Wojciech Kaminski- The EPRL intertwiners and correct partition function

8/3/2019 Wojciech Kaminski- The EPRL intertwiners and correct partition function

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The EPRL intertwiners andcorrect partition function.Wojciech Kaminski

Zakopane, 28.02.10

p.


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Spin-foams models

Is the EPRL map injective, doesnt it kill any SU(2)intertwiner?

p.


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Spin-foams models


Is the EPRL map isometric, does it preserve the scalar

product between the SU(2) intertwiners?

p.


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Spin-foams models


Is the EPRL map isometric, does it preserve the scalar

product between the SU(2) intertwiners?

If not, what is a complete, correct form of the partitionfunction written directly in terms of the SU(2)intertwiners, the preimages of the EPRL map?

p.


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Spin-foams

Spin-foam approach to LQG is an analog of the

Feynman path integral (Rovelli and Reisenberger)

p.


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Spin-foams



Spin-foam is a history of spin-network.

p.


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Spin-foams




p.


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Spin-foams




Evolution without adding vertices.

p.


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Spin-foams




Evolution by adding vertices.

p.


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Induced boundary spin-network

Spin network is obtained by transversal section of a

spin foam.

p.


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spin foam.

Colouring are induced on the spin-network from the

spin foam.

p.


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spin foam.

Colouring are induced on the spin-network from the

spin foam.

p.


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Euclidean 4d QG as spin-foam

4d QG is regarded as a BF theory with constraints.

Spin-networks consists of gauge invariant functions.

p.


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In each step we add one vertex with evolution of BF

theory, obtaining a transition amplitude between spinnetworks.

p.

E lid 4d QG i f


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In each step we add one vertex with evolution of BF

theory, obtaining a transition amplitude between spinnetworks.

Constraints are imposed as projections on edges(nodes of spin-network). p.


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Details

The simplicity constraints are imposed on the

elements of

H, locally at each vertex.

InvSimp(1 ...k k+1 ... N)

p.


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Details


elements of


InvSimp(1 ...k k+1 ... N)In each subsequent spin-network intertwiners shouldbe in this subspace,

p.

D il


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Details


elements of


InvSimp(1 ...k k+1 ... N)In each subsequent spin-network intertwiners shouldbe in this subspace,

To sum with respect to the spin-network histories with

the amplitude as a weight, one fixes an orthonormal

basis in each space of simple intertwiners.

p.

Si l i t t i


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Simple intertwiners

There are three main proposals for the simple intertwiners:

1. that ofBarrett-Crane (BC) corresponding to the

Palatini action,

2. that ofEngle-Pereira-Rovelli-Livine (EPRL)

corresponding the Holst action with the value of the

Barbero-Immirzi parameter = 1,3. that ofFreidel-Krasnov (FK) also corresponding to

the Holst action with the value of the Barbero-Immirzi

parameter = 1,

p.

EPRL i t t i


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EPRL intertwiner

Given intertwiner Inv(k1 ...) with the spins kI

jI =1 2

kIkI = j

+ + j, if || < 1 and kI = |j+ j|, if || > 1. This follows from

adjusted/improved constraints.

p.

EPRL i t t i


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EPRL intertwiner


jI =1 2

kI

p.

EPRL i t t i


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EPRL intertwiner


jI =1 2

kI

p.

EPRL i t t i


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EPRL intertwiner


jI =1 2

kI

p.

EPRL intertwiner


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EPRL intertwiner


jI =1 2

kI

The map EPRL

EPRL() : = (P+ P)c1 ... cn.

P projections onto SU(2) invariants,

ci Clebsch-Gordon coefficients ki

j+i

ji

.

p.

Orthonormal basis


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Orthonormal basis

One can show that the map EPRL() isinjective, so one can labelled EPRL intertwiners by

the SU(2) intertwiners.

p.

Orthonormal basis


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Orthonormal basis

One can show that the map EPRL() isinjective, so one can labelled EPRL intertwiners by

the SU(2) intertwiners.The proof is based on the observation that in suitable

basis matrix of the EPRL is upper triangular andinjective on diagonal entries.

p.

