4

k2 +m2×

×

(( qk

)2+ Θ((q/2k)− 1)

[−3(q/k)[(q/2k)2 − 1]1/2 +

(4k/q

)[(q/2k)2 − 1]3/2

])(151)

or

Πijq = −e2δij

∫k<2π/Ld

dk

6π2

k3

k2 +m2

(( qk

)2− 2Θ((q/2k)− 1)[(q/2k)2 − 1]1/2

[q/k+ 2k/q

]). (152)

31

Writing Πµνq =

∑Πijq viqv

j−q = Π(q2)(q2δµν − qµqν), we have

Π(q2 = 0) = −e2

∫k<2π/Ld

dk

6π2

k3

k2 +m2

1

k2= − e2

12π2ln(1 + (2π/(mLd))

2)

(153)

and

Π(q2)−Π(q2 = 0) = −e2

∫k<2π/Ld

dk

3π2

k3

k2 +m2

1

q2

(Θ((q/2k)− 1)[(q/2k)2 − 1]1/2(q/k + 2k/q)

).

(154)Note that since Ld is considered to be a finite physical length scale, Π(0) is a finite quantity.Making the substitution z = 2k/q we have

Π(q2)−Π(q2 = 0) = e2

∫ 1

0

dz

24π2

q2

q2z2/4 +m2[z−2 − 1]1/2z2(1 + z2/2). (155)

Define x± = 12 [1± (1− z2)1/2] then z2 = 4x±(1− x±) and [z−2 − 1]1/2dz = ∓2[z−2 − 1]dx±. Using

the lower sign converts the integral over z into an integral over x− from 0 to 1/2, while usingthe upper sign converts the integral over z into an integral over x+ from 1/2 to 1. Adding bothexpressions and dividing by 2 we have

Π(q2)−Π(q2 = 0) = e2

∫ 1

0

dx

24π2

q2

q2x(1− x) +m2(1− 2x)2(1 + 2x(1− x)) (156)

Using

d

dxln(m2 + q2x(1− x)

)=

q2(1− 2x)

q2x(1− x) +m2(157)

and

d

dx

((1− 2x)(1 + 2x(1− x))

)= −12x(1− x) (158)

we can integrate by parts to obtain

Π(q2)−Π(q2 = 0) = − e2

2π2

∫ 1

0dx x(1− x) ln

(m2

m2 + q2x(1− x)

). (159)

This is the standard result[46] in Euclidean spacetime with our sign conventions. In contrast toconventional approaches, we obtained this result without encountering any divergences along theway. We still have to renormalize as in the standard approaches, however all counter terms arefinite and so the bare charge, for example, is also finite.

C. Self Energy

The Green’s function can be written

G(k) =

∫d4xd4x′/V (4)e−ik(x−x′) < TΨ(x)Ψ†(x′) > (160)

where Ψ(x) can be expanded in the orthonormal basis Ψkws which can be written in terms of thenon-normalized eigenfunctions, Ψkws. as

Ψkws = Ψkws +∑k′w′s′

Nkws;k′w′s′Ψk′w′s′ (161)

32

where the coefficients Nkws;k′w′s′ are determined by the orthonormalization conditions. We can splitthis into the bare propagator plus corrections from the Aµ dependence of the eigenvalue, λkws, thenon-normalized eigenfunction,Ψkws, and the diagonal part of the orthonormalization coefficients,Nkws ≡ Nkws;kws, as follows

δG = G(k)−G0 = δλG+ δΨG+ δNG, (162)

where

δΨG = −′∑

kwswsws

1

ikw +mξkws

< kws|(ie /A)|kws >< kws|(ie /A)†|kws >(wk − wk)(wk − wk)

ξ†kws (163)

δλG =∑ws

i(λkws − kw)

(ikw +m)2ξkwsξ

†kws (164)

and

δNG = −∑ws

1

ikw +m

(2Nkws

)ξkwsξ

†kws. (165)

Although not explicitly written, the above equations should be understood to include a factor ofΘ((2π/Ld)

2 − q2)

for each photon q momenta. Each of equations (164) and (165) also includes afactor of Θ

((2π/Ld)

2 − k2), while equation (163) includes a factor of Θ

((2π/Ld)

2 − k2)

which canbe written as

Θ((2π/Ld)

2 − k2)

= Θ((2π/Ld)

2 − k2)

+[Θ((2π/Ld)

2 − k2)−Θ

((2π/Ld)

2 − k2)]. (166)

