j.w. barrett- spin networks, 6j-symbols and the ponzano-regge model of quantum gravity
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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)
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