On the relation between covariant and canonical Quantum...

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On the relation between covariant and canonical Quantum Gravity [arXiv:gr-qc/1307.5885] Antonia Zipfel with Thomas Thiemann Antonia Zipfel (FUW) ILQGS, 15.04.2014 1 / 30 Grant: 2012/05/E/ST2/03308

Transcript of On the relation between covariant and canonical Quantum...

On the relation between covariant and canonicalQuantum Gravity [arXiv:gr-qc/1307.5885]

Antonia Zipfelwith Thomas Thiemann

Antonia Zipfel (FUW) ILQGS, 15.04.2014 1 / 30

Grant: 2012/05/E/ST2/03308

Motivation

Canonical LQG

Ts(A) = Tr[∏l

hjl(A)∏n

ιn]

Hkin =⊕γ∈ΣHkin,γ

Covariant LQG

v

m

ji

jk

jj

Z [κ] =∑c∏v

Av∏f

Af × B

H∂κ ≃Hkin,γ

Is it possible to merge covariant and canonical LQG?

Antonia Zipfel (FUW) ILQGS, 15.04.2014 2 / 30

The general idea

v

m

ji

jk

jj

⟨Ts ∣Z [κ]∣Ts′⟩

∂κ = γs ∪ γs′

Rovelli,Reisenberger,Barrett, Crane, Engle,

Pereira, Livine, Freidel, Krasnov,

Kaminski, Kisielowski, Lewandowski, . . .

Heuristic Idea

δ(H) = ∫ exp (itH)dt ↭∑κ

Z [κ]±

Feynmandiagrams

[Reisenberger, Rovelli]

“Physical Scalar Product”

η[Ts](Ts′) ∶= ∑κ∶γs′→γs

⟨Ts ∣Z [κ]∣Ts′⟩

Antonia Zipfel (FUW) ILQGS, 15.04.2014 3 / 30

Main Results

The extension of the [Engle,Pereira,Rovelli, Livine]-[Freidel, Krasnov]-model by[Kaminski, Kisielowski, Lewandowski] enables to investigate whether spin foams can beused to construct a rigging map for any of the presently definedHamiltonian constraint operators.We suggest a ‘rigging map’ closely following the ideas of [Reisenberger, Rovelli].

In the analysis of the resulting object we are able to identify an elementaryspin foam transfer matrix that allows to generate any finite foam as a finitepower of the transfer matrix.

It transpires that the postulated map does not define a projector on thephysical Hilbert space. However, it might be possible to construct a properrigging map in terms of a modified transfer matrix.

Antonia Zipfel (FUW) ILQGS, 15.04.2014 4 / 30

Outline

1 A spin foam rigging mapRigging maps and spin foams - a brief reviewProposal for a spin foam ‘rigging map’Properties

2 Why the would-be ‘rigging map’ is not a proper rigging mapTime orderingFactorizationWhy η is not a rigging map

3 Discussion and outlookWhat is going wrong?Open questions

Antonia Zipfel (FUW) ILQGS, 15.04.2014 5 / 30

A spin foam rigging map Rigging maps and spin foams - a brief review

Rigging maps

How to solve the constraints of canonical QG?

Generalized solution of C

.. is a state l ∈ D∗kin s.t [C∗l] (f ) ∶= l(C †f ) = 0 ∀f ∈ Dkin

Dkin ⊂Hkin dense domain, algebraic dual D∗kin

Construction principle? What is the scalar product?

