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Relating LQC with LQG
– Algebraic Aspects –
Maximilian Hanusch
University of Wurzburg
With special focus on jointwork with J. Engle and Th. Thiemannat Florida Atlantic University
NSF Grant PHY-1505490
July 4th, 2017
From LQG to LQC
Spacetime Symmetry: pA, Eq Ñ pAred, Eredq
Fix algebra Aclass D P of phase space functions on pAred, Eredq
D CylrPs|AredP Γ|Ered Γ Fluxes XS,f
as well as Poisson bracket t, u on Aclass. P class of curves
Quantum algebra with commutation relations determined by t, u
Holonomy-Flux A: generated by pD, pPWeyl-algebra W: generated by pD, exppi pPq
Single out representation of A by physical conditions.
Diffeomorphism invariant state on A,W LOST, AC, EHT
Unique Diff-invariant state: GNS-rep. Std. rep. phys. intuition
Bianchi I Homogeneous Isotropic
LQG AWE
AWE
ABL Fleischhack
Unqs LOST AC
EHT
EHT EHT
QA A W
A
A A
Ared A c1, c2, c3
c1, c2, c3
c c
P analytic axes
axes
linear analytic
D Cyl CAPpR3q
CAPpR3q
CAPpRq CAPpRq`C0pRq
P Γ p1, p2, p3
p1, p2, p3
p p
Diff AutpPq Dil3V
Dil3
Dil1 Dil1
same representation same representation
From LQG to LQC
Spacetime Symmetry: pA, Eq Ñ pAred, Eredq
Fix algebra Aclass D P of phase space functions on pAred, Eredq
D CylrPs|AredP Γ|Ered Γ Fluxes XS,f
as well as Poisson bracket t, u on Aclass. P class of curves
Quantum algebra with commutation relations determined by t, u
Holonomy-Flux A: generated by pD, pPWeyl-algebra W: generated by pD, exppi pPq
Single out representation of A by physical conditions.
Diffeomorphism invariant state on A,W LOST, AC, EHT
Unique Diff-invariant state: GNS-rep. Std. rep. phys. intuition
Bianchi I Homogeneous Isotropic
LQG AWE
AWE
ABL Fleischhack
Unqs LOST AC
EHT
EHT EHT
QA A W
A
A A
Ared A c1, c2, c3
c1, c2, c3
c c
P analytic axes
axes
linear analytic
D Cyl CAPpR3q
CAPpR3q
CAPpRq CAPpRq`C0pRq
P Γ p1, p2, p3
p1, p2, p3
p p
Diff AutpPq Dil3V
Dil3
Dil1 Dil1
same representation same representation
From LQG to LQC
Spacetime Symmetry: pA, Eq Ñ pAred, Eredq
Fix algebra Aclass D P of phase space functions on pAred, Eredq
D CylrPs|AredP Γ|Ered Γ Fluxes XS,f
as well as Poisson bracket t, u on Aclass. P class of curves
Quantum algebra with commutation relations determined by t, u
Holonomy-Flux A: generated by pD, pPWeyl-algebra W: generated by pD, exppi pPq
Single out representation of A by physical conditions.
Diffeomorphism invariant state on A,W LOST, AC, EHT
Unique Diff-invariant state: GNS-rep. Std. rep. phys. intuition
Bianchi I Homogeneous Isotropic
LQG AWE
AWE
ABL Fleischhack
Unqs LOST AC
EHT
EHT EHT
QA A W
A
A A
Ared A c1, c2, c3
c1, c2, c3
c c
P analytic axes
axes
linear analytic
D Cyl CAPpR3q
CAPpR3q
CAPpRq CAPpRq`C0pRq
P Γ p1, p2, p3
p1, p2, p3
p p
Diff AutpPq Dil3V
Dil3
Dil1 Dil1
same representation same representation
From LQG to LQC
Spacetime Symmetry: pA, Eq Ñ pAred, Eredq
Fix algebra Aclass D P of phase space functions on pAred, Eredq
D CylrPs|AredP Γ|Ered Γ Fluxes XS,f
as well as Poisson bracket t, u on Aclass. P class of curves
Quantum algebra with commutation relations determined by t, u
Holonomy-Flux A: generated by pD, pPWeyl-algebra W: generated by pD, exppi pPq
Single out representation of A by physical conditions.
