From groups and knots to BH entropy

From groups and knots to BH entropy Hanno Sahlmann

Department of PhysicsUniversity of Erlangen-Nürnberg

0. Introduction

2

Loop Quantum gravityQuantum theory of gravity.

Some of the assumptions:

• space-time 3+1 dimensional

• Einstein gravity

• quantization of geometry (not topology, not diff manifold,…)

• general covariance (—> use background structures as little as possible)

3

Some of the consequences:

• QFT of geometry

• discrete, combinatorial picture: “Atoms” of space(-time)

• not easy to interpret operationally

1. Quantized space

4

Spin networksPenrose (1971):

Spin network:

• directed graph (may be embedded in 3-dim manifold)

• SU(2) irrep (spin je) associated to each edge e

• invariant tensor at each vertex

5

Iv 2 Inv

Oe at v

je

!

Quantum (spatial) geometry <--> QM of spin:Trivial example: existence of invariant tensor, quantum triangle

Chromatic evaluation: Penrose invents way to associate number to spin network via graphical calculusExample:

Uses it to give inner product on spin networks, beginning of quantum theory of geometry

6

j1j2

j3

j1j2

j3

! j1 + j2 + j3 � 2max(j1, j2, j3) !

j2

j4j1

j3 j5j6

!⇢

j1 j3 j5j2 j4 j6

�

Note for later:Virtual edges can be used to always go to trivalent case:

ck can be interpreted in terms of geometry.

7

Beautiful thing:In loop quantum gravity:

spin networks = states of non-commutative spatial geometry

is result of quantization of gravity.

k

j1j2

j3 j4

j1j2

j3 j4

!X

k

ck

Geometric operators: [Rovelli+Smolin, Ashtekar+Lewandowski]

Area

more generally:

8

jj

bA = 8⇡�l2ppj(j + 1)

�v =

X

e above S

~Je �X

e below S

~Je

!2

bA|v = 4⇡�`2Pp

�v

Spectrum is purely discrete

Area is non-commutative

9

[ bA , bA ] 6= 0

Volume: Action on intertwiner spaces

Picture: Vertices as atoms of space

More quantitatively [Baez, … , Bianchi, Dona,Speziale]

n valent vertex <——> quantized flat polyhedron

Some examples for monochromatic 4-vertex (from Bachelors thesis of K.Eder)

10

k

j1j2

j3 j4

j1j2

j3 j4

=X

k

ck

bVR |ki =X

l

V (v)kl |li

=X

k

ck |ki

11

j0 5 10 15 20 25

volumeeigenvalues

0

10

20

30

40

50

60

intertwiner state

12

j0 1 2 3 4 5 6 7 8 9 10

angle

inrad

0

0.5

1

1.5

2

2.5

3

3.5

4

<12>

<13>

Derivations (sketch!)

• space-time split

• covering group SO(3,1) —> SL(2,C)

• partial gauge fixing (time gauge) SL(2,C) —> SU(2)

14

Ashtekar’s discovery: GR can be formulated such that (non-reduced) phase space is that of SU(2) Yang-Mills

S[e,!] ⇠Z

⇤(e ^ e) ^ F (!) +1

�e ^ e ^ F (!)

· eIa : Tetrad · !IJa : SO(3,1) connection

· ⇤ : “internal Hodge”

⇠Z

dt

Z

⌃t

Eai A

ia � !i0

0 Gi + ei0Ca + e00Ct

phase space formulation

• (A,E) phase space coordinates

• (first class) constraints G,Ci, C

Quantum algebra

15

IJ

I[E ,f , ] = 8⇡�l2P f(p)I

[[E ,f , E ,g], ] = (8⇡�l2P)2 f(p)Ig(p)J

[h↵, h↵0 ] = 0

A 7�! he = Pexp

Z

e�A

E 7�! ES,f =

Z

SEifi

Representation:

Combinatorial description

No background geometry used anywhere. Spatial diffeomorphisms unitarily

represented.

Uniqueness: [HS, LOST, Fleischhack]This is the only cyclic rep with spatial diffeo invariant vacuum.

