Loop Quantum Gravity

Conceptual FoundationsReduced Phase Space Quantisation

Summary, Open Questions & Outlook

Loop Quantum GravityReduced Phase Space Approach

Thomas Thiemann1,2

1 Albert Einstein Institut, 2 Perimeter Institute for Theoretical Physics

Bad Honnef 2008

Thomas Thiemann Loop Quantum Gravity (LQG)

Contents

Conceptual Foundations

Reduced Phase Space Quantisation

Contents

Classical Canonical FormulationProblem of TimeCanonical Quantisation Strategies

Classical Canonical Formulation

Canonical formulation: M ∼= R×σ

Starting Point:

Well posed (causal) initial value formulation for geometry and matter

⇒ Globally hyperbolic spactimes (M, g)

⇒ Topological restriction: M ∼= R × σ [Geroch, 60’s]

No classical topology change, possibly quantum?

Starting Point:

Consequences: [ADM 60’s]

Consider arbitrary foliations Y : R × σ → M

Require spacelike leaves of foliation Σt := Y(t, σ)

Pull all fields on M back to R × σ

Obtain velocity phase space of spatial fields (e.g. 3 – metric qab andextrinsic curvature Kab ∝ ∂qab/∂t)

Legendre transform Kab 7→ pab singular (due to Diff(M) invariance):Spatial diffeomorphism and Hamiltonian constraints ca, c

Canonical Hamiltonian

Hcanon =

d3x n c + va ca =: c(n)+~c(v)

Hcanon =

Remarks:

Algebraic structure of c, ca Foliation independent

Symplectic structure of geometry and matter fields Foliationindependent

Foliation dependence encoded in lapse, shift n, va

Foliation independence (Diff(M) invariance) ⇒ Hcanon ≈ 0

10 Einstein Equations equivalent to

∂tqab = {Hcanon, qab}, ∂tpab = {Hcanon, p

ab}, c = 0, ca = 0

In particular, building gµν , nµ from qab, n, va one obtainsqµν = gµν + nµnν and

{Hcanon, qµν} = [Luq]µν , uµ = nnµ+(~v)µ

recovery of Diff(M) (on shell).

Remarks:

ab}, c = 0, ca = 0

Remarks:

ab}, c = 0, ca = 0

Remarks:

ab}, c = 0, ca = 0

Remarks:

ab}, c = 0, ca = 0

Remarks:

ab}, c = 0, ca = 0

Consistency:

First class (Dirac) hypersurface deformation algebra D

{~c(v),~c(v′)} = −~c([v, v′]){~c(v), c(n)} = −c(v[n])

{c(n), c(n′)} = −~c(q−1[n dn′ − n′ dn])

Universality: purely geometric origin, independent of matter content[Hojman, Kuchar, Teitelboim 70’s]

spatial diffeos generate subalgebra but not ideal

D no Lie algebra (structure functions)

Consistency:

{~c(v),~c(v′)} = −~c([v, v′]){~c(v), c(n)} = −c(v[n])

{c(n), c(n′)} = −~c(q−1[n dn′ − n′ dn])

Consistency:

{~c(v),~c(v′)} = −~c([v, v′]){~c(v), c(n)} = −c(v[n])

{c(n), c(n′)} = −~c(q−1[n dn′ − n′ dn])

Consistency:

{~c(v),~c(v′)} = −~c([v, v′]){~c(v), c(n)} = −c(v[n])

{c(n), c(n′)} = −~c(q−1[n dn′ − n′ dn])

Problem of Time

Interpretation:

Hcanon constrained to vanish, no true Hamiltonian

Hcanon generates gauge transformations, not physical evolution

qab, pab, ... not gauge invariant, not observable

{Hcanon,O} = 0 for observable, gauge invariant O

Problem of time: Dynamical interpretation?

Problem of Time

Interpretation:

Problem of Time

Interpretation:

Problem of Time

Interpretation:

Problem of Time

Interpretation:

Solution: Deparametrisation [Brown & Kuchar 90’s]

In GR, gauge invariant definition of curvature etc. only relative togeodesic test observers [Wald 90’s]

Test observers = mathematical idealisation

Brown – Kuchar dust action: 4 scalar fields T,SJ minimially coupledhowever: geometry backreaction taken seriously

Natural: Superposition of ∞ # of point particle actions

EL Equations: Dust particles move on unit geodesics, T(x) = propertime along geodesic trough x, SJ(x) labels geodesic

Dark matter candidate (NIMP)

Deparametrisation:

c := cD + cND, ca = cDa + cND

a ⇒ c = P + h, h =q

[cND]2 − qabcNDa cND

For close to flat geometry h ≈ cND ≈ hSM hard to achieve!