It is readily verified that the term in square brackets does not contribute to equation (163) in thelimit that k/(2π/Ld) → 0. We will therefore drop this term since our main goal for this sectionis to verify that the standard result is obtained in the limit that k << 2π/Ld. The quantityΘ((2π/Ld)

2 − k2)

is unity in this limit so the only step functions that need to be retained are theΘ((2π/Ld)

2 − q2)

factors.In the Feynman gauge, the functional integral over the photon field gives

/A−qw/k + k

2k/Aq →

1

q2

−w/k + 2k

k(167)

where q = k − k. Note that terms proportional to qµ can be evaluated by replacing qµ withkµkαqα/k

2 since the part of q orthogonal to k integrates to zero in the expressions for δG. Usingstandard properties of the γ matrices, we have

δΨG+ δNG = e2 1

V (4)

′∑kwswsws

h(w, w, w)

(wk − wk)(wk − wk)

(δw,w

ikw +m− 1

ikw +m

)1

q2(168)

where

h(w, w, w) =1

2

((w + w)/k/k + (ww + 1)

)(1− ww k

2k(1 + k · q/k2)

). (169)

Each of w, w and w are ±1. Keeping track of the discrete number of possibilities, we find

δΨG+δNG = e2 1

V (4)

′∑kw

iw

ikw +m

1

k2 − k2

1

k2 +m2

(h−(k− k)(ikw+m)−h+(k+ k)(−ikw+m)

) 1

q2

(170)

33

where h+ = h(w, w, w) and h− = h(−w,−w, w). Note that both δΨG and δNG contain divergencesrelated to the fact that we determined our eigenfunction perturbative expansion without diagonal-izing in the degenerate subspace. As discussed in Section IV B, such divergences must cancel andwe see from equation (170) that indeed they do cancel here. After some algebra, we find

δΨG+ δNG = e2 1

V (4)

∑′ 1

(k2 +m2)2

( 1

k2 +m2− 1

q2 + 2k · q)×

×(i/k(4(k2 −m2) + 4(k2 + k · q)m

2 − k2

k2

)− 4m(k2 + k · q + q2)

) 1

q2. (171)

We also have

δλG = −2e2 1

V (4)

∑′ 1

(k2 +m2)2

1

q2 + 2k · q(i/k(k2 − k · q)m

2 − k2

k2+ 2m(k2 − k · q)

) 1

q2. (172)

Combining the separate terms, we find

δG = 2e2 1

V (4)

∑′ 1

(k2 +m2)2

1

k2 +m2

(−i/k(4m2−(k2 +k ·q)m

2 − k2

k2)+2m(m2 +k ·q)

) 1

q2. (173)

We will need the following integral which follows from shifting the integration variable:

I =

∫d4q

(2π)4

Θ((2π/Ld)2 − q2)

((k + q)2 +m2)(q2 + µ2)=

∫d4q

(2π)4

∫ 1

0dz

Θ(q2m(x, z)− q2)

(q2 + ∆2)2(174)

where ∆2 = −(1−z)2k2+(1−z)(k2+m2)+zµ2 and qm(x, z) satisfies q2m−2(1−z)kqmx+(1−z)2k2 =

(2π/Ld)2 and x is the cosine of the angle between k and q. The parameter µ2 is a fictitious small

photon mass that is introduced to facilitate comparison with standard results. The limit µ2 → 0is understood. The solution for qm is

qm(x, z) =2π

Ld

[(1− (k(1− z))2(1− x2)

(2π/Ld)2

)1/2

+kx(1− z)

2π/Ld

]. (175)

We have

I =1

8π3

∫ 1

−1dx(1− x2)1/2

∫ 1

0dz

[ln(q2

m(x, z) + ∆2(z)

∆2(z)

)− q2

m(x, z)

q2m(x, z) + ∆2(z)

]. (176)

We will also need

J =

∫d4q

(2π)4

Θ((2π/Ld)2 − q2)

((k + q)2 +m2)(q2 + µ2)

k · qk2

=

∫d4q

(2π)4

∫ 1

0dz

Θ(q2m(x, z)− q2)

(q2 + ∆2)2(kqx/k2 − (1− z))

(177)or

J =1

4π3k

∫ 1

−1dx(1− x2)1/2x

∫ 1

0dz

[qm −

3

2∆ tan−1(qm/∆) +

qm2

∆2

∆2 + q2m

]− I2 (178)

where I2 is the integral for I with a factor of (1− z) inserted into the integrand. So we have

