Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Spin networks, 6j-symbols and thePonzano-Regge model of quantum gravity

J.W. Barrett

School of Mathematical SciencesUniversity of Nottingham

Coventry, 24 April 2008

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Outline

Introduction

Planar spin networksDefinitions3d triangulations

The Ponzano-Regge modelObservables and regularisationThe model via connectionsFunctional integral

Particle quantizationCoupling 3d gravity to particlesParticles without gravity

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Introduction

• The Ponzano-Regge model is a simple model of 3dquantum gravity.

• Explicit calculations can be carried out.

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Spin networks

Irreducible representations

a, b, . . . ∈ Irrep(SU(2)) ∼= {0,12, 1, . . .}

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a b

c

d

a⊗ b → c ⊗ d

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Standard networks

a id : a → a

a a⊗ a → C

a C → a⊗ a

a = a

a= (−1)2a(2a + 1) = ∆a

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

3j and 6j symbols

3j-symbol ab

c C → a⊗ b ⊗ c

Normalisation ac

b = 1

6j-symbola

b c

f e

d

=

{a b cd e f

}

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Asymptotics of SU(2) 6j-symbol

1959 (Wigner)1968 (Ponzano and Regge){

j1 j2 jJ j3 j ′

}' cos(Einstein action)√

vol

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Network reduction

f

e d

c

a b

=

c

a b

{a b cd e f

}

a b c

e

f

=∑

d

c

d

f

a b

∆d

{a b ec f d

}

TheoremA planar network is a sum of products of 6j -symbols and ∆j .

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

The dual triangulationA closed planar spin network is dual to a triangulation of S2.

TheoremThe network reduction constructs a triangulation of B3. Then∂B3 is dual to the network.

Tetrahedra correspond to 6j-symbols, interior edges to ∆j .

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Ponzano-Regge model for a compact manifold

• Weight factor

W =∏

interior edges

(−1)2j(2j + 1)∏

tetrahedra

{j1 j2 j3j4 j5 j6

}

• State sum – interior edges.

Z =∑j1j2...

W

• Agrees with formula for network reduction.• For other triangulations, sum is infinite and requires

regularisation.

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Infinite sums

Triangulations of B3 not given by network reduction:• With interior vertices• Interior edges dual to Bing’s house

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Physics of the Ponzano-Regge model

• Geometric interpretation – 3d quantum gravity

Length of edge = j +12.

• Euclidean geometries Z '∑

eiSE

• Minkowski geometries Z '∑

e−|SL|

• Analogue of Wheeler-deWitt equation for manifold withboundary.

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Observables in a closed manifold

< f >=∑

spinsf (spins) W

• a1, a2, . . . distances.

f (j1, j2, . . .) =∏

k δ(jk + 12 − ak ) a

a1

2

• θ1, θ2, . . . masses

f (j1, j2, . . .) =∏

k

sin((2jk + 1) θk

2

)(2jk + 1) sin θk

2

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

RegularisationExtend the graph of observables Γ by trees T meeting allvertices. Set these spins to zero.

θθ

1

2

Theorem (with I. Naish-Guzman)The Ponzano-Regge state sum is well-defined and independentof triangulation and regularisation if H2(M \ Γ, ρ) = 0 for everyflat SU(2) connection ρ with holonomy in the conjugacy classesgiven by θ1, θ2, . . ..

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Group variables

Oriented triangle 7→ g ∈ SU(2).

LemmaFor an oriented closed manifold

W =∏

triangles

∫dg

∏edges

(2j + 1)Trj(h).

h = g1g2g3 . . .

gg

g

1

3

2

h

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Partition function by integration

Summing over spins,

Z =∏

triangles

∫dg

∏edges

δ(h).

This still requires regularisation.

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Proof of theorem

With observables Γ andregularisation T , θ

θ1

2

< f > =∏

triangles

∫dg

∏Γ

π

sin2 θ/2δ(θ − c(h)

) ∏Not T ,Γ

δ(h)

=

∫moduli

tor(M \ Γ)π

sin2 θ/2δ(θ − c(h)

)c(h) = conjugacy class of h.

h onto ⇐⇒ H2(M \ Γ, ρ) = 0.

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Examples

θ θθ

1

0

2

θ

Z =π

sin2 (12θ0

) δ(θ0)

Z =1

|A(eiθ)|θ < π/3

A = Alexander Polynomial.

H2 6= 0 if θ > π/3.

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

3d Gravity Functional Integral.1986 (Achucarro and Townsend)

3d gravity ⇐⇒ Chern-Simons.

1989 (Witten) Finite QFT.

Z =

∫eiSE de dω

SE =

∫M

e ∧ (dω + ω ∧ ω) + Λ e ∧ e ∧ e.

Λ > 0 SU(2)× SU(2).

Λ = 0 ISU(2). One loop exact. Flat SU(2) connections weightedwith analytic torsion. No observables

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Quantization principle.

Principle: quantizing gravity automatically includes integrationover space of particle trajectories.

X (g)g*

X

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Feynman Amplitudes.

x jxi

xi ∈ R3, rij = |xi − xj |

I =

∫ ∏edges

G(|xi − xj |)∏

k

dxk =

∫J (rij)

∏edges

G(rij) drij

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Coupling Particles to Quantum Gravity

• Replace kinetic part of Feynman amplitude I with 3dgravity state sum. ∫

dr −→∑

spins

J (rij) −→ Ponzano-Regge observable.

• Fourier transform:

position space ⇐⇒ momentum space

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Gravitational effects.

a b

c

6=a b

c

Introduction Planar spin networks The Ponzano-Regge model Particle quantization

Quantum Flat Space.

• (Baratin and Freidel) Kinetic part of Feynman amplitude I isa new spin foam model. No gravity.

• Model is Ponzano-Regge for the Poincaré group.• Works in 3d or 4d• (JWB) Equivalent functional integral∫

eiS de dω db dc

S =

∫M

b ∧ (dω + ω ∧ ω) + c ∧ (de + ω ∧ e)