16

=½ ½ 1

+

=jj

=k

k

=

j j = 2j+1

j k = 0

[Rovelli+Smolin, Ashtekar+Lewandowski]

bE ,f

Analytic description:

Spin nets are associated to gauge invariant functionals.

Inner product:

Spin nets are orthogonal (compare Peter-Weyl).

Hilbert space

17

j1 j2j3

[Ashtekar+Lewandowski]

[A] = ⇡j1(he1 [A])m1n1 ⇡j2(he2 [A])m2

n2 ⇡j3(he3 [A])m3n3 ◆m1m2m3◆

n1n2n3

h � | 0�i =

Z

SU(2)|�|

Y

i

dhi (h1, h2, . . .) 0(h1, h2, . . .)

HAL = lim�!1

H� = L2(A/G, dµAL)

projective/inductive limits

Note: Higher dimensions

Can embed gravity in D+1 in SO(D+1) -Yang Mills phase space.

Canonical pair: – A: SO(D+1) connection – π: tensor density corresponding momentum

with:– simplicity constraints ⇒ π encodes d-bein– Gauss, diffeo, Hamilton constraints

18

[Bodendorfer, Thiemann, Thurn 2013]

2. Quantized space-time (see also D. Oriti’s talk later this afternoon!)

19

Spin foam [Ponzano-Regge, Baez, Rovelli+Reisenberger,…]

For group G:

• oriented 2-complex (with boundary) C

• labeling of faces with irreps of G

• labeling of edges with interwiners between face-reps

Amplitude:

Interpretation: Transition amplitude.

Typically:

20

Z =X

{f}!IrrepG

X

◆

Y

f

Af (⇡)Y

v

Av(⇡e, ◆n)

Z : { C, boundary data } �! C

EPRL-FK vertex amplitude:

• T: vertex spin network

• Y: map SU(2) spin nets —> SL(2,C) spin netsFor choice of SU(2) subgroup

21

Av(je, ◆n) = (Y T{je,◆n})(ISL(2,C))

Hp,k =1M

j=k

Hjp,k Y : ⇡j(·)mm0 7�! ⇡j

�j,j(·)m

m0

Asymptotics

Vertex amplitude asymptotes to Regge-action when evaluated on boundary coherent state

where Ψ is peaked on boundary geometry of 4-simplex, and SRegge is Regge action.

Vertex: Atom of space-time?

22

X

je

X

◆n

AEPRLv (je, ◆n)

t(je, ◆n) ' eiSRegge + e�iSRegge

Derivations (sketch!)

1) Canonical

• Diff constraint: Roughly speaking

• Hamilton constraint

23

j jj= ≠

+

X

a,b

↵ab

X

a,b

↵ab

X

a,b

↵ab

using formal expansion

Projector on kernel

Has expansion of matrix elements labeled by 2-complexes

24

�(~x) =

Zexp(i

~

k · ~x)d~k =

Z(1 + i

~

k · ~x+

1

2

(i

~

k · ~x)2 + . . .)d

~

k

Pphys =

Y

x

�(

bC(x)) =

ZDN exp(i

ZdxN(x)

bC(x))

= 1+ i

ZDN

ZdxN(x)

bC(x)

� 1

2

ZDN DN

0Z

dx dx

0N(x)

bC(x)N

0(x

0)

bC(x

0) + . . .

2) covariant• with phi enforcing simplicity

• BF theory topological, Z can give exact path integral

• Y constructed in such a way that simplicity holds in a sense for the SL(2,C) representations

25

S[B,!,�] =

ZB ^ F (!) + � B ^B

B = ⇤(e ^ e) +1

�e ^ e

Z =X

{⇡f}

X

{Ie}

Y

f

Af (⇡f )Y

v

(I)

Hp,k =1M

j=k

Hjp,k Y : ⇡j(·)mm0 7�! ⇡j

�j,j(·)m

m0

But:

• divergencies in the Lorentzian case• summing over spin-foams• renormalization?

