Approaches to Quantum Gravity

490 S. Majid
direction, even though there is no time coordinate, and the new parameter μ ̸= 0
appears as the freedom to change its normalisation. The partial derivatives ∂ i are
defined by
and act diagonally on plane waves as
k i
dψ(x) = (∂ i ψ)dx i + (∂ 0 ψ)
∂ i = ı
λ|⃗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 for
the model (24.1). The map φ reproduces (24.2) by its • product and commutes
with ∂ i (but not ∂ 0 ), which means that actions such as (24.32) proposed in [3] as
an effective theory for 3D Quantum Gravity essentially coincide with the NCG
effective actions such as (24.31)asin[2]. Here ∫ = ∑ j∈N ( j + 1)Tr j is the sum of
traces in the spin j/2 representation. The noncommutative action has an extra term
involving ∂ 0 , which can be suppressed only by assuming that the 4D Hodge ∗-
operator is degenerate. Moreover, the map φ sees only the integer spin information
in 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 invariance
of 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 of
view of Euclidanized 3D Quantum Gravity is as a renormalization group flow
direction associated to blocking of the spins in the Ponzano–Regge model [4].
Alternatively, one can imagine this noncommuative spacetime arising in other
nonrelativistic limits of a 4D theory, with the extra ‘time’ direction x 0 adjoined
by [21]
= dx 0 , [x 0 , x i ]=0, [x 0 , dx i ]=ı λ2
μ dx i,
[x 0 ,]=ı λ2
μ
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 λ2
μ ) = 1 + λ 2⃗ ∂ 2 ψ(⃗x, x 0 ) (24.33)