Remarkably {c(n), c(n′)} = 0 [Brown & Kuchar 90’s]

⇒ Explicit relational solution [Bergmann 60’s, Rovelli 90’s, Dittrich 00’s]

First symplectic reduction wrt ca [Kuchar 90’s] e.g.

qab(x) → qJK(s) := [qab(x)SaJ(x)S

bK(x)]SJ(x)=sJ , Sa

JSJ,b = δa

b , SaJSK

,a = δKJ

For any spatially diffeo inv., dust indep. f get observable

Of(τ ) := exp({Hτ , .}) · f, Hτ :=

d3x (τ − T(x)) hND(x)

Physical time evolution

Of(τ ) = {Hphys,Of(τ )}, Hphys :=

d3x hND(x)

Deparametrisation:

a ⇒ c = P + h, h =q

bK(x)]SJ(x)=sJ , Sa

JSJ,b = δa

b , SaJSK

,a = δKJ

Of(τ ) := exp({Hτ , .}) · f, Hτ :=

d3x hND(x)

Deparametrisation:

a ⇒ c = P + h, h =q

bK(x)]SJ(x)=sJ , Sa

JSJ,b = δa

b , SaJSK

,a = δKJ

Of(τ ) := exp({Hτ , .}) · f, Hτ :=

d3x hND(x)

Deparametrisation:

a ⇒ c = P + h, h =q

bK(x)]SJ(x)=sJ , Sa

JSJ,b = δa

b , SaJSK

,a = δKJ

Of(τ ) := exp({Hτ , .}) · f, Hτ :=

d3x hND(x)

Deparametrisation:

a ⇒ c = P + h, h =q

bK(x)]SJ(x)=sJ , Sa

JSJ,b = δa

b , SaJSK

,a = δKJ

Of(τ ) := exp({Hτ , .}) · f, Hτ :=

d3x hND(x)

Deparametrisation:

a ⇒ c = P + h, h =q

bK(x)]SJ(x)=sJ , Sa

JSJ,b = δa

b , SaJSK

,a = δKJ

Of(τ ) := exp({Hτ , .}) · f, Hτ :=

d3x hND(x)

Closed observable algebra due to automorphism property ofHamiltonian flow

{Of(τ ),Of′(τ )} = O{f,f′}(τ )

Reduced phase space Q’ion conceivable since e.g.

QJK(s) := OqJK(s)(0), PJK(s) := OpJK(s)(0)

⇒ {PJK(s),QLM(s′)} = δ(JL δ

K)M δ(s, s

Closed observable algebra due to automorphism property ofHamiltonian flow

{Of(τ ),Of′(τ )} = O{f,f′}(τ )

Reduced phase space Q’ion conceivable since e.g.

QJK(s) := OqJK(s)(0), PJK(s) := OpJK(s)(0)

⇒ {PJK(s),QLM(s′)} = δ(JL δ

K)M δ(s, s

Physics of the Dust:

Dust = Gravitational Higgs, Non-Dust = Gravitational Goldstone Bosons

Conservative Hamiltonian system w/o constraints but true Hamiltonian

Hamiltonian EOM wrt Hphys of physical Non-Dust dof agree with GaugeTransformations wrt Hcanon of unphysical Non-Dust dof under properfield substitutions, e.g. qab(x) ↔ Qjk(s)

No constraints but energy – momentum current conservation law

{Hphys,OhND(s)} = 0, {Hphys,OcNDj (s)} = 0,

Effectively reduces # of propagating dof by 4, hence in agreement withobservation (gravitational waves) [Giesel,Hofmann,T.T.,Winkler 00’s]

In terms of c dust fields are perfect (nowhere singular) clocks

Canonical Quantisation Strategies

Objective: Irreducible representation of the ∗−algebra (or C∗) Aphys ofDirac observables supporting bHphys

Strategy 1: Constraint Q’ion (CQ) = Q’ion before reduction

Strategy 2: Reduced phase space Q’ion (RQ) = Q’ion after reduction

Complementary Advantages and Disadvantages

CQ+: Reps. of Akin easy to findCQ-: Phys. HS = Kernel(constraints) constructioncomplicated (group averaging)RQ+: Directly phys. HS w/o redundant dof in Akin