δG = 2e2 1

(k2 +m2)2

(i/k((I + J)(m2 − k2)− 4m2I

)+ 2m(m2I + k2J). (179)

From this the self energy is determined to be

Σ = 2e2(i/k(I + J) + 2mI

). (180)

34

We have already assumed that k << 2π/Ld in dropping the term in square brackets in equation(166). Taking m << 2π/Ld as well and expanding accordingly, we find that to leading order

Σ =e2

8π2

∫ 1

0dz(2m+ iz/k

)ln

[z(2π/Ld)

2

(1− z)m2 + zµ2 + z(1− z)k2

]. (181)

So we recover the standard result[46] for momentum and mass small compared to 2π/Ld. Thereare corrections to continuum spacetime quantum field theory, however, when the momentum is notsmall compared to 2π/Ld.

IX. DISCUSSION AND CONCLUSION

We started from the notion that underlying the Lorentzian spacetime that we experience is adiscrete pre-spacetime that must be averaged in a statistical mechanical sense to determine theLorentzian spacetime. The discrete structure in this picture is considered to consist of extended 4-dimensional objects that make up the pre-spacetime and define a metric with Euclidean signature.Quantum field theory in our approach can be described as the statistical mechanics of fields sincethe Euclidean domain is considered to be fundamental. Lorentzian time is not fundamental, butrather is an emergent quantity that is defined by the Wick rotation procedure. We can think oftime in this picture as measuring the evolution of a system that is perturbed from equilibrium.

We developed a continuous approximation to the discrete pre-spacetime structure where thecutoff is the only feature of the discrete structure that is retained. Such an approach gives a welldefined description of low energy physics without any UV divergences. An alternate approach,which we did not pursue here, would be to work directly in the discrete Euclidean pre-spacetime.One can, in principle, conduct a lattice gauge theory calculation for each “lattice” (not necessarilyperiodic) in an ensemble of lattices and average the results. This will give an SO(4) covariant resultin Euclidean spacetime and will generate a Lorentz covariant result after Wick rotation. The maindisadvantage of such an approach for gauge theories is that it is not analytically tractable unlessone restricts the lattices to simple tractable lattices. For quantum gravity, the main difficulty isthat we don’t know what the discrete action actually is or even precisely what the relevant degreesof freedom are at a fundamental level, though according to our approach, the relevant degrees offreedom must relate to the configurations of the discrete extended 4-dimensional building blocks ofspacetime. As long as we are interested in low energy physics, however, a continuous approximationto the discrete structure is reasonable.

We have shown that UV-finite gauge theories can be constructed by expanding fields in a finiteset of gauge covariant basis functions. We have derived partition functions for Yang-Mills theoryand for quantum general relativity using this approach. Let us summarize the properties of thesepartition functions. First, the partition functions are finite since they involve only a finite numberof degrees of freedom and the functional integrals are well defined, at least perturbatively, since thequadratic part of the action is positive. Second, the partition functions generate unitary evolutionsince each partition function follows from a Hamiltonian where unitarity is manifest. In the case ofYang-Mills theory, renormalization can be carried out as is usually the case except that divergentquantities in the conventional approaches are replaced with quantities such as 2π/Ld which woulddiverge if Ld were taken to zero. To verify our formalism, we have presented detailed calculationsfor vacuum polarization and electron self-energy in QED and we have obtained standard resultsfor the finite parts of these quantities with no UV-divergences arising in our approach. In thecase of quantum gravity, the partition function generates an expansion in powers of 2π/Ld. If thisis assumed to be a small quantity, we obtain a well defined perturbative expansion for quantumgravity that preserves general covariance and, according to the derivation of the partition function,

35

generates unitary time evolution. Although detailed calculations for quantum gravity beyond thelevel of quadratic fluctuations remain to be worked out, our approach would appear to provide apromising framework for understanding quantum gravity.

The partition function for quantum gravity yields Einstein’s equation with a cosmological con-stant as a stationary phase approximation. The cosmological constant enters as a Lagrange mul-tiplier which must be introduced in order to vary the metric while holding the 4-volume fixed. Itis necessary to hold the 4-volume fixed when varying the metric since the theory is formulatedin terms of a finite number of degrees of freedom for any given 4-volume. We have limited ourdiscussion in the present paper to the basic formulation of the theory and some basic calculationsin QED. We will discuss in more detail elsewhere[43], the relationship between the Lagrange mul-tiplier, the chemical potential and the cosmological constant. As discussed in [43], the existence ofa cosmological constant with a magnitude many orders of magnitude below the Planck scale canbe interpreted as an indication that discrete degrees of freedom underly spacetime. We will alsodiscuss the application of the discrete spacetime approach to black holes and the early universeelsewhere[48] and we will argue following [5] that a spectral index observed to be about 0.96 canbe interpreted as an indication that discrete degrees of freedom underly spacetime.

