Florian Conrady- Spin foams with timelike surfaces

8/3/2019 Florian Conrady- Spin foams with timelike surfaces

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Spin foams with timelike surfaces

Florian Conrady

Perimeter Institute

ILQG seminarApril 6, 2010

FC, Jeff Hnybida, arXiv:1002.1959 [gr-qc]FC, arXiv:1003.5652 [gr-qc]

Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April 6 2010 1 / 60
http://goforward/http://find/http://goback/


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Outline

1 Motivation

2 Coherent states

3 Three ways to simplicity

4 Spin foams

5

Summary



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Motivation



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Restriction of triangulations

In the Lorentzian EPRL model, normals U of tetrahedra are timelike.

timelike U

All tetrahedra are Euclidean and triangles can be only spacelike.



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What we did

We extended the EPRL model to also include spacelike normals U.

timelike U

spacelike

U

Lorentzian tetrahedra are allowed and triangles can be spacelike ortimelike.



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Three cases

normal U

timelike U

spacelike

U

spacelike

U

triangle spacelike spacelike timelike

EPRL



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Canonical perspective

Are restricted triangulations preferred froma Hamiltonian point of view?

In the examples I know of the transition from space tospacetime leads to a 4d lattice with timelike (or null) edges. causal dynamical triangulations Ambjorn, Jurkiewicz, Loll evolution schemes for Lorentzian Regge calculus

Barrett, Galassi, Miller, Sorkin, Tuckey, Williams



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Canonical perspective

Are restricted triangulations preferred froma Hamiltonian point of view?

In the examples I know of the transition from space tospacetime leads to a 4d lattice with timelike (or null) edges. causal dynamical triangulations Ambjorn, Jurkiewicz, Loll

evolution schemes for Lorentzian Regge calculusBarrett, Galassi, Miller, Sorkin, Tuckey, Williams

The Hamiltonian approach to Lorentzian spin foams creates a

sequence of 3d spatial lattices. Han, Thiemann cannot (yet) be directly compared with 4d triangulations

discussed here



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Coherent states



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Little groups

normal U

timelike U

spacelike

U

spacelike

U

gaugefix U = (1, 0, 0, 0) U = (0, 0, 0, 1) U = (0, 0, 0, 1)

littlegroup SO(3) SO(1,2) SO(1,2)


EPRL



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Little groups

normal U

timelike U

spacelike

U

spacelike

U

gaugefix U = (1, 0, 0, 0) U = (0, 0, 0, 1) U = (0, 0, 0, 1)

littlegroup SU(2) SU(1,1) SU(1,1)


EPRL



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Representation theory

SL(2,C) SU(2)

generators Ji

, Ki

J1

, J2

, J3

Casimirs C1 = J2 K2

C2 = 4J KJ2

unitaryirreps

H(,n) Dj

R, n Z+ j Z+/2

C1 =12 (n

2 2 4)C2 = n

J2 = j(j + 1)



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Representation theory

SU(2) SU(1,1)

generators J1, J2, J3 J3, K1, K2

Casimirs J2 Q = (J3)2 (K1)2 (K2)2discrete series continuous series

unitaryirreps

Dj Dj Csj Z+/2 j = 12 , 1, 32 . . . j = 12 + is,

0 < s 1/2 D+

j

0

ds

Cs

n/2

j>1/2 D

j

0

ds

Cs

(,n) =

n/2j>1/2

m=j

+j m+j m +n/2

j>1/2

m=j

j mj m

+ =1,2

0

ds (s)

m= ()s m()s m

(see chapter 7 in Ruhls book)



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Nonnormalizability

States

|j m

in the continuous series irrep

Cs are normalizable, but the

corresponding states ()s m in H(,n) are not.

()s m()s m = (s s)(s) mm



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SU(1,1) coherent states discrete series

Definition

|j g Dj(g)|j j , g SU(1, 1)

|j N Dj(g(N))|j j , N H SU(1, 1)/U(1)

Upper/lower hyperboloid

H

=

{N

|N2 = 1 , N0 0

}



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Expectation values

So far the coherent states are the ones introduced by Perelomov.

They have the property that

j N

|J

|j N

= j N, N

S2

j N|F|j N = j N, N H

in accordance with the fact thatJ

2

= j(j + 1) 0 andF

2

= j(j 1) 0.



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SU(1,1) coherent states continuous series

Question

What are the appropriate coherent

states for the continuous series?

In this case Q = F2 =

(s2 + 1/4) < 0, so the classical vector N should

be spacelike.

Perelomov uses the state |j m = 0, resulting in a zero classical vector.



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Question

What are the appropriate coherent

states for the continuous series?

In this case Q = F2 =

(s2 + 1/4) < 0, so the classical vector N should

be spacelike.