RQ-: Reps. of Aphys often difficult to find

With dust, reduced phase space q’ion simpler, avoid difficultrepresentation of D

Algebra of Kinematical FunctionsAlgebra of Physical ObservablesPhysical Hilbert SpacePhysical coherent statesPhysical HamiltonianSemiclassical Limit

Algebra of Kinematical Functions

Gauge Theory Formulation:

Due to fermionic dof need to start with Palatini/Holst action [Ashtekar 80’s],

[Barbero, Holst, Immirzi 90’s]

After solving 2nd class (simplicity) constraints obtain

{Eaj (x),A

kb(y)} = κδa

bδkj δ(x, y)

Non-dust, gravitational contributions to the constraints

cgeoj = DaEa

cgeoa = Tr

`FabEb

cgeo =Tr(Fab [Ea,Eb])√

| det(E)|+ ....

{Eaj (x),A

kb(y)} = κδa

bδkj δ(x, y)

cgeoj = DaEa

cgeoa = Tr

`FabEb

| det(E)|+ ....

{Eaj (x),A

kb(y)} = κδa

bδkj δ(x, y)

cgeoj = DaEa

cgeoa = Tr

`FabEb

| det(E)|+ ....

Algebra of Physical Observables

Simply define (similar for EIj(s))

AjI(s) := O

ajI(s)

(0), ajI(s) := [Aj

aSaI ](x)S(x)=s,

Then{EI

j(s),AkJ(s

′)} = κδkj δ

IJ δ(s, s

No constraints but phys. Hamiltonian (Σ = S(σ))

p| − ηµν Tr (τµ F ∧ {A,V}) Tr (τν F ∧ {A,V}) | =:

Zd3s H(s)

Physical total volume

p| det(E)|

Symmetry group of H: S = N ⋊ Diff(Σ)

N : Abelian normal subgroup generated by H(s), active Diff(Σ)

AjI(s) := O

ajI(s)

(0), ajI(s) := [Aj

aSaI ](x)S(x)=s,

Then{EI

j(s),AkJ(s

′)} = κδkj δ

IJ δ(s, s

Zd3s H(s)

p| det(E)|

AjI(s) := O

ajI(s)

(0), ajI(s) := [Aj

aSaI ](x)S(x)=s,

Then{EI

j(s),AkJ(s

′)} = κδkj δ

IJ δ(s, s

Zd3s H(s)

p| det(E)|

AjI(s) := O

ajI(s)

(0), ajI(s) := [Aj

aSaI ](x)S(x)=s,

Then{EI

j(s),AkJ(s

′)} = κδkj δ

IJ δ(s, s

Zd3s H(s)

p| det(E)|

AjI(s) := O

ajI(s)

(0), ajI(s) := [Aj

aSaI ](x)S(x)=s,

Then{EI

j(s),AkJ(s

′)} = κδkj δ

IJ δ(s, s

Zd3s H(s)

p| det(E)|

AjI(s) := O

ajI(s)

(0), ajI(s) := [Aj

aSaI ](x)S(x)=s,

Then{EI

j(s),AkJ(s

′)} = κδkj δ

IJ δ(s, s

Zd3s H(s)

p| det(E)|

Physical Hilbert Space

Lattice – inspired canon. gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli,

Smolin 88]

Magnet. dof.: Holonomy (Wilson – Loop)

A(e) = P exp(

Electr. dof: flux

Ej(S) =

Sǫabc Ea

j dxb ∧ dxc

Poisson – brackets:

{Ej(S),A(e)} = G A(e1) τj A(e2); e = e1 ◦ e2, e1 ∩ e2 = e ∩ S

Smolin 88]

A(e) = P exp(

Electr. dof: flux

Ej(S) =

Sǫabc Ea

j dxb ∧ dxc

Smolin 88]

A(e) = P exp(

Electr. dof: flux

Ej(S) =

Sǫabc Ea

j dxb ∧ dxc

��

Lattice – inspired gauge theory variables [Gambini & Trias 81], [Jacobson, Rovelli, Smolin 88]

A(e) = P exp(

Electr. dof: flux

Ef(S) =

Sǫabc Ea

j dxb ∧ dxc

Reality conditions:

A(e) = [A(e−1)]T, Ej(S) = Ej(S)

Defines abstract Poisson∗−algebra Aphys.

Bundle automorphisms G ∼= G ⋊ Diff(Σ) act by Poisson automorphismson Aphys e.g. αg = exp({

Rλjcj, .}), g = exp(λjτj)

αg(A(e)) = g(b(e)) A(e)g(f(e))−1, αϕ(A(e)) = A(ϕ(e))

A(e) = P exp(

Electr. dof: flux

Ef(S) =

Sǫabc Ea

j dxb ∧ dxc

Reality conditions:

A(e) = [A(e−1)]T, Ej(S) = Ej(S)

A(e) = P exp(

Electr. dof: flux

Ef(S) =

Sǫabc Ea

j dxb ∧ dxc

Reality conditions:

A(e) = [A(e−1)]T, Ej(S) = Ej(S)

HS Reps.: In QFT no Stone – von Neumann Theorem!!!

Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04]

Diff(Σ) inv. states on hol. – flux algebra Aphys unique.

wave functions of Hphys

ψ(A) = ψγ(A(e1), ..,A(eN)), ψγ : SU(2)N → C

Holonomy = multiplication – operator

[A(e) ψ](A) := A(e) ψ(A)

Flux = derivative – operator

[Ej(S) ψ](A) := i~ {Ej(S), ψ(A)}Scalar product

< ψ,ψ′ >:=

SU(2)NdµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′

γ(h1, .., hN)

[A(e) ψ](A) := A(e) ψ(A)

< ψ,ψ′ >:=

γ(h1, .., hN)

[A(e) ψ](A) := A(e) ψ(A)

< ψ,ψ′ >:=

γ(h1, .., hN)

[A(e) ψ](A) := A(e) ψ(A)

< ψ,ψ′ >:=

γ(h1, .., hN)

[A(e) ψ](A) := A(e) ψ(A)

< ψ,ψ′ >:=

γ(h1, .., hN)

[A(e) ψ](A) := A(e) ψ(A)

< ψ,ψ′ >:=

γ(h1, .., hN)

Spin Network ONB Tγ,j,I

v4,I4 v5,I5

v10,I10

e2,j2 e3,j3

e6,j6 e7,j7

e10,j10

e11,j11

e12,j12 e13,j13

e14,j14

e15,j15

e16,j16

e17,j17

e18,j18

e19,j19

e20,j20

e21,j21

e22,j22

Does rep. support bH with correct semiclassical limit?

Gauss constraint solved by restriction of intertwiners I

Hphys not separable

Hphys = ⊕γ Hγ , Hγ = span{Tγ,j,I; j 6= 0, I}Diff(Σ) does not downsize it since symmetry group, not gauge group

Unitary representation U(ϕ)Tγ,j,I := Tϕ(γ),j,I

If U(ϕ) F U(ϕ)−1 = F (e.g. F = H; all operationally definedobservables) then “superselection” (subgraph preservation)

F Hγ ⊂ Hγ ⇒ F = ⊕γ Fγ

This imposes strong constraints on regularisation of bH and removesmost ambiguities usually encountered for bC !

1. Construct Hγ ∀ γ2. Compute < ψγ ,Hψγ >=< ψγ ,Hγψγ > f. semiclass. ψγ

Hphys not separable

Physical coherent states

Choose cell complex γ∗, dual graph γ s.t. e ↔ Se

Choose classical field configuration (A0(x),E0(x)), compute

ge := exp(iτjEj0(Se)) A0(e) ∈ GC

Define [Hall 90’s], [Sahlmann, T.T., Winkler 00’s]

ψA0,E0 := ⊗e ψe, ψe(he) :=X

dim(π) e−teλπ χπ(geh−1e )

Minimal uncertainty states, that is, ∀ e ∈ E(γ)

< ψA0,E0,A(e)ψA0,E0 >= A0(e), < ψA0,E0 ,Ej(Se)ψA0,E0 >= Ej0(Se)

<∆A(e)> < ∆Ej(Se))>=12| < [A(e), Ej(Se))]> |

Overlap Function

0.2-0.1

arcsin(ℑ(A(e)))

E(S)/L2E(S)/L2

Remarks:

Notice: Σ just differential manifold, no Riemannian space!

No a priori meaning to how densely γ embedded

In particular, final operator bH cannot depend on short distance regulatorused at intermediate stages of construction

Notice: Operator family (bHγ) defines Continuum operator

Expect that good semiclassical states depend on graphs which are verydensely embedded wrt background metric to be approximated

Choose graph to be countably infinite (for compact Σ large finite graphsufficient)

Remarks:

Physical Hamiltonian

Example: Cubic graph

~nSeSe

Comparison with YM theory on cubic lattice

Yang – Mills on (R4, η) [Kogut & Susskind 74]

Hγ =~

2 g2 ǫ

v∈V(γ)

E(Sav)

2 + [2 − A(αav) − A(αa

v)−1]

Gravity on R × Σ [T.T. 96 – 05, Giesel & T.T. 06]

Hγ =~

v∈V(γ)

˛˛˛

ηµµ

Tr (τµ A(αav) A(ea

v) [A(eav)−1,Vv])

#2˛˛˛

Volume operator

|ǫabcTr`E(Sa

v) E(Sbv) E(Sc

Lattice spacing ǫ disappears, automat. UV finite.

In a precise sense: ǫ replaced by ℓP

Hγ =~

2 g2 ǫ

v∈V(γ)

E(Sav)

2 + [2 − A(αav) − A(αa

v)−1]

Hγ =~

v∈V(γ)

˛˛˛

ηµµ

v) [A(eav)−1,Vv])

#2˛˛˛

Volume operator

|ǫabcTr`E(Sa

v) E(Sbv) E(Sc

Hγ =~

2 g2 ǫ

v∈V(γ)

E(Sav)

2 + [2 − A(αav) − A(αa

v)−1]

Hγ =~

v∈V(γ)

˛˛˛

ηµµ

v) [A(eav)−1,Vv])

#2˛˛˛

Volume operator

|ǫabcTr`E(Sa

v) E(Sbv) E(Sc

Hγ =~

2 g2 ǫ

v∈V(γ)

E(Sav)

2 + [2 − A(αav) − A(αa

v)−1]

Hγ =~

v∈V(γ)

˛˛˛

ηµµ

v) [A(eav)−1,Vv])

#2˛˛˛

Volume operator

|ǫabcTr`E(Sa

v) E(Sbv) E(Sc

Hγ =~

2 g2 ǫ

v∈V(γ)

E(Sav)

2 + [2 − A(αav) − A(αa

v)−1]

Hγ =~

v∈V(γ)

˛˛˛

ηµµ

v) [A(eav)−1,Vv])

#2˛˛˛

Volume operator

|ǫabcTr`E(Sa

v) E(Sbv) E(Sc

Semiclassical Limit

Theorem [Giesel & T.T. 06] For any (A0,E0), suff. large γ

1. Exp. Value < ψA0,E0 ,bHψA0,E0 >= H(A0,E0) + O(~)

2. Fluctuation< ψA0,E0 ,bH2ψA0,E0 > − < ψA0,E0 ,

bHψA0,E0 >2= O(~)

Corollary

i. Quantum Hamiltonian correctly implemented

ii. For sufficiently small τ

eiτbH/~ ψA0,E0 ≈ ψA0(τ),E0(τ)

Semiclassical Limit

bHψA0,E0 >2= O(~)

Corollary

Semiclassical Limit

bHψA0,E0 >2= O(~)

Corollary

Semiclassical Limit

bHψA0,E0 >2= O(~)

Corollary

Semiclassical Limit

bHψA0,E0 >2= O(~)

Corollary

SummaryOpen QuestionsOutlook

Summary

LQG dynamically severly constrained (uniqeness result)

correct semiclassical limit of bH established

Final picture equivalent to background independent, Hamiltonian“floating” lattice gauge theory

Summary

Open Questions

Proposal for removing graph dependence (preservation), nonseparability, controlling fluctuations of all dof:Algebraic Quantum Gravity (AQG) [Giesel, T.T. 06]

Implementation of classical N symmetrybH stable coherent states?

Better understanding of validity/physics of dust, other types of matter?

Scrutinise LQG/AQG by further consistency checks

Open Questions

Outlook

If LQG/AQG pass consistency tests then:

All LQG techniques developed so far can be imported to phys. HS level!

Physical semiclassical techniques to make contact with standard model

phys. Hamiltonian defines S – Matrix, scattering theory, Feynman rules

conservative system, hence possible improvement of vacuum problemin QFT on time dep. backgrounds (cosmology)

Outlook

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