Ongoing work:

Course graining, renormalization [Dittrich, Oriti, Bahr, Ben Geloun, Steinhaus,…], Tensor models [Gurau, Benedetti,…]

26

3. Why could LQG be right?

27

Classical limit: – covariant formulation ---> Regge gravity– canonical formulation ---> coherent states describing approx. classical metrics

Cosmology:– resolution of singularities– consistent picture of inflationary phase of universe

3d gravity:- canonical and covariant picture equivalent- standard picture obtained

Black holes: – Entropy for large class of black holes

28

4. Black holes in LQG

29

Isolated horizons:

Quasilocal notion of BH horizon, strong enough for BH thermodynamics.

Boundary condition at horizon

Symplectic structure aquires CS boundary term.

Quantum theory:

LQG in bulk, SU(2) CS on boundary. Spin nets can puncture

horizon and endow it with area.

Punctures = CS particles.

30

1/2

1/25/2

3/2

2

1/2

1/2 1

[Ashtekar et al, Engle+Noui+Perez]

[Ashtekar+Baez+Krasnov, many others]

F(

I(A�) =(1� �2)⇡

aH⌃(

I

H(j1,j2...) = HCS(j1, j2 . . .)⌦Hbulk

Entropy:

For given puncture structure

Infinite dimensional. For entropy ignore bulk part.

Count: All sequences j1, j2, … such that

with multiplicity approx

Result:

reproduces Bekenstein-Hawking for correct choice of β.Similar results for more general types of BH with same choice of β.

31

HaH=

M

4⇡�P

i

pji(ji+1)⇡aH

H(j1,j2...)

4⇡�X

i

pji(ji + 1) aH

dim Inv(j1 ⌦ j2 ⌦ . . .)

lnN(a) =e�

2⇡�

aH4l2P

� 3

2ln(aH/l2P ) +O(a0H)

Intrinsic description?

Quantum boundary condition: Want

But F not well defined in LQG. Thus exponentiate, using non-abelian Stokes’ theorem:

Then demand quantum boundary condition:

Can WS be defined? Does the equation have solutions?

32

bF(

I(A�) =(1� �2)⇡

aHb⌃(

I

h@S [A] = P exp�ZZ

SF [A] d

2s W S := P exp�ZZ

SE [A] d

2s

tr(h@S) = tr(WS)

F = hFh�1[A] E = c h(⇤E)h�1

Non-Abelian Stokes theorem (I. Ya. Aref'eva 1980):

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S

h@S [A] = P exp�ZZ

SF [A]d

2p

ep

p

P exp�ZZ

SF [A] d

2p := P exp�ZZ

Sh�1ep F (p)hep d

2p

= I+Z

Sh�1ep F (p)hep d

2p+

ZZ

p,p02S2:pp0

h�1ep F (p)heph

�1e0p

F (p)he0p d2pd2p0 + . . .

http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=18687

Key object:

Can we make it well defined?

First step: LQG E is operator (matrix) valued distribution, factorizes:

S

[HS+Thiemann]

hsW S :=P exp�

ZZ

SE [A,E](s) d2s

= I2 + 8⇡ic

Z

SAdhs(⇤ bE(s))

+ 8⇡ic

Z

S2

Ks,s0Adhs(⇤ bE(s))Adhs0 (⇤ bE(s0))

+ . . .

34

bEaI (s) = bEa(s) bEI(s) : [ bEI(s), bEJ(s)] = ✏IJ

K bEK(s)

bEa(s)he[A] = ea(s)he[A], ea(s) =

Zdt ea(t)�3(s, e(t)),

Two problems:

1) Delta functions at integration boundaries. Solution: Standard procedure gives factor 1/n!

2) Ordering problem: How to order the EI ? Solution: Harish-Chandra/Duflo isomorphism

earlier suggested in somewhat different context [Alekseev et al, Freidel]

35

Harish-Chandra/Duflo mapGiven semisimple Lie algebra .

Quantization map

which is an Isomorphism

g

⌥ : Sym(g) �! U(g)

SymG(g) ! Z(U(g))

⌥ = � � j 12 (@)

j

12 (x) = det

12

✓sinh 1

2adx12adx

◆= 1 +

1

48kxk2 + . . .