Diffeomorphism invariant state on A,W LOST, AC, EHT
Unique Diff-invariant state: GNS-rep. Std. rep. phys. intuition
Bianchi I Homogeneous Isotropic
LQG AWE
AWE
ABL Fleischhack
Unqs LOST AC
EHT
EHT EHT
QA A W
A
A A
Ared A c1, c2, c3
c1, c2, c3
c c
P analytic axes
axes
linear analytic
D Cyl CAPpR3q
CAPpR3q
CAPpRq CAPpRq`C0pRq
P Γ p1, p2, p3
p1, p2, p3
p p
Diff AutpPq Dil3V
Dil3
Dil1 Dil1
same representation same representation
From LQG to LQC
Spacetime Symmetry: pA, Eq Ñ pAred, Eredq
Fix algebra Aclass D P of phase space functions on pAred, Eredq
D CylrPs|AredP Γ|Ered Γ Fluxes XS,f
as well as Poisson bracket t, u on Aclass. P class of curves
Quantum algebra with commutation relations determined by t, u
Holonomy-Flux A: generated by pD, pPWeyl-algebra W: generated by pD, exppi pPq
Single out representation of A by physical conditions.
Diffeomorphism invariant state on A,W LOST, AC, EHT
Unique Diff-invariant state: GNS-rep. Std. rep. phys. intuition
Bianchi I Homogeneous Isotropic
LQG AWE
AWE
ABL Fleischhack
Unqs LOST AC
EHT
EHT EHT
QA A W
A
A A
Ared A c1, c2, c3
c1, c2, c3
c c
P analytic axes
axes
linear analytic
D Cyl CAPpR3q
CAPpR3q
CAPpRq CAPpRq`C0pRq
P Γ p1, p2, p3
p1, p2, p3
p p
Diff AutpPq Dil3V
Dil3
Dil1 Dil1
same representation same representation
From LQG to LQC
Spacetime Symmetry: pA, Eq Ñ pAred, Eredq
Fix algebra Aclass D P of phase space functions on pAred, Eredq
D CylrPs|AredP Γ|Ered Γ Fluxes XS,f
as well as Poisson bracket t, u on Aclass. P class of curves
Quantum algebra with commutation relations determined by t, u
Holonomy-Flux A: generated by pD, pPWeyl-algebra W: generated by pD, exppi pPq
Single out representation of A by physical conditions.
Diffeomorphism invariant state on A,W LOST, AC, EHT
Unique Diff-invariant state: GNS-rep. Std. rep. phys. intuition
Bianchi I Homogeneous Isotropic
LQG AWE AWE ABL Fleischhack
Unqs LOST AC EHT EHT EHT
QA A W A A A
Ared A c1, c2, c3 c1, c2, c3 c c
P analytic axes axes linear analytic
D Cyl CAPpR3q CAPpR3q CAPpRq CAPpRq`C0pRq
P Γ p1, p2, p3 p1, p2, p3 p p
Diff AutpPq Dil3V Dil3 Dil1 Dil1
same representation same representation
Ashtekar Variables e I e Iα dxα co-tetrad
ω ωIJα dxα SOp1, 3q
Spin connection Γia, and K i
a ωi0a
Aia c δia
E ai
8πGγ3V0rCs pδ
ai
The Holst Action:
L 132πG
³pM,resq
εIJKL e
I ^ eJ ^ ΩKL 2γ e I ^ eJ ^ ΩIJ
rL p 9c Nb
3V0rCskγ3
|p|12c2 for p 3V0rCs|v |v8πGγ
rL 3V0rCs|v |v8πGγ
ddx0 pc wq 3NV0rCs|v |
8πGγ2pc2 γ2w2 2cwq
Legendre transform: Aia Γi
a γK ia (Ashtekar connection)
E ai |det e jb | e
ai (Dreibein)
Cosmology: M R R3 with R3 SOp3qñ ttu R3
Set S of invariant pω, eq parametrized by ppc ,wq, vq.
Integrate rL L|S over volume V0rCs of fixed cell C.
Perform Dirac constrained analysis. (Details in Jon’s talk !)
Conjugate variables c and p.
