Relating LQC with LQG { Algebraic...

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$Page 1: Relating LQC with LQG { Algebraic Aspectsktwig.fuw.edu.pl/pliki/Loops17PlenariesPresentations/Loops17_Hanusch.pdfRelating LQC with LQG { Algebraic Aspects {Maximilian Hanusch University$

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

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

From LQG to LQC

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

From LQG to LQC

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

From LQG to LQC

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

From LQG to LQC

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

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.

a ωi0a

Aia c δia

E ai

8πGγ3V0rCs pδ

ai

The Holst Action:

L 132πG

³pM,resq

εIJKL e

rL p 9c Nb

3V0rCskγ3

E ai |det e jb | e

ai (Dreibein)

a ωi0a

Aia c δia

E ai

8πGγ3V0rCs pδ

ai

The Holst Action:

L 132πG

³pM,resq

εIJKL e

rL p 9c Nb

3V0rCskγ3

E ai |det e jb | e

ai (Dreibein)

a ωi0a

Aia c δia

E ai

8πGγ3V0rCs pδ

ai

The Holst Action:

L 132πG

³pM,resq

εIJKL e

rL p 9c N

b3V0rCskγ3

E ai |det e jb | e

ai (Dreibein)

a ωi0a

Aia c δia

E ai

8πGγ3V0rCs pδ

ai

The Holst Action:

L 132πG

³pM,resq

εIJKL e

rL p 9c N

b3V0rCskγ3

a γK ia

(Ashtekar connection)

E ai |det e jb | e

ai

(Dreibein)

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 ÞÑ³SrE 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

` C0pRq

Arcsia c δia

ai

rE 12! εabc E

i Ba b τ i

Aclass : Cyl Γ

D C p

Poisson Bracket:

` C0pRq

Arcsia c δia

ai

rE 12! εabc E

i Ba b τ i

Aclass : Cyl Γ

D C p

Poisson Bracket:

ψ : c ÞÑ ΨpArcsq P CAPpRq ` C0pRq

Arcsia c δia

ai

rE 12! εabc E

i Ba b τ i

Aclass : Cyl Γ

D C p

Poisson Bracket:

Arcsia c δia

ai

rE 12! εabc E

i Ba b τ i

Aclass : Cyl Γ D C p

Poisson Bracket:

Arcsia c δia

ai

rE 12! εabc E

i Ba b τ i

Poisson Bracket:

Arcsia c δia

ai

rE 12! εabc E

i Ba b τ i

Poisson Bracket:

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

V0rCs ÞÑ λ3V0rCs

Diffeomorphisms

V0rCs ÞÑ λ3V0rCs

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

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

pp P I

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

with J gen. by

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

pp P I

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

with J gen. by

ab b b b a i ta, bu

Ψ0 1A

pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq

L continuous, Φ-invariant ùñ LpΨq ³A GpΨq dµAL OL+S

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

pp P I

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

with J gen. by

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

pp P I

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

with J gen. by

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

Uniqueness of the Invariant State

LOST

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

pp P I

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

Quantum holonomy-flux -algebra A T pD` C pqJ with J gen. by

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

LOST

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

pp P I

ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

ab b b b a i ta, bu

Ψ0 1A

pXS,f P I ùñ ω uniquely determined by L : Ψ ÞÑ ωppΨq

L continuous, Φ-invariant ùñ LpΨq ³A GpΨq dµAL OL+S

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

LOST

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

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

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

LOST

EHT

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

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

L : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0

ùñ Lp 9ψq 0 @ ψ P D

ðù Lp 9ψq 0 @ ψ P D

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

LOST

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

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

ùñ Lpψq ³RBohr

pG πAPqpψq dµB

L : C0pRq ÞÑ 0;

Lpc ÞÑ eiµcq δµ,0

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

LOST

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

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

pp P I ùñ ω uniquely determined by L : ψ ÞÑ ωp pψq L continuous, Φλ-invariant ùñ Lpψq

³RBohr

pG πAPqpψq dµB

ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

LOST

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

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

³RBohr

pG πAPqpψq dµB

ùñ ωp pψ ppq 0 ωppp pψqL : C0pRq ÞÑ 0; Lpc ÞÑ eiµcq δµ,0

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

LOST

EHT

ω pO pXS,f

0

ab b b b a i ta, bu

Ψ b Ψ1 ΨΨ1

Ψ0 1

ψ b ψ1 ψψ1

ψ0 1

p b ψ ψ b p i 9ψ

ψ b ψ1 ψψ1

ψ0 1

³RBohr

pG πAPqpψq dµB

with J gen. by

ab b b b a i ta, bu

Ψ0 1A

D dAP dAP ` d0

J pd0q IHS HF dAP

³nn ψptq ψ

1ptq dt

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

THE END

Relating LQC with LQG { Algebraic...

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Transcript of Relating LQC with LQG { Algebraic...