⌥(kEk2) = �SU(2) +1

8I

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It’s a refinement of symmetric quantization (PBW)

where

Example: For SU(2)

Kirillov-Kostant brackets

[TI , TJ ] = 0, {TI , TJ} = fKIJTK

[TI , TJ ] = fKIJTK

@ITJ = �IJ

�

This makes WS well defined (albeit hard to determine explicitly)

General properties:For suitably chosen path systems

Under gauge transformations

For SU(2):

S

b

WS1+S2 = WS1WS2 , W †S = W�S

Ug WS U�1g = g(b)WSg(b)

�1

37

j

j’

S

trj (WS) |j0,mi = sin [⇡c(2j + 1)(2j0 + 1)]

sin [⇡c(2j0 + 1)]| {z }=:�jj0

|j0,mi j, j0 6= 0

WS |0, 0i = I |0, 0i

This is subject to quantization ambiguity

Note: Eigenvalues can be written in terms of quantum integers

They are related to

– Verlinde coefficients of SU(2)k rational CFT (k=1/c)

– Trace of the square of the R-matrix of Uq(su(2)) on j O j’

and are precisely what to expect for holonomy around particle in SU(2) CS [Witten]

x

38

�

j,j

0 =[(2j + 1)(2j0 + 1)]

q

[2j0 + 1]q

[x]q

:=q

x � q

�x

q � q

�1q = e

⇡ic

Back to black holes

Quantum boundary condition: For any S in H and

If we had a solution

• formula for simple loops:

• inv under small diffeos fixing punctures: • reps on H only different mod k• nontrivial monodromy of punctures• some fluxes transversal to surface well defined• must be in new rep of HF algebra

S

j

H

tr(h@S) = tr(WS)

W (j)↵ =

(�jji ↵ around pi ↵ trivial

W� = W�(�)

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c = 1/k = �⇡�(1� �2)`2P/2aH

Question: Do we have a representation? (WIP)

Have a good representation for the bulk and simple loops in H. Extension to non- simple loops? Difficult to answer due to

1) Action of WS Complicated:

2)There are many Mandelstam identities to be satisfied. Checked some things, ex.

but not all.

Further question: Do we have SU(2) CS? (WIP)

– DOF remaining on horizon point to ISU(2) with particles– would be nice: 3d Euclidean quantum gravity with particles on horizon– Boundary conditions do not seem to fit constraints of ISU(2)-CS 100%

S

j1

j2

8

I > I k>WS =X

k

ck

40

tr 12(WS1) tr 1

2(WS2) |j1j2i= tr 1

2(WS1+S2) tr 1

2(WS1W�S2) |j1j2i

Jones polynomial

Invariant of oriented knots. Usually defined via skein relations:

Examples:

\ 41

L+ L0 L-

J(�) =q � q�1

q1/2 � q�1/2= q1/2 + q�1/2

J(H+) = 1 + q�1 + q�2 + q�3

�qJ(L+) + (q12 � q�

12 )J(L0) + q�1J(L�) = 0

H+

Jones polynomial and SU(2)

Witten: Jones polynomial

is expectation value of j=1/2 trace of holonomies in SU(2) CS

To regularize the integral: Introduce framing. Standard framing --> Jones polynomial.

Elegant proof via CQFT.

\ 42

J(↵) = htr(h↵)iCS =

Z

Atr(h@S )[A]eiSCS[A] dµ[A]

SCS =k

4⇡

Z

Mtr

✓A ^ dA+

2

3A ^A ^A

◆Quantum Field Theory and the Jones Polynomial 377

~ M

b M L M R

ML

@ MR

@x Fig. 7a-c. A link C on a general three manifold M is sketched in a. A small sphere S has been drawn about an inconvenient crossing; it cuts M into a simple piece (the interior of S) and a complicated piece. In b, the picture is rearranged to exhibit the cutting of M more explicitly; the two pieces now appear on the left and right as ML (the complicated piece whose details are not drawn) and MR (the interior of S). The key to the skein relation is to consider replacing M R with some substitutes, as shown in c

last section, the physical Hilbert spaces Nd~L and NgR associated with the boundaries of ML and MR are two dimensional.