We have emphasized that it is necessary to understand quantum field theory and quantumgravity in terms of a local thermodynamic equilibrium approximation. Although we have dis-cussed only local physics, non-local physics can in principle be determined by piecing togetherneighboring regions. For example, although we have defined the Green’s function in terms of asingle region in local thermodynamic equilibrium, we can in principle piece together many suchregions to construct a Green’s function describing propagation over arbitrarily large distances.This can be approximately achieved by modeling interactions locally and extending the 2-pointfunctions to arbitrary distances by solving the generally covariant differential equation satisfied bythe propagators.

Our approach provides a sensible perturbative expansion for quantum gravity if the length scaleLd is large compared to the Planck scale. Since the GUT scale is only a few orders of magnitudebelow the Planck scale, an interesting possibility that Ld could be at or near the GUT scale.This would open the possibility that any unified theory should be based on a discrete underlyingspacetime rather than a continuous spacetime as is typically assumed and also suggests that gravitymay need to be included with the other three forces in a GUT scheme. Another interesting possibleconsequence of having Ld near the GUT scale is a modification to the flow of the coupling constantsas the GUT scale is approached. As is well known, the strong, weak and QED coupling constantsalmost but don’t quite meet at the GUT scale according to conventional calculations. This istypically considered to be indirect evidence for supersymmetry which can modify the couplingconstant flow so that the three coupling constants meet at a common scale. The renormalizationgroup calculations which determine the coupling constant flow are based on a continuous spacetimedescription. An interesting possibility is that the coupling constant flow is modified by discretespacetime effects as the energy scale approaches the GUT scale. This does not occur for theQED coupling in the calculations that we have presented here at one loop order. However, thesecalculations are based on a continuous description of a discrete structure where the cutoff is theonly feature of the discrete structure that is retained. One would expect corrections that are notincluded in such a continuous description could become important for energy scales on the orderof 1/Ld. If discrete spacetime effects modify the flow of the coupling constants as the GUT scaleis approached, it is possible that supersymmetry enters at a higher energy scale so that a smallershift in the coupling constant flow is needed or it is possible that supersymmetry is not needed atall for the coupling constants to meet at a common energy scale.

36

Appendix A: Local Frame for Wick Rotation

Here we show that locally about any point any Lorentzian metric can be put in the form ofequation (2). We start with normal coordinates (for example, Riemann normal coordinates) andconsider a series of coordinate transformations of the form

xα = xα +Aαβγδxβxγ xδ. (A1)

We will keep only terms up to quadratic order in coordinate distances and we show that terms notincluded in equation (2) can be eliminated with a suitable choice of Aαβγδ. The proof proceeds inseveral steps in which separate terms containing one or two powers of time are eliminated. Somesteps generate additional time-dependent terms that are eliminated in subsequent steps. Each stepcan otherwise be treated independently since we are dropping cubic and higher order terms. Wegenerically denote coefficients of the coordinate changes by A (with appropriate superscripts andsubscripts) in each step, but the A’s in each step are independent.

1. Elimination of dxidxj t2 and dtdx[ixj] t termsConsider

xi = xi + t2xjAij (A2)

The metric picks up a term of the form

4dt dxi xj t Aij + 2t2 dxi dxj Aij (A3)

where indicies on Aij are raised and lowered with δij . A(ij) can be chosen to eliminate the dxidxjt2

terms and A[ij] can be chosen to eliminate dtdx[ixj]t terms.

2. Elimination of dtdx(ixj)t termsConsider

t = t+ txixjAij (A4)

The metric picks up an x-dependent dt2 term that we don’t need to eliminate and a term of theform

−4dtdx(ixj)tAij (A5)

Aij can be chosen to eliminate the dtdx(ixj)t terms.

3. Elimination of dt2 t2 termsConsider

t = t+ t3A (A6)

The metric picks up a term proportional to dt2 t2 A and so A can be chosen to eliminate the dt2 t2

terms in the metric.

4. Elimination of dt2 t x termsConsider

t = t+ t2xiAi (A7)

This gives a term proportional to dt2 t xiAi which can be used to eliminate dt2 t x terms in themetric. It also generates a dt dxiAi t

2 term which will be eliminated in the next section.