Ashtekar Variables e I e Iα dxα co-tetrad
ω ωIJα dxα SOp1, 3q
Spin connection Γia, and K i
a ωi0a
Aia c δia
E ai
8πGγ3V0rCs pδ
ai
The Holst Action:
L 132πG
³pM,resq
εIJKL e
I ^ eJ ^ ΩKL 2γ e I ^ eJ ^ ΩIJ
rL p 9c Nb
3V0rCskγ3
|p|12c2 for p 3V0rCs|v |v8πGγ
rL 3V0rCs|v |v8πGγ
ddx0 pc wq 3NV0rCs|v |
8πGγ2pc2 γ2w2 2cwq
Legendre transform: Aia Γi
a γK ia (Ashtekar connection)
E ai |det e jb | e
ai (Dreibein)
Cosmology: M R R3 with R3 SOp3qñ ttu R3
Set S of invariant pω, eq parametrized by ppc ,wq, vq.
Integrate rL L|S over volume V0rCs of fixed cell C.
Perform Dirac constrained analysis. (Details in Jon’s talk !)
Conjugate variables c and p.
Ashtekar Variables e I e Iα dxα co-tetrad
ω ωIJα dxα SOp1, 3q
Spin connection Γia, and K i
a ωi0a
Aia c δia
E ai
8πGγ3V0rCs pδ
ai
The Holst Action:
L 132πG
³pM,resq
εIJKL e
I ^ eJ ^ ΩKL 2γ e I ^ eJ ^ ΩIJ
rL p 9c Nb
3V0rCskγ3
|p|12c2 for p 3V0rCs|v |v8πGγ
rL 3V0rCs|v |v8πGγ
ddx0 pc wq 3NV0rCs|v |
8πGγ2pc2 γ2w2 2cwq
Legendre transform: Aia Γi
a γK ia (Ashtekar connection)
E ai |det e jb | e
ai (Dreibein)
Cosmology: M R R3 with R3 SOp3qñ ttu R3
Set S of invariant pω, eq parametrized by ppc ,wq, vq.
Integrate rL L|S over volume V0rCs of fixed cell C.
Perform Dirac constrained analysis. (Details in Jon’s talk !)
Conjugate variables c and p.
Ashtekar Variables e I e Iα dxα co-tetrad
ω ωIJα dxα SOp1, 3q
Spin connection Γia, and K i
a ωi0a
Aia c δia
E ai
8πGγ3V0rCs pδ
ai
The Holst Action:
L 132πG
³pM,resq
εIJKL e
I ^ eJ ^ ΩKL 2γ e I ^ eJ ^ ΩIJ
rL p 9c N
b3V0rCskγ3
|p|12c2 for p 3V0rCs|v |v8πGγ
rL 3V0rCs|v |v8πGγ
ddx0 pc wq 3NV0rCs|v |
8πGγ2pc2 γ2w2 2cwq
Legendre transform: Aia Γi
a γK ia (Ashtekar connection)
E ai |det e jb | e
ai (Dreibein)
Cosmology: M R R3 with R3 SOp3qñ ttu R3
Set S of invariant pω, eq parametrized by ppc ,wq, vq.
Integrate rL L|S over volume V0rCs of fixed cell C.
Perform Dirac constrained analysis. (Details in Jon’s talk !)
Conjugate variables c and p.
Ashtekar Variables e I e Iα dxα co-tetrad
ω ωIJα dxα SOp1, 3q
Spin connection Γia, and K i
a ωi0a
Aia c δia
E ai
8πGγ3V0rCs pδ
ai
The Holst Action:
L 132πG
³pM,resq
εIJKL e
I ^ eJ ^ ΩKL 2γ e I ^ eJ ^ ΩIJ
rL p 9c N
b3V0rCskγ3
|p|12c2 for p 3V0rCs|v |v8πGγ
rL 3V0rCs|v |v8πGγ
ddx0 pc wq 3NV0rCs|v |
8πGγ2pc2 γ2w2 2cwq
Legendre transform: Aia Γi
a γK ia
(Ashtekar connection)
E ai |det e jb | e
ai
(Dreibein)
Cosmology: M R R3 with R3 SOp3qñ ttu R3
Set S of invariant pω, eq parametrized by ppc ,wq, vq.
Integrate rL L|S over volume V0rCs of fixed cell C.
Perform Dirac constrained analysis. (Details in Jon’s talk !)
Conjugate variables c and p.
