Approaches to Quantum Gravity

Three-dimensional spin foam Quantum Gravity 301
where X = X i σ i . The group elements can be concretely written as
g = (P 4 + ıκ P i σ i ), P4 2 + κ2 P i P i = 1, P 4 ≥ 0. (16.27)
We restrict ourselves to the “northern hemisphere” of SU(2) P 4 > 0 since this is
enough to label SO(3) elements, and the plane waves are simply E g (X) = e i ⃗P(g)· ⃗X .
We define a non-commutative ⋆-product on R 3 which is defined on plane
waves by
(E g1 ⋆ E g2 )(X) ≡ E g1 g 2
(X). (16.28)
This ⋆-product can be more explicitly written in terms of the momenta as
e ı ⃗P 1· ⃗X ⋆ e ı ⃗P 2· ⃗X = e ı( ⃗P 1 ⊕ ⃗P 2 )· ⃗X , (16.29)
where
⃗P 1 ⊕ ⃗P 2 =
√
√
1 − κ 2 | ⃗P 2 | 2 P ⃗ 1 + 1 − κ 2 | ⃗P 1 | 2 P ⃗ 2 (16.30)
− κ ⃗P 1 × ⃗P 2 , (16.31)
and × is the 3d vector cross product. By linearity this star product can be extended
to any function of R 3 which can be written as a linear combination of plane waves.
It can also be extended to any polynomial function of X by taking derivatives of
E g with respect to P around P = 0. Using this, it can be easily shown that this
star product describes a non-commutative spacetime with the non-commutative
coordinates satisfying
[X i , X j ]=iκɛ ijk X k ,
[X i , P j ]=i √ 1 − κ 2 P 2 δ ij − iκɛ ijk P k . (16.32)
The non-commutativity of the space time is directly related to the fact that
momentum space is curved. Indeed in a quantum mechanics X ∼ i∂ P the coordinate
is a derivation on momentum space, and derivatives of a curved space do
not commute. 3 That a non-commutative spacetime structure arises in the quantization
of 3d gravity was first proposed by ’t Hooft [19], although the details are
different. The existence of plane waves pairing R 3 with SO(3) allows us to develop
a new Fourier transform [4; 18] F : C(SO(3)) → C κ (R 3 ) mapping functions on
the group to functions on R 3 having momenta bounded by 1/κ:
∫
φ(X) =
dg ˜φ(g)e 1
2κ tr(Xg) . (16.33)
3 The left ⋆-multiplication by X is realized as a right invariant derivative on momentum space S 3 .