37

5. Elimination of dt dxi t2 termsTransformations of the form xi = xi + t3Ai can be used to eliminate dt dxi t2 terms.

6. Elimination of Ckij t xk dxi dxj terms

Here Ckij = Ck(ij) is a set of constants. First note the following

Ckijxkdxidxj = Ckijx

(kdxi)dxj +1

2Ckijx

kdxidxj − 1

2Ckijx

idxkdxj . (A8)

Since Ckij = Ck(ij), the last term can be rewritten as −12Ckijx

(idxj)dxk. Relabeling indicies in thisterm and moving the second term to the left hand side we have

Ckijxkdxidxj =

(2C(ki)j − Cj(ki)

)x(kdxi)dxj (A9)

The coordinate transformation

xi = xi + txj xkAijk (A10)

generates the metric term

4tx(kdxj)dxiAijk + 2dtdxixjxkAijk (A11)

The second term will be addressed in the next section. The first term can be used to eliminate theCkijt x

k dxi dxj term with the choice

Ajki = −1

4

(2C(ki)j − Cj(ki)

)(A12)

Steps 1-6 puts the metric in the form of equation (2). We cannot eliminate a term in the metricof the form dtdxixjxkBijk where Bijk is a set of constants since this term contains physicallymeaningful information on rotation in the spacetime geometry. We can however eliminate a partof this term.

7. Elimination of symmetric part of dtdxixjxkBijkWe can keep a term like dtdxixjxkBijk in the metric and still have a real metric in both Lorentzianand Euclidean spacetime by requiring Bijk → iBijk under Wick rotation. Such a term generatesan R0i term which is proportional to a velocity by the Einstein equation, so it is sensible to requireBijk to pick up a factor of i under Wick rotation. However, the completely symmetric part of Bijkcan be eliminated by a coordinate transformation of the form

t = t+ xixj xkAijk. (A13)

This generates a term in the metric proportional to

dtdx(ixj xk)Aijk (A14)

The coefficients Aijk can be chosen to set B(ijk) = 0. Note that Aijk is completely determined bythe condition B(ijk) = 0, so if we impose B(ijk) = 0 as a gauge condition, the metric of equation(2) is uniquely determined up to transformations of the spatial coordinates.

Appendix B: Exact Summation for the Partition Function for Yang-Mills Theory and forQuantum Gravity at Quadratic Order

In this appendix, we derive an expression for the sum over frequencies appearing in equations(51) and (105). The partition function for quantum gravity can be written as

lnQ = −′∑k

n=m∑n=−m

ln(k2 + w2n) (B1)

38

where wn = 2πn/β and m, which is the integer part of ((2π/Ld)2−k2)1/2β/(2π), is the maximum

value of n where the step function is satisfied. The prime on the sum over k indicates that thesum is restricted to k2 ≤ (2π/Ld)

2.

The partition function is a finite sum of finite terms and is therefore obviously finite andso regularization is not necessary in order to evaluate Equation (B1). However, zeta functionregularization allows us to derive the standard results for the partition function from Equation(B1) up to small corrections due to the cutoff. We define the generalized zeta function as

ζ(s) =′∑k

n=m∑n=−m

λ−sk,n. (B2)

Note that this is a sum of a finite number of finite analytic functions of s and so ζ(s) is an analyticfunction of s. The partition function can be written

lnQ = ζ ′(0). (B3)

We now add and subtract an infinite number of terms to obtain

ζ(s) =′∑k

n=∞∑n=−∞

λ−sk,n − 2′∑k

n=∞∑n=m+1

λ−sk,n. (B4)

The first term is the usual zeta function for a boson field and thus gives the usual partitionfunction in Equation (B3). The second term gives two types of corrections: 1) the zero pointenergy is shifted, and 2) there are additional terms that vanish in the limit Ldk → 0. To see this,write the second term as

ζ(s) =

′∑k

ζk(s) = 2

′∑k

n=∞∑n=m+1

λ−sk,n =

′∑k

1

Γ(s)

∫ ∞0

ts−1Y (t)dt (B5)

where Y (t) is given by

Y (t) = 2

n=∞∑n=m+1

exp(−λk,nt) = 2

n=∞∑n=m

exp(−(k2 + (2πn/β)2)t). (B6)

By shifting the origin of n this is

Y (t) = 2

n=∞∑n=1

exp(−(k2 + (2π(n+m)/β)2)t) (B7)

which gives

1

2Γ(s)ζk(s) =

n=∞∑n=1

∫ ∞0

ts−1 exp(−(k2 + (2π(n+m)/β)2)t)dt. (B8)