The Classical Holonomy-Flux Algebra
γ : r0, 1s ÑM embedded analytic
f : S ÞÑ su SmVf on Surface S
ψ : c ÞÑ ΨpArcsq P CAPpRq
` C0pRq
xS,f : p ÞÑ XS ,f pE rpsq λS ,f p
Arcsia c δia
E rpsai 8πGγ3V0rCs pδ
ai
D P tdAP, dAP ` d0u for
dAP tψ P CAPpRq X C8pRq | ψpnq P CAPpRq @n P Nu CAPpRq
d0 tψ P C0pRq X C8pRq | ψpnq P C0pRq @n P Nu C0pRq
Phase space formed by pairs pA,E q with functions:
Cyl Ψ: A ÞÑ hi ,jγ pAq
Flux XS,f : E ÞѳSrE pf q
rE 12! εabc E
ai dxb ^ dxc b τ i for E E a
i Ba b τ i
Aclass : Cyl Γ
D C p
Γ: C-span of (XS ,f : Cyl Ñ Cyl derivation)
XS1,f1 , rXS1,f1 ,XS2,f2s, r. . . rXS1,f1 ,XS2,f2s, . . . ,XSk ,fk s
D: Suitable dense subalgebra of CAPpRq ` C0pRq
Poisson Bracket:
tpΨ,Y q, pΨ1,Y 1qu pY pΨ1q Y 1pΨq, rY ,Y 1sq
tpψ, zpq, pψ1, z 1pqu pz 9ψ1 z 1 9ψ, 0q
The Classical Holonomy-Flux Algebra
γ : r0, 1s ÑM embedded analytic
f : S ÞÑ su SmVf on Surface S
ψ : c ÞÑ ΨpArcsq P CAPpRq
` C0pRq
xS,f : p ÞÑ XS ,f pE rpsq λS ,f p
Arcsia c δia
E rpsai 8πGγ3V0rCs pδ
ai
D P tdAP, dAP ` d0u for
dAP tψ P CAPpRq X C8pRq | ψpnq P CAPpRq @n P Nu CAPpRq
d0 tψ P C0pRq X C8pRq | ψpnq P C0pRq @n P Nu C0pRq
Phase space formed by pairs pA,E q with functions:
Cyl Ψ: A ÞÑ hi ,jγ pAq
Flux XS,f : E ÞѳSrE pf q
rE 12! εabc E
ai dxb ^ dxc b τ i for E E a
i Ba b τ i
Aclass : Cyl Γ
D C p
Γ: C-span of (XS ,f : Cyl Ñ Cyl derivation)
XS1,f1 , rXS1,f1 ,XS2,f2s, r. . . rXS1,f1 ,XS2,f2s, . . . ,XSk ,fk s
D: Suitable dense subalgebra of CAPpRq ` C0pRq
Poisson Bracket:
tpΨ,Y q, pΨ1,Y 1qu pY pΨ1q Y 1pΨq, rY ,Y 1sq
tpψ, zpq, pψ1, z 1pqu pz 9ψ1 z 1 9ψ, 0q
The Classical Holonomy-Flux Algebra
γ : r0, 1s ÑM embedded analytic
f : S ÞÑ su SmVf on Surface S
ψ : c ÞÑ ΨpArcsq P CAPpRq
` C0pRq
xS,f : p ÞÑ XS ,f pE rpsq λS ,f p
Arcsia c δia
E rpsai 8πGγ3V0rCs pδ
ai
D P tdAP, dAP ` d0u for
dAP tψ P CAPpRq X C8pRq | ψpnq P CAPpRq @n P Nu CAPpRq
d0 tψ P C0pRq X C8pRq | ψpnq P C0pRq @n P Nu C0pRq
Phase space formed by pairs pA,E q with functions:
Cyl Ψ: A ÞÑ hi ,jγ pAq
Flux XS,f : E ÞѳSrE pf q
rE 12! εabc E
ai dxb ^ dxc b τ i for E E a
i Ba b τ i
Aclass : Cyl Γ
D C p
Γ: C-span of (XS ,f : Cyl Ñ Cyl derivation)
XS1,f1 , rXS1,f1 ,XS2,f2s, r. . . rXS1,f1 ,XS2,f2s, . . . ,XSk ,fk s
D: Suitable dense subalgebra of CAPpRq ` C0pRq
Poisson Bracket:
tpΨ,Y q, pΨ1,Y 1qu pY pΨ1q Y 1pΨq, rY ,Y 1sq
tpψ, zpq, pψ1, z 1pqu pz 9ψ1 z 1 9ψ, 0q
The Classical Holonomy-Flux Algebra
γ : r0, 1s ÑM embedded analytic
f : S ÞÑ su SmVf on Surface S
ψ : c ÞÑ ΨpArcsq P CAPpRq ` C0pRq
xS,f : p ÞÑ XS ,f pE rpsq λS ,f p
Arcsia c δia
E rpsai 8πGγ3V0rCs pδ
ai
D P tdAP, dAP ` d0u for
dAP tψ P CAPpRq X C8pRq | ψpnq P CAPpRq @n P Nu CAPpRq
d0 tψ P C0pRq X C8pRq | ψpnq P C0pRq @n P Nu C0pRq
Phase space formed by pairs pA,E q with functions:
Cyl Ψ: A ÞÑ hi ,jγ pAq
Flux XS,f : E ÞѳSrE pf q
rE 12! εabc E
ai dxb ^ dxc b τ i for E E a
i Ba b τ i
Aclass : Cyl Γ
D C p
Γ: C-span of (XS ,f : Cyl Ñ Cyl derivation)
XS1,f1 , rXS1,f1 ,XS2,f2s, r. . . rXS1,f1 ,XS2,f2s, . . . ,XSk ,fk s
D: Suitable dense subalgebra of CAPpRq ` C0pRq
Poisson Bracket:
tpΨ,Y q, pΨ1,Y 1qu pY pΨ1q Y 1pΨq, rY ,Y 1sq
tpψ, zpq, pψ1, z 1pqu pz 9ψ1 z 1 9ψ, 0q
The Classical Holonomy-Flux Algebra
γ : r0, 1s ÑM embedded analytic
f : S ÞÑ su SmVf on Surface S
ψ : c ÞÑ ΨpArcsq P CAPpRq ` C0pRq
xS,f : p ÞÑ XS ,f pE rpsq λS ,f p
Arcsia c δia
E rpsai 8πGγ3V0rCs pδ
ai
D P tdAP, dAP ` d0u for
dAP tψ P CAPpRq X C8pRq | ψpnq P CAPpRq @n P Nu CAPpRq
d0 tψ P C0pRq X C8pRq | ψpnq P C0pRq @n P Nu C0pRq
Phase space formed by pairs pA,E q with functions:
Cyl Ψ: A ÞÑ hi ,jγ pAq
Flux XS,f : E ÞѳSrE pf q
rE 12! εabc E
ai dxb ^ dxc b τ i for E E a
i Ba b τ i
Aclass : Cyl Γ D C p
Γ: C-span of (XS ,f : Cyl Ñ Cyl derivation)
XS1,f1 , rXS1,f1 ,XS2,f2s, r. . . rXS1,f1 ,XS2,f2s, . . . ,XSk ,fk s
D: Suitable dense subalgebra of CAPpRq ` C0pRq
Poisson Bracket:
tpΨ,Y q, pΨ1,Y 1qu pY pΨ1q Y 1pΨq, rY ,Y 1sq
tpψ, zpq, pψ1, z 1pqu pz 9ψ1 z 1 9ψ, 0q
The Classical Holonomy-Flux Algebra
γ : r0, 1s ÑM embedded analytic
f : S ÞÑ su SmVf on Surface S
ψ : c ÞÑ ΨpArcsq P CAPpRq ` C0pRq
xS,f : p ÞÑ XS ,f pE rpsq λS ,f p
Arcsia c δia
E rpsai 8πGγ3V0rCs pδ
ai
D P tdAP, dAP ` d0u for
dAP tψ P CAPpRq X C8pRq | ψpnq P CAPpRq @n P Nu CAPpRq
d0 tψ P C0pRq X C8pRq | ψpnq P C0pRq @n P Nu C0pRq
Phase space formed by pairs pA,E q with functions:
Cyl Ψ: A ÞÑ hi ,jγ pAq
Flux XS,f : E ÞѳSrE pf q
rE 12! εabc E
ai dxb ^ dxc b τ i for E E a
i Ba b τ i
Aclass : Cyl Γ D C p
Γ: C-span of (XS ,f : Cyl Ñ Cyl derivation)
XS1,f1 , rXS1,f1 ,XS2,f2s, r. . . rXS1,f1 ,XS2,f2s, . . . ,XSk ,fk s