F J3

K1

K2 Build coherent states from eigenstates of K1 or K2!



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Eigenstates ofK1

K1

|j

=

|j

,

< /2


( )


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Smearing in s

At the level of the SL(2,C) irrep H(,n), one also needs a smearing in s.

We define a smeared projector onto the irrep with spin j = 1/2 + is:

Ps() =1,2

m=

0

ds (s) f(s s) ()s m()s m


SU(1 1) h i i


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Definition

()s g 0

ds (s) f(s s)

d

1

f( s) D(,n)(g)()s +


P

s

() = =1,2

SU(1,1)

dg ()s g()

s g


Th i li i


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Three ways to simplicity


Th t i li it


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Three ways to simplicity

Derivation of extended simplicity constraints by three methods

1. weak imposition of constraints2. master constraint

(as advocated by EPRL)

3. restriction of coherent state basis (inspired by FK model)


Cl i l t t h d


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Classical tetrahedron

Describe a tetrahedron by four bivectors

J = B +1

B,

where B is constrained to be simple.

Simplicity constraint: unit fourvector U such that

U

B = 0 .


Classical simplicity constraints


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Classical simplicity constraints

Express B in terms of the total bivector J:

B = 22 + 1

J 1

J

Starting point for quantization:

U

J 1

J

= 0


First case: normal U timelike


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First case: normal U timelike

timelike U

In the gauge U = (1, 0, 0, 0), the simplicity constraint takes the form

J +1

K = 0

The little group is SU(2), so we use states of the SU(2) decomposition!


Spacelike U


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Spacelike U

spacelike

U

In the gauge U = (0, 0, 0, 1), the simplicity constraint becomes

F +1

G = 0

The little group is SU(1,1), so we use states of the SU(1,1) decomposition!


Basic example of secondclass constraints


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Basic example of secondclass constraints

Phase space and constraints

(qi, pi) , i = 1, 2 q1 q2 = 0

{qi, pi} = ij p1 p2 = 0

Change of variables

q = 12 (q1 q2) q = p = 0

p

= 1

2

(p1

p2)

a = 12 (p iq) a = 0


Weak imposition of constraints


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Impose a|

= 0 on physical states, implying

|a| = |a| = 0 |, | Hphys

Physical Hilbert Hphys space is spanned by |n+ |0, n+ N0.


Restriction of coherent state basis


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Overcomplete basis of coherent states |+ |, a| = |:

H=

1

2 d+ d |++| ||Restrict basis to states whose expectation values satisfy the constraint, i.e.to labels = 0.

Projector on Hphys

Pphys 1 d+ |++| |00|


Master constraint


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Master constraint

Master constraint operator

M = aa =1

2

p2 + q

2

+1

2

Define Hphys as the subspace of states with minimal eigenvalue w.r.t. M.

Hphys spanned by |n+ |0, n+ N0.


From classical to quantum simplicity


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From classical to quantum simplicity

J + 1K = 0

F + 1G = 0

4C3 = n

BIJ(B)IJ = 0

(

n) + n = 0

BIJ(B)IJ = 0 is the diagonal simplicity constraint.

C3 is the Casimir of the little group determined by the normal U.




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Consider the case U = (1, 0, 0, 0) and suppose that

Hphys =jJ

Dj H(,n) ,

where J is a subset of the total set of spins {j |j n/2}.Require that

|C| = 0 |, | Hphys.

Unless Hphys is trivial, this implies, that for some j n/2,

j m J3 + 1K3j m = j m J + 1Kj m = 0for all admissible m, m.




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pBy using K = [K3, J] and

K3|j m = (. . .)|j+ 1 m mAj |j m + (. . .)|j 1 m , Aj = n4j(j + 1)

,

one obtains

Aj = or 4C3 = 4j(j + 1) = n .

In conjunction with the constraint B B = 0, this gives 4j(j + 1) = n2 if = n and 4j(j + 1) = 2 if n = .

Approximate solution

= n , j = n/2

Hphys = Dn/2 H(n,n)




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p

Next consider U = (0, 0, 0, 1). Suppose first that the weak constraintholds for some irrep Dj of the discrete series:

j mF0 +

1

G0j m = j mF

+1

Gj m = 0 m, m ,

with F F2 iF1 and G G2 iG1. According to Mukunda

K3

|j m = (. . .)|j + 1 m mAj |jm + (. . .)|j 1 m , Aj = n

4j(j 1) ,

and G = [G0, F], which leads to Aj = or 4j(j 1) = n.


= n , n 2 , j = n/2




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p

For an irrep

Cs of the continuous series, the equations are the same except

that Aj is replaced by

Aj = n

4j(j + 1)= n

4(s2 + 1/4).