Proof 1


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Proof 1

Suitable basis is given by an intermediate spin k12

p.1

Proof 1


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Proof 1

Suitable basis is given by an intermediate spin k12 and j12.

Instead of projecting we consider contractions with the el-

ements ofInvSU(2)

SU(2).

p.1

Proof 1


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Proof 1

Suitable basis is given by an intermediate spin k12 and j12.

+ Instead of projecting we consider contractions with the el-

ements ofInvSU(2)SU(2).

p.1

Proof 2


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Proof 2

Such a basis is graded by k12 and j+12 + j

12 respectively,

+

p.1

Proof 2


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Proof 2

Such a basis is graded by k12 andj+12 +j

12 respectively,

+ then

(k12), j+12 j12 = 0, k12 > j+12 +j12.

p.1

Proof 2


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Proof 2

Such a basis is graded by k12 andj+12 +j

12 respectively,

+ then

(k12), j+12 j12 = 0, k12 > j+12 +j12.We can restrict our attention to k12 = j

+12 +j

12

p.1

Proof 3


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Proof 3

The contraction of

+ k12 = j

+12 +j

12

p.1

Proof 3


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Proof 3

...is equivalent to contraction of

+

p.1

Proof 3


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Proof 3


+ Imposing additional constraints on j12 allows forinductive procedure.

p.1

Proof 3


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Proof 3


+ Imposing additional constraints on j12 allows forinductive procedure.

There are some technical details...

p.1

Additional factor in the amplitude


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p

However, the map is not isometric for || = 1.

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p

However, the map is not isometric for || = 1. In orderto cure the lack of unitarity one should replace the former

spin -foam amplitude

Z[] =

jf ,e

f face

dim j+f , jf

v vertex

Av({(e)})({(bdr)}),

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p


spin -foam amplitude by

Z[] =

jf ,in/oute

f face

dim j+f , jf

e edge

A(ine , oute )

v vertex

Av({(e)})({(bdr)}),

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p


spin -foam amplitude by

Z[] =

jf ,in/oute

f face

dim j+f , jf

e edge

A(ine , oute )

v vertex

Av({(e)})({(bdr)}),

A(1, 2) is the inverse of the matrix (1), (2) so

PEPRL =

in/out

Ain,out|EPRL(out)EPRL(in)| p.1

Asymptotic behavior ofEPRL


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Even if the map EPRL is not unitary the asymptoticbehavior ofA may occure to be rather simple.

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=1

2 , j1 = 2, j2 = 4, j3 = 4, j1 = 2; a, b {2, . . . , 6}:

53723

175616

2265

57

50176

50935

1053696

355

250880

22655750176

117853

501760 12805

3010567

45 1177168

31378960

50935

1053696 12805

3010567

741949

3512320781

11

752640

513

5376

35525088

45 1177168

78111752640

5832560

0

0

3

137

8960

513

53760 13

40

p.1



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=1

2 , j1 = 2, j2 = 4, j3 = 4, j1 = 2; a, b {2, . . . , 6}:

0.3 0 0 0 0

0 0.23 0 0 00 0 0.21 0 0

0 0 0 0.22 0

0 0 0 0 0.32

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Examples of low j indicate that matrix A isaproximately diagonal (conjecture).

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Examples of low j indicate that matrix A isaproximately diagonal (conjecture).

If this is true A does not change (spoil) asymptoticbehavior of the spin foam amplitude.

p.1

Summary


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The map

EPRL()

is injective for all ,

p.1

Summary


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The map

EPRL()


Simple examples shows that it is not unitary for

||

= 1.

p.1

Summary


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The map

EPRL()


Simple examples shows that it is not unitary for

||

= 1.

The basis labelled by SU(2) intertwiners is notorthonormal and we should introduce an additional

factor A in the spin-foam amplitude

1|A12 = EPRL(1)|EPRL(2)

p.1

Wojciech Kaminski- The EPRL intertwiners and correct partition function

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