D: Suitable dense subalgebra of CAPpRq ` C0pRq
Poisson Bracket:
tpΨ,Y q, pΨ1,Y 1qu pY pΨ1q Y 1pΨq, rY ,Y 1sq
tpψ, zpq, pψ1, z 1pqu pz 9ψ1 z 1 9ψ, 0q
The Classical Holonomy-Flux Algebra
γ : r0, 1s ÑM embedded analytic
f : S ÞÑ su SmVf on Surface S
ψ : c ÞÑ ΨpArcsq P CAPpRq ` C0pRq
xS,f : p ÞÑ XS ,f pE rpsq λS ,f p
Arcsia c δia
E rpsai 8πGγ3V0rCs pδ
ai
D P tdAP, dAP ` d0u for
dAP tψ P CAPpRq X C8pRq | ψpnq P CAPpRq @n P Nu CAPpRq
d0 tψ P C0pRq X C8pRq | ψpnq P C0pRq @n P Nu C0pRq
Phase space formed by pairs pA,E q with functions:
Cyl Ψ: A ÞÑ hi ,jγ pAq
Flux XS,f : E ÞѳSrE pf q
rE 12! εabc E
ai dxb ^ dxc b τ i for E E a
i Ba b τ i
Aclass : Cyl Γ D C p
Γ: C-span of (XS ,f : Cyl Ñ Cyl derivation)
XS1,f1 , rXS1,f1 ,XS2,f2s, r. . . rXS1,f1 ,XS2,f2s, . . . ,XSk ,fk s
D: Suitable dense subalgebra of CAPpRq ` C0pRq
Poisson Bracket:
tpΨ,Y q, pΨ1,Y 1qu pY pΨ1q Y 1pΨq, rY ,Y 1sq
tpψ, zpq, pψ1, z 1pqu pz 9ψ1 z 1 9ψ, 0q
Diffeomorphisms
Full theory: SUp2q-bundle P over Σ
AutpPq Q Φ ñ Aclass Cyl Γ acts t, u - preserving via
Ψ ÞÑ Ψ Φ XS ,f ÞÑ XφpSq,f φ1
φ : Σ Ñ Σ diffeomorphism induced by Φ
Cosmology: M R R3 with R¡0 ñ ttu R3 SOp1, 3q
V0rCs ÞÑ λ3V0rCs
S Q pω, eq ÞÑ prΦλsω, rΦλseq P S ppc ,wq, vq p 3V0rCs|v |v8πGγ
pc, vq ÞÑ pλ1c , λ1vq ùñ pc , pq ÞÑ pλ1c , λpq
Aclass Q pψ, zpq ÞÑψ Φ
λ : c ÞÑ ψpλ1 cq, λ zp
Preserve t, u; thus, carry over to the quantum algebra.
Diffeomorphisms
Full theory: SUp2q-bundle P over Σ
AutpPq Q Φ ñ Aclass Cyl Γ acts t, u - preserving via
Ψ ÞÑ Ψ Φ XS ,f ÞÑ XφpSq,f φ1
φ : Σ Ñ Σ diffeomorphism induced by Φ
Cosmology: M R R3 with R¡0 ñ ttu R3 SOp1, 3q
V0rCs ÞÑ λ3V0rCs
S Q pω, eq ÞÑ prΦλsω, rΦλseq P S ppc ,wq, vq p 3V0rCs|v |v8πGγ
pc, vq ÞÑ pλ1c , λ1vq ùñ pc , pq ÞÑ pλ1c , λpq
Aclass Q pψ, zpq ÞÑψ Φ
λ : c ÞÑ ψpλ1 cq, λ zp
Preserve t, u; thus, carry over to the quantum algebra.
Diffeomorphisms
Full theory: SUp2q-bundle P over Σ
AutpPq Q Φ ñ Aclass Cyl Γ acts t, u - preserving via
Ψ ÞÑ Ψ Φ XS ,f ÞÑ XφpSq,f φ1
φ : Σ Ñ Σ diffeomorphism induced by Φ
Cosmology: M R R3 with R¡0 ñ ttu R3 SOp1, 3q
V0rCs ÞÑ λ3V0rCs
S Q pω, eq ÞÑ prΦλsω, rΦλseq P S ppc ,wq, vq p 3V0rCs|v |v8πGγ
pc, vq ÞÑ pλ1c , λ1vq ùñ pc , pq ÞÑ pλ1c , λpq
Aclass Q pψ, zpq ÞÑψ Φ
λ : c ÞÑ ψpλ1 cq, λ zp
Preserve t, u; thus, carry over to the quantum algebra.