The strategy is now the same as the strategy which led to the multiplicativity relation (4.1). The Feynman path integral on ML determines a vector X in YgL" The Feynman path integral on MR determines a vector ~, in ~'~R. The vector spaces ~ z and ;/gR (which are associated with the same Riemann surface S 2 with opposite orientation) are canonically dual, and the partition function or Feynman path integral Z (L) is equal to the natural pairing

Z(L) = (Z, ~ ) . (4.8)

We cannot evaluate (4.8), since we know neither;( nor ~u. The one thing that we do know, at present, is that (for the groups and representations we are considering) this pairing is occurring in a two dimensional vector space. A two dimensional vector space has the marvelous property that any three vectors obey a relation of

from Witten 1989

Jones polynomial from WS

Using

can replace holonomies under the CS path integral by WS, obtaining relations among expectation values:

Enough to define functional in some cases. Note: Choice of S introduces framing.

\

[HS+Thiemann]

�8⇡i

k✏abc

�

�AceiSCS[A] = Fabe

iSCS[A]

hL tr⇢(h@S )i =Z

AL[A] tr⇢(h@S )[A]eiSCS[A] dµ[A]

=

Z

AL[A] tr⇢(WS)e

iSCS[A] dµ[A]

=

Z

A(tr⇢(W�S)L)[A]eiSCS[A] dµ[A]

= h(tr⇢(W�S)L)i.

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Using Seifert surfaces for the WS one can calculate some expectation values. For example for the Hopf link:

reproducing the known values for Kauffman bracket and Jones polynomial and their generalization.

As far as we know: First QFT calculation of such without using CFT

Nontrivial knots?

hH+(j1, j2)iCS = �j1j2�j20 =sin

⇥⇡k (2j1 + 1)(2j2 + 1)

⇤

sin⇥⇡k

⇤

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α1 α2j2

Seifert surfaces are not simply connected so Stokes does not work.

Contraction discs: Simply connected but have selfintersections. Definition of WS?

Skein relations: Idea

Need: Quantization of the exponential map

Surprisingly hard to determine. We have a result (Sahlmann+Zilker arXiv) but we do not like it.

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5 Application to quantum Chern Simons theory and black holes

In [4, 5], the Duflo map was used to calculate certain expectation values in CS theory. Inthat application, the expectation values were calculated in a piecemeal fashion, turningone loop ˆS into an operator WS at a time, and calculating its action under the pathintegral. Now that we have extensions of the Duflo map at our disposal, we can aim forskein relations among the expectation values. The argument goes as follows.

We consider the path integral expectation value of the traces of holonomies along thecomponents of a link L:

ÈFL

ÍCS =ˆ

Aexp(iS

CS

[A]) F

L

[A] dµ

AL

[A]. (31)

Consider two holonomy strands passing each other as in figure 1(i). As the expectationvalue does not depend on smooth deformations of L, we can deform the one strand inthe manner shown in fig. 1(ii).1 By applying the non-abelian Stokes theorem (for detailssee [4]), we can replace the curved section of the deformed strand by a certain orderedexponential integral I

S

of the curvature of A over a surface S bounded by the curvedsection,

(i) (ii) (iii) (iv)

S

Figure 1: Manipulation of a crossing of two holonomy strands, using the operators W

S

ÈFL

ÍCS =ˆ

Aexp(iS

CS

[A]) ( ÂF

L

)I

J

[A] (IS

)J

I

[A] dµ

AL

[A], (32)

see also (iii). ( ÂF

L

)I

J

is obtained from the original functional by removing the holonomyalong ˆS. In the next step, I

S

can be replaced by a functional di�erential operator actingon the action term. For the action

1Strictly speaking, the deformation depicted in (ii) is smooth only so long as the circle around the

other holonomy strand does not get closed completely. If it is not completely closed, however, the

replacement in step (iii) (see below) is only an approximation. This approximation can be made

arbitrarily good, classically, and we will assume in the following that this is also true in the quantum

theory.