A solution exists only when = n/ < 1 and then

s2 + 1/4 =2

4

=n2

42

.

Overall result for U = (0, 0, 0, 1)

= n , n 2 or = n/ < 1Hphys = D+n/2 Dn/2 Hphys = C1

2

n2/21 C

12

n2/21




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Case U = (1, 0, 0, 0).

Resolve the identity on H(,n) in terms of SU(2) coherent states,

(,n) =

j=n/2

(2j + 1)

S2

d2N

jN

jN

,

and require that

jN

J +1

K

jN

= 0 .

This implies Aj = , that is, 4j(j + 1) = n.




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The second condition is either obtained from B B = 0 or alternativelyfrom the requirement of minimal uncertainty in K.From = Aj it follows that

J2 = 12

K2 + O(|J|) and J2 = 1J K ,

so that

(K)2 = 1

(1 2)J K 12

C1 + O(|J|)

=

4 1 1

2C2 +2

C

1 + O(|J|)=

4B B + O(|J|) .




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Projector on Hphys

Pphys = (n + 1)

S2

d2N

n/2N

n/2N


Master constraint


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For the case U = (0, 0, 0, 1) the Master constraint reads

M = F +1

G2

= 1 +1

2F2

1

22C1

1

2C2 = 0

The diagonal constraint B

B = 0 is equivalent to

1 12

C1 +

2

C2 = 0 .

By combining the two one arrives at the desired second condition

4F2 = n .


Master constraint


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In the case of the discrete series, the constraints are therefore

n +n

= 04j(j 1) = n


= n , n

2 j = n/2


Master constraint


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For states of the continuous series,

n +

n

= 0

4

s2 +1

4

= n

Solution

=

n

constr. on irreps j = n/2 j = n/2 s2

+ 1/4 = 2

/4

area spectrum

j(j + 1)

j(j 1) s2 + 1/4 = n/2 EPRL


Spin foams


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Spin foam theory


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Complex:

simplicial complex : 4simplex , tetrahedron , triangles t, . . .

dual complex : vertex v, edge e, face f, . . .

Variables (same as in EPRL):

connection ge SL(2,C)irrep label nf Z+

Additional variables:

Ue = (1, 0, 0, 0) or (0, 0, 0, 1): normal of tetrahedron dual to ef = 1: spacelike/timelike triangle dual to f


Uniform notation


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To cover the different cases we introduce a uniform notation.

Little group

H(, U) SU(2) , if = 1 , U = (1, 0, 0, 0) ,

SU(1, 1) , if = 1 , U = (0, 0, 0, 1) , , if = 1 , U = (1, 0, 0, 0) .

Spin

j = n/2 , if = 1 , U = (1, 0, 0, 0) ,

n/2 , if = 1 , U = (0, 0, 0, 1) ,

12 + i2

n2/2 1 , if = 1 , U = (0, 0, 0, 1) .


Uniform notation


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Coherent states

|()j h =

|j h

, if = 1 , U = (1, 0, 0, 0) ,

|j h , if = 1 , U = (0, 0, 0, 1) ,|()j h , if = 1 , U = (0, 0, 0, 1) .

Projector

Pj(, U, ) = dj(, U)

H(,U)

dh ()j h ()

j h


Vertex amplitude


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e e

v

f

gev gve

Av((f, nf); hef, ef, )

=

SL(2,C)

e

dgev

f

(ef)jefhef

D(f,nf)(gevgve) (ef)jefhef



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Compatibility of path integral and operator formalism


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So far we were mainly guided by the path integral pictureand defined the spin foam sum by simply summing overall choices of the normal U.

From the canonical perspective, however, one would alsorequire that the sum over intermediate states on a trian-

gulations boundary is a projector.

For a give choice of U, this is certainly the case.

Is it also true when we sum over U?




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normal U

timelike U

spacelike

U

spacelike

U


(, n) = n = n, n

2 n =

>

EPRL




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,

= 0

,

= 0

,

= 0

,

= 0

The sum over tetrahedral states is a projector if we exclude .

P = + red>0

The boundary Hilbert space is genuinely larger than the EPRL one.


Summary


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Summary


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extension of EPRL model tetrahedra can be Euclidean and Lorentzian triangles can be spacelike and timelike larger boundary Hilbert space

discrete area spectrum of timelike surfaces

definition of associated spin foam model

coherent states for timelike triangles


Outlook


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regions with timelike boundaries! Rovelli et al., Oeckl

extension of results on EPRL model? asymptotics, graviton propagator . . . ?

canonical LQG on general boundaries?

comparison with previous work on timelike surfacesPerez, Rovelli

Alexandrov, Vassilevich

Alexandrov, Kadar


Florian Conrady- Spin foams with timelike surfaces

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