Approaches to Quantum Gravity

Algebraic approach to Quantum Gravity II 489etc., using classical integration and calculus, but with the • product in place of theusual product of functions. This assumes that ∫ = ∫ d 4 Xφ( ) and that the quantumdifferentials become classical through φ as is the case for the simplest NCG models(including θ-spacetimes and the 3D quantum double model (24.2)). In the caseof the bicrossproduct spacetime model the quantum integration is indeed definedby the normal ordering φ and we have seen (24.25)–(24.26) that spatial quantumdifferentials indeed relate to the classical ones, but the ∂ 0 direction relates underφ to a finite difference in the imaginary time direction. Hence a noncommutativeaction will not have a usual • form (24.32) but will involve finite differences for∂ 0 . One also has the problem that the quantum calculus and hence the NCG actionis not necessarily λ-Poincaré covariant (even though the spacetime itself is), thereis an anomaly for the Poincaré group at the differential level. One can replace thecalculus by a 5D covariant one but then one has to interpret this extra direction.We expect it (see below) to relate to the renormalization-group flow in the QFT onthe spacetime. Again the physics of these issues remains fully to be explored at thetime of writing.24.6 Other noncommutative spacetime modelsThe 4D bicrossproduct model is the simplest noncommutative spacetime modelthat could be a deformation of our own world with its correct signature. There areless developed models and we outline them here.We start with (24.2) for which U λ (poinc 2,1 ) = U(so 2,1 )⊲⊳SO 2,1 and from the general theory we know that it acts on U(so 2,1 )as a 3D noncommutative spacetime. Its Euclideanised version U(su 2 ) is the algebra(24