12

exp3

≠8fii

k

Ÿ

ij

E

i

T

j

4= cos

3 4fiÔ2k

||E||4

12 ≠ 8fii

k

sin1

4fiÔ2k

||E||2

4fiÔ2k

||E||Ÿ

ij

E

i

T

j

=Œÿ

m=0

(≠1)m

14fiÔ2k

22m

(2m)! ||E||2m12 ≠ 8fii

k

Œÿ

m=0

(≠1)m

14fiÔ2k

22m

(2m + 1)! ||E||2m

Ÿ

ij

E

i

T

j

.

(25)

We will now consider the application of the di�erent quantisation maps and for furthercomparison we also express the results in the basis used in [7].

For Q = Q

D

we obtain

Q

D

5exp

3≠8fii

k

Ÿ

ij

E

i

(Tj

)A

D

46C

B

= cos32fi

k

4”

A

D

”

C

B

+ 4i

3

S

Usin32fi

k

4≠

1 ≠ cos1

2fi

k

2

2fi

k

T

Vÿ

i

(Ti

)A

D

(Ti

)C

B

=

S

Ucos32fi

k

4≠ i

3 sin3

fi

k

4+ i

31 ≠ cos

12fi

k

2

2fi

k

T

V”

A

B

”

C

D

≠

S

Ucos32fi

k

4+ i

3 sin32fi

k

4≠ i

31 ≠ cos

12fi

k

2

2fi

k

T

V‘

AC

‘

BD

(26)

In the case Q = Q

D

we have

Q

D

5exp

3≠8fii

k

Ÿ

ij

E

i

(Tj

)A

D

46C

B

= cos32fi

k

4”

A

D

”

C

B

+ 2i sin32fi

k

4 ÿ

i

(Ti

)A

D

(Ti

)C

B

=5cos

32fi

k

4≠ i

2 sin32fi

k

46”

A

B

”

C

D

≠5cos

32fi

k

4+ i

2 sin32fi

k

46‘

AC

‘

BD

.

(27)

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To take home:• Quantum field theory without background metric• Canonical and covariant approach• space-time geometry from combinatorics, representation theory

Understanding beginning to emerge about

• quantum geometry• black holes, cosmology

Many open questions

• interpretation of transition amplitudes/solutions of constraints• divergencies in the spinfoam sum• …

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3d gravitysuccessfully approached with canonical and covariant LQG methods.

Euclidean works best but Lorentzian also possible.

Spin foam picture:

– Divergencies understood (twisted betty number)– SU(2) --> SUq(2) gives cosmological constant,

– Equivalence to CS treatment can be shown – Inclusion of particle “Feynman diagrams” gives effective NC field theory

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j2

j 4j 1

j3 j5j6

[Ponzano+Regge `68, many others later]

[Smerlak+Bonzom ‘10 ]

[Turaev+Viro, Mizoguchi +Tada]

[Freidel+Louapre]

[Freidel+Livine]

Z =X

{f}!j

Y

f

(2jf + 1)Y

v

Se↵ =

Zd3x

(@� ? @�)(x)� 1

2

sin2 m

2(� ? �)(x) +

�

3!(� ? � ? �)(x)

�

Canonical approach:

Hamilton constraint imposes flatness.

Can recover spin foam picture. In particular, for physical inner product

Connections to other canonical approaches can be made.

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j2

j 4j 1

j3 j5j6

from Perez 06

[Perez+Noui]

C =Y

↵

�(Wj(↵))

Loop quantum cosmologySymmetric sector of gravity as testbed for LQG

– What happens at the big bang singularity?– Can GR be reproduced far away from the singularity?

Some aspects derivable from full theory.

Example: FRW with massless scalar

Canonical variables

Quantization in terms of – holonomies– Fluxes

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[Bojowald,…]

[Ashtekar+Pawlowski+Singh]

M = R⇥ R3, ds2 = dt2 � a

2(t)(dx2 + dy2 + dz2)

AIa = c !I

a, EaI = p

pq0

0eaI

b=exp(ilc)

b= p

Ashtekar, Pawlowski and Singh, Phys. Rev. Lett. 96 (2006)

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From groups and knots to BH entropy

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