Rigging mapGiven η ∶Hkin → D

∗kin s. t. ⟨η(f ), η(f ′)⟩phys = [η(f ′)](f )

Antonia Zipfel (FUW) ILQGS, 15.04.2014 6 / 30

A spin foam rigging map Rigging maps and spin foams - a brief review

Euclidean spin foam models

1 Action: SBF = ∫MTr[(B +

1β∗B) ∧ F ] + Simplicity constraint

M space-time, B bivector, F curvature of spin(4)-connection on M, β Barbero-Immirzi parameter

2 Discretize: T triangulation ofM with dual 2-complex κT ↝ SBF [κT]

3 Quantize ∫ DP eiSBF [κT] ↝ ABF [κT, ψin, ψout] = ⟨ψin,ZBF [κT ]ψout⟩

4 Implement simplicity constraint

Result

AEPRL−FK

[κT, ψin, ψout] = ⟨ψin,Z [κT]ψout⟩ =

∑j±f,ιEPRLe

∏f

Af ∏e∈κint

Qe ∏v∈κint

Av(j±f , ι

EPRLe ) B(ψin, ψout)

j± = ∣1±β∣2 j, ιEPRL intertwiner coupling j and (j+, j−)

Antonia Zipfel (FUW) ILQGS, 15.04.2014 7 / 30

A spin foam rigging map Proposal for a spin foam ‘rigging map’

Merging canonical and covariant LQG - Prerequisites

Can we construct η[Ts](Ts′) = ∑κ∶γs′→γs

⟨Ts ∣Z [κ]∣Ts′⟩ ?

Need identification of boundary states ψin, ψout with states in Hkin

Canonical LQGContinuum theory

Quantize point separatingsub-algebra of Poisson algebra↝ all possible graphs in Σ

Hamiltonian ↝ 3-valent nodes

EPRL-FK Model (Eucl.)Discrete theory

κ dual to triangulation T ofM,↝ boundary graphs dual totriangulation of Σ

at least 4-valent nodes

Other issues: Path integral measure? Fate of observables?Implementation of simplicity constraint?

Antonia Zipfel (FUW) ILQGS, 15.04.2014 8 / 30

A spin foam rigging map Proposal for a spin foam ‘rigging map’

Abstract foams - The KKL-model

Spin foams - a tool to determine the physical scalar product ?

KKL-model[Kaminski, Kisielowski, Lewandowski]

Achievements:1 Z [κ] defined for arbitrary 2-complexes

allowing arbitrary boundary graphs ∂κ

2 For certain choice of β: H∂κ ≃Hkin,∂κ

In the following we will consider abstract foams

Abstract foam κ: p.l. homeomorphic to a 2-complex {f , e, v}whose boundary ∂κ is the disjoint union ofclosed graphs bordering κ

γ borders κ: exist 1-to-1 affine map γ × [0,1]→ κ

Antonia Zipfel (FUW) ILQGS, 15.04.2014 9 / 30

A spin foam rigging map Proposal for a spin foam ‘rigging map’

Proposed rigging-map

Abstract equivalence

Two embedded spin nets belong to the same abstract equivalence class[s]A if they are embeddings of the same abstract spin net sA.

A spin foam rigging map

η[Ts](Ts′) = ∑[s′]A∈NA

η[s]A,[s′]AL[s′]A with η[s]A,[s′]A = ∑κA(sA,s′A)

Z [κA(sA, s′A)]

and L[s′]A = η[s′]A ∑s∈[s′]A

⟨Ts , ⋅ ⟩

Z [κA(sA, s′A)] ∶= ∑

jf ,ιe

∏f

Af ∏e∈κint

Qe ∏v∈κint

Av(jf , ιe)∏l∈∂κ

δjl ,jfl ∏n∈∂κ

διn,ιen

[Kaminski, Kisielowski, Lewandowski], [Ding, Han, Rovelli],[Bahr, Hellmann, Kaminski, Kisielowski, Lewandowski]

Antonia Zipfel (FUW) ILQGS, 15.04.2014 10 / 30

A spin foam rigging map Properties

Properties of Z [κ]

Z [κA(sA, s′A)] ∶= ∑

jf ,ιe

∏f

Af ∏e∈κint

Qe ∏v∈κint

Av(jf , ιe)∏l∈∂κ

δjl ,jfl ∏n∈∂κ

διn,ιen

[Kaminski, Kisielowski, Lewandowski], [Ding, Han, Rovelli],[Bahr, Hellmann, Kaminski, Kisielowski, Lewandowski]

Can be defined s.t. Z [κA(sA, s′A)] is invariant if . . .

