Approaches to Quantum Gravity

490 S. Majiddirection, even though there is no time coordinate, and the new parameter μ ̸= 0appears as the freedom to change its normalisation. The partial derivatives ∂ i aredefined byand act diagonally on plane waves ask idψ(x) = (∂ i ψ)dx i + (∂ 0 ψ)∂ i = ıλ|⃗k| sin(λ| ⃗k|), ∂ 0 = ı μ λ (cos(λ| ⃗k|) − 1) = ı μ ∂2 ⃗2 + O(λ 2 ).Finally, there is a classicalisation map [4]φ(ψk ⃗ (x)) = X μ eıpμ , p 0 = cos(λ|⃗k|), p i = sin(λ| ⃗k|)k i .λ|⃗k|One can also label the noncommutative plane waves directly by p μ as we did forthe model (24.1). The map φ reproduces (24.2) by its • product and commuteswith ∂ i (but not ∂ 0 ), which means that actions such as (24.32) proposed in [3] asan effective theory for 3D Quantum Gravity essentially coincide with the NCGeffective actions such as (24.31)asin[2]. Here ∫ = ∑ j∈N ( j + 1)Tr j is the sum oftraces in the spin j/2 representation. The noncommutative action has an extra terminvolving ∂ 0 , which can be suppressed only by assuming that the 4D Hodge ∗-operator is degenerate. Moreover, the map φ sees only the integer spin informationin the model which is not the full NCG, see [4].Note that μ cannot be taken to be zero due to an anomaly for translation invarianceof the DGA. This anomaly forces an extra dimension much as we saw for(24.1) before. The physical meaning of this extra direction ∂ 0 from the point ofview of Euclidanized 3D Quantum Gravity is as a renormalization group flowdirection associated to blocking of the spins in the Ponzano–Regge model [4].Alternatively, one can imagine this noncommuative spacetime arising in othernonrelativistic limits of a 4D theory, with the extra ‘time’ direction x 0 adjoinedby [21] = dx 0 , [x 0 , x i ]=0, [x 0 , dx i ]=ı λ2μ dx i,[x 0 ,]=ı λ2μ and new partial derivatives ∂ μ on the extended algebra. Then the ‘stationary’condition in the new theory is dψ = O(dx i ) or ∂ 0 ψ = 0, i.e.(√ )ψ(⃗x, x 0 + ı λ2μ ) = 1 + λ 2⃗ ∂ 2 ψ(⃗x, x 0 ) (24.33)