Writing τ = (n+m)2

β2 t and expanding the k2 part of the exponent, we have

1

2Γ(s)ζk(s) =

∞∑n=1

∫ ∞0

dτβ2s∞∑l=0

(βk)2l(−1)lτ s+l−1

(n+m)2l+2sl!e−4π2τ . (B9)

39

Carrying out the integral and changing the order of summation (we can always choose s to be ina region of the complex plane where this is valid and obtain the result for other s via analyticcontinuation), we have

1

2Γ(s)ζk(s) =

∞∑l=0

∞∑n=1

β2s (βk)2l(−1)lΓ(s+ l)

(n+m)2l+2sl!(2π)2s+2l. (B10)

Noting that

∞∑n=1

1

(n+m)2l+2s= ζ(2l + 2s,m+ 1) (B11)

where ζ(s, q) =∑∞

n=0(q + n)−s is the Hurwitz zeta function, we have

1

2Γ(s)ζk(s) = (

β

2π)2sΓ(s)ζ(2s,m+ 1) +

∞∑l=1

β2s (βk)2l(−1)lΓ(s+ l)

l!(2π)2s+2lζ(2l + 2s,m+ 1). (B12)

This gives

ζ′k(0) = 2

∂

∂s

(ζ(2s,m+ 1)(

β

2π)2s

)s=0

+ 2

∞∑l=1

(βk)2l(−1)l

l(2π)2lζ(2l,m+ 1). (B13)

Using the Leurch identity (Gradshteyn and Ryzhik [49] 9.533-3)

ζ′(0,m+ 1) = ln Γ(m+ 1)− 1

2ln(2π) = ln Γ(m)− 1

2ln(2π) + ln(m) (B14)

and

∞∑l=1

(βk)2l(−1)l

l(2π)2lζ(2l,m+ 1) = −2 ln Γ(m+ 1) + ln Γ(m+ 1− iβk/(2π)) + ln Γ(m+ 1− iβk/(2π))

= 2 ln |Γ(m+ iβk/(2π))/Γ(m)|+ ln(1 + (βk/(2πm))2), (B15)

where the last expression is valid for βk real, we obtain

ζ′k(0) =4 ln Γ(m)− 2 ln(2π)− 4(m+ 1/2) ln(β/2π)+

4 ln |Γ(m+ iβk/(2π))/Γ(m)|+ 2 ln(1 + (βk/(2πm))2) + 4 ln(m). (B16)

We can write this as

ζ′k(0) =4

(ln Γ(m)− (m− 1/2) ln(m) +m− 1

2ln(2π)

)− 4(m− 1/2) ln(β/(2πm))

+ 4 ln |Γ(m+ iβk/(2π))/Γ(m)|+ 2 ln(1 + (βk/(2πm))2) + 4 ln(m)− 4 ln(β/2π)− 4m.(B17)

Using ln(1 + (βk/(2πm))2)− ln((β/(2πm))2) = ln((2π/Ld)2), this becomes

ζ′k(0) = ln((2π/Ld)

2)− 4m(1 + ln(β/(2πm))) + 4

(ln Γ(m)− (m− 1/2) ln(m) +m− 1

2ln(2π)

)+ 4 ln |Γ(m+ iβk/(2π))/Γ(m)|+ ln(1 + (βk/(2πm))2). (B18)

40

The partition function is then given by

lnQ = −′∑k

(βk + 2 ln(1− e−βk) + ζ

′k(0)

). (B19)

The first term in equation (B18) is an irrelevant constant that can be eliminated by symmetricallykeeping 1/2 of the boundary terms in the sum. The second term is proportional to β and thereforerepresents a shift in the zero point energy. The remaining terms vanish in the limit Ld → 0 forfixed p and therefore do not contribute to the low energy physics. Thus the conventional Riemannzeta function regularization gets the low energy physics correct. Note that we cannot assumethat the high energy physics determined by the partition function is physically meaningful since,as discussed in Section IV, we are smearing out the discrete structure by using a finite basis ofcontinuous functions rather than averaging over discrete structures and therefore equation (B19)should not be taken as a physically meaningful result for k approaching 2π/Ld.

Note that the partition function given in equation (B1) is also the partition function for a non-interacting complex massless scalar field. A mass term can easily be included by replacing k2 with(k2 + M2) in equation (B1). Note also that now that we have the result, the identity implied byequations (B19),(B18) and (B1) can be readily proven by induction on m.

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41

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