. . . faces labeled by trivial representation are added or removed

. . . internal edges splitting a face are added or removed

. . . 2-valent vertices are added or removed

Z [κA](sA, s′A) ∶Hkin,γ(s′

A) →Hkin,γ(sA) is cylindrically consistent!

Furthermore: ∀ γ ∃ κ0γ s.t. Z [κ0

γ] ∶Hkin,γ →Hkin,γ with Z [κ0γ] = 1γ

Antonia Zipfel (FUW) ILQGS, 15.04.2014 11 / 30

A spin foam rigging map Properties

Minimal foams

NoteSince adding faces with jf = 0, splitting faces and edges leave theamplitude/spin net function invariant one should only sum over equivalenceclasses in the rigging map .

DefinitionAn abstract foam/graph is called minimal iff it cannot be obtained fromanother foam/graph by subdivisions.

RemarkMinimal representative is not unique . . .

. . . but amplitude does not depend on choice

Can always fix a representative for each class and only sum over those.

Antonia Zipfel (FUW) ILQGS, 15.04.2014 12 / 30

A spin foam rigging map Properties

Gluing

v1v2

v3

v1v2

v3

Suppose κ1 ∩ κ2 = ∂κ1 ∩ ∂κ2 = γ then

∑s(γ)

Z [κ1(s, s)]Z [κ2(s, s′)] = Z [κ1 ♯κ2(s, s

′)]

Antonia Zipfel (FUW) ILQGS, 15.04.2014 13 / 30

Why the would-be ‘rigging map’ is not a proper rigging map

Outline

1 A spin foam rigging mapRigging maps and spin foams - a brief reviewProposal for a spin foam ‘rigging map’Properties

2 Why the would-be ‘rigging map’ is not a proper rigging mapTime orderingFactorizationWhy η is not a rigging map

3 Discussion and outlookWhat is going wrong?Open questions

Antonia Zipfel (FUW) ILQGS, 15.04.2014 14 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Time ordering

Spliting

v1v2

v3

v1v2

v3

Idea: Use this to split big complexes to get a better control on

η[Ts](Ts′) ∶= ∑κ∶γs′→γs

⟨Ts ∣Z [κ]∣Ts′⟩

Antonia Zipfel (FUW) ILQGS, 15.04.2014 15 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Time ordering

Time ordering

v1v2

v3

Definitionv ∈ κint s.t. ∃n ∈ γi and ∃e ∈ κint withs(e) = n and t(e) = v

↝ vertex of first generation

Inductively: Vertex of nth generation

Only final graph ⇒ count backwards

∂κ = ∅ all v ∈ κ of first generation

TheoremA minimal foam κ can be uniquely split into minimal foams κi , called ‘onetime step’ foams, containing only vertices of i th generation with respect tothe original foam κ = κ1 ♯⋯ ♯κN where N is the maximal generation of κ.

Antonia Zipfel (FUW) ILQGS, 15.04.2014 16 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Time ordering

Sketch of the Proof I

To prove the theorem we need the following observations:

LemmaIf e ∈ κint has a vertex of nth generation then the other vertex of e is eitherof generation n − 1,n or n + 1 or a vertex in a final graph.

Introduce additional vertices so that the second vertex of e is either ofgeneration n − 1,n or n + 1.

LemmaIf a face has an edge e joining a vertex of nth and (n + 1)th generation thenit contains at least one other edge connecting vertices of nth and (n + 1)th

generation (or v of generation m and v ′ in final graph).

Antonia Zipfel (FUW) ILQGS, 15.04.2014 17 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Time ordering

Sketch of the Proof II

Proof of the theorem

2nd 2nd

1th

1th

2nd2nd

1th

1th

En,n+1 ∶= {e ∈ κint ∶joining vertices of nth and (n + 1)th generation}

Number of boundary edges e ∈ En,n+1 of f is even

Split faces until a face contains at most 2 edges inE1,2

Split all edges in E1,2 and join new vertices by edges

↝ κ = κ1 ♯κ′

Repeat the procedure until κ = κ1 ♯⋯ ♯κN

Antonia Zipfel (FUW) ILQGS, 15.04.2014 18 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Factorization

Factorization of the rigging map I

If η is a proper rigging map then . . .