Uniqueness of the Invariant State LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I
ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq
L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ
with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I
ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq
L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ
with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I
ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq
L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ
with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq
L continuous, Φ-invariant ùñ LpΨq ³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I
ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq
L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ
with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I
ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq
L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ
with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I
ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq
L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I
ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq
L continuous, Φ-invariant ùñ LpΨq ³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψq
L : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ pp
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψq
L : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0
ùñ Lp 9ψq 0 @ ψ P D
ðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ pp
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant
ùñ Lpψq ³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψq
L : C0pRq ÞÑ 0;
Lpc ÞÑ eiµcq δµ,0
ùñ Lp 9ψq 0 @ ψ P D
ðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ pp
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant ùñ Lpψq
³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0
ùñ Lp 9ψq 0 @ ψ P D
ðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ pp
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant ùñ Lpψq
³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0
ùñ Lp 9ψq 0 @ ψ P D
ðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
Uniqueness of the Invariant State
LOST
EHT
+A spanned by pΨ and pO pXS ,f
ω pO pXS,f
0
ab b b b a i ta, bu
Ψ b Ψ1 ΨΨ1
Ψ0 1
+A spanned by pψ and pψ ppp b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
p b ψ ψ b p i 9ψ
ψ b ψ1 ψψ1
ψ0 1
Diffeomorphism Invariance: ωpΦλpaqq ωpaq for each λ ¡ 0, a P A
pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant ùñ Lpψq
³RBohr
pG πAPqpψq dµB
% standard representation of LQC on L2pRBohr, dµBq
Quantum holonomy-flux -algebra A T pD` C pqJ
with J gen. by
ωpppppq ωpΦλpppppqq λ2ωpppppq ùñ ωpppppq 0
ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0 ùñ Lp 9ψq 0 @ ψ P Dðù Lp 9ψq 0 @ ψ P D
Quantum holonomy-flux -algebra A T pCyl` ΓqJ with J gen. by
Ψ b Ψ1 ΨΨ1 Ψ b Y Y
ab b b b a i ta, bu
Ψ0 1A
State: ω : AÑ C linear with ωp1q 1 and ωpaaq ¥ 0 for a P A
GNS-construction: |ωpabq|2 ¤ ωpaaq ωpbbq
H : AI with xras, rbsy : ωpabq and I ta P A |ωpaaq 0u
Representation: % : AÑ BpHq, %paq : rbs ÞÑ rabs
Diffeomorphism Invariance: ωpΦpaqq ωpaq for each Φ, a P A
pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq L continuous, Φ-invariant ùñ LpΨq
³A GpΨq dµAL OL+S
% standard representation of LQG on L2pA,dµALq
Standard Fleischhack
D dAP dAP ` d0
A AS AF AS ` J pd0q% %S %F %S ` 0
J pd0q IHS HF dAP
Both % : AÑ dAP H with xψ,ψ1y limn12n
³nn ψptq ψ
1ptq dt
%p pψq : χ ÞÑ πAPpψq χ %pz ppq : χ ÞÑ z i 9χ
The Homogeneous Case Ashtekar-Campiglia + EHT
Homogeneity + Triad diagonal + Dirac constrained analysis:
Conjugate variables c i , pi for i 1, 2, 3 with
Aia c iδia E i
a 8πGγV0rCs pi δ
ia.
Aclass D C p1 C p2 C p3 D generated by x ÞÑ eiµxj
tpψ, zpi q, pψ1, z 1pjqu pzBiψ
1 z 1Bjψ, 0q
Preserved under: Φ~λ: px1, x2, x3q ÞÑ pλ1x1, λ2x2, λ3x3q (act on Cell)
Quantum Algebra: WAC AEHT
(ψ P D) pψ, expi µi ppi pψ, ppi
%AC|pD %EHT|pD : pψ ÞÑ rχ ÞÑ ψ χs
%ACpexpi µj ppjqpψq exp
i µjBj
ψ %EHTpppjqpψq i Bjψ