η[Tsf ](HTsi )!= 0 ∀Tsi ∈Hkin

Ô⇒ η[T∅](HTsi ) = (∑n(ηc[T∅](T∅))n)ηc[T∅](HTsi )

!= 0

where ∑κ in ηc is restricted to connected foams

Thus to test the rigging map it suffices to only consider connected foams!!

But when splitting a connected κ into ‘one time step’ foamsκ1 ♯⋯ ♯κn = κ then κi are not necessarily connected

However, if T∅ is excluded and either γ(sf ) and or γ(si) areconnected then any foam κ1 ♯⋯ ♯κn is connected

Test wether ηc[Tsf ] annihilates H for any sf with connected graph γ(sf )

Antonia Zipfel (FUW) ILQGS, 15.04.2014 19 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Factorization

Factorization of the rigging map II

Spin foam transfer matrix

Kγ,γ′ set of ‘one time step’ foam and Pγ ∶Hkin →Hkin,γ

Z ∶= ∑γ,γ′

Pγ′ [ ∑κ∈Kγ,γ′

Z(κ)] Pγ

We then obtain for γ(sf ) connected . . .

ηc[Tsf ](Tsi ) = ∑κ∈Kγ(si ),γ(sf )

⟨Tsf , Z(κ1 ♯⋯ ♯κN) Tsi ⟩ =∞∑N=0

⟨Tsf , ZNTsi ⟩

RemarksAbove factorization would lead to ordering ambiguities fordisconnected foamsZ is symmetric

Antonia Zipfel (FUW) ILQGS, 15.04.2014 20 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Why η is not a rigging map

A naive argument

If η would be a proper rigging map then

ηc[Tsf ](HTsi ) = ⟨Tsf ,∞∑N=0

ZNHTsi ⟩!= 0

But . . .

A ∶=∞∑N=0

ZN⇒ A = 1 + ZA

leads to a contradiction:

0 = AH = (1 + ZA)H = H

Of course, this expression is only formal as A is very likely diverging andtherefore requires a regularization

Antonia Zipfel (FUW) ILQGS, 15.04.2014 21 / 30

Why the would-be ‘rigging map’ is not a proper rigging map Why η is not a rigging map

Regularization of Z

Cut-Off: Kγ,γ′ is infinite, therefore introduce cut-off (J,Nf ,Ne)

Weight: Suppose there exist a weight ω(κ ♯κ′) = ω(κ)ω(κ′) s.t.Z ′

∶= ∑

κ∈Kγ,γ′ω(κ)Z [κ] has finite norm

Even if Z ′ is now densely defined (Z ′)n is not necessarily densely defined.

Recall: Z is symmetric and so is Z ′

Suppose: Z ′ can be extended to an self-adjoint operator withprojection valued measure E and let

Then: Z ′ acts on the states ψq ∶= ∫q−q dE(λ) ψ, ψ ∈Hkin, by

multiplication with λ ∈ [−q,q] where 0 < q < 1

But: Aψq = ∫q−q dE(λ) (1 − λ)−1 ψ ↝ A = 1 + ZA ☇

Antonia Zipfel (FUW) ILQGS, 15.04.2014 22 / 30

Discussion and outlook

Outline

1 A spin foam rigging mapRigging maps and spin foams - a brief reviewProposal for a spin foam ‘rigging map’Properties

2 Why the would-be ‘rigging map’ is not a proper rigging mapTime orderingFactorizationWhy η is not a rigging map

3 Discussion and outlookWhat is going wrong?Open questions

Antonia Zipfel (FUW) ILQGS, 15.04.2014 23 / 30

Discussion and outlook What is going wrong?

What is going wrong?

Option I: Z is itself a projector, e.g. BF-theory

Option II: Vertex amplitude too local

Option III: Necessity to restrict to one Plebanski sector?

Option IV: Wrong assumption on the weight!

Antonia Zipfel (FUW) ILQGS, 15.04.2014 24 / 30

Discussion and outlook What is going wrong?

What is going wrong?

Option I: Z is itself a projector, e.g. BF-theory

1 It can be shown that for β = 1 the operator ∑κ∈KMelon

Z [κ] annihilates the

Euclidean constraint. Here, κ ∈ KMelon are foams with just one internal

vertex and boundary graphs of the form: and

[Alesci, Thiemann, A.Z.]

2 Z can be interpreted as an object obtained from some sort of coarsegraining. In [Dittrich, Hellmann, Kaminski] a similar object was derived forholonomy spin foam models via coarse graining. For BF-theory thisalready incorporates the dynamics.

Antonia Zipfel (FUW) ILQGS, 15.04.2014 25 / 30

Discussion and outlook What is going wrong?

What is going wrong?

Option II: Vertex amplitude too local

The would-be rigging map can be factorized since foams can besplit, that is, κ = κ1 ♯⋯ ♯κN

Suppose AvQe(v , v′)Av ′ /∝ AvQe(v ,m)AmQe(m, v

′)Av ′

then splitting is no longer possible.

In this sense: vertex amplitude too local

Other hints into this direction: E.g. [Dittrich], [Hellmann, Kaminski],. . .

Antonia Zipfel (FUW) ILQGS, 15.04.2014 26 / 30

Discussion and outlook What is going wrong?

What is going wrong?

Option III: Necessity to restrict to one Plebanski sector?

2πδ(C) = limT→∞∫

T

−Te itC = lim

T→∞∫T

0[e itC + e−itC ]

= limT→∞

limn→∞

n

∑k=1

T

n[e iCT /n

]k

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Uk

+ [e−iCT /n]k

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶(U†)k

But: Z is symmetric ↝ Z = U +U†

⇒ ∑k∈N

Z k= ∑

k∈N(U +U†

)k≠ δ(H)

A similar problem occurs in the asymptotic expansion.Can it be solved by the proper vertex proposal [Engle]?

Antonia Zipfel (FUW) ILQGS, 15.04.2014 27 / 30

Discussion and outlook What is going wrong?

What is going wrong?

Option IV: Wrong assumption on the weight!

We assumed that Z can be regularized by introducing a weight ω thatobeys ω(κ1)ω(κ2) = ω(κ1 ♯κ2). Is that sensible?

1 Regularization might require symmetry factors that destroy splitting:

δ(x) = ∫ dk eikx = ∫ dk(∑n

in

n!(kx)n)

For example in standard GFT λ∣γ(0)∣

∣sym(γ)∣2 Spin foam amplitudes suffer from bubble divergencies. However, if

foams with bubbles are excluded then gluing might no longer bepossible, since even if κ1, κ2 contain no bubbles, κ1 ♯κ2 might havebubbles.

Antonia Zipfel (FUW) ILQGS, 15.04.2014 28 / 30

Discussion and outlook Open questions

Open questions

RegularizationExplore regularization methods of generalized GFT [Oriti]

Explore latest results on regularization of the EPRL-model[Bonzom,Carrozza,Oriti, Puchta, Riello, Rovelli, Smerlak,. . . ]

Connection to coarse graining [Dittrich, . . . ]?

Proper vertexGeneralize proper vertex proposal [Engle] to n-valent vertices and test wetherthe modified rigging map still has the same problems.

Technical aspectsSingle time step ↝ better control on the sumFormalism to design single time step foams;[Kisielowski, Lewandowski, Puchta]

Antonia Zipfel (FUW) ILQGS, 15.04.2014 29 / 30

Discussion and outlook Open questions

Thank you for your attention!Contents

Antonia Zipfel (FUW) ILQGS, 15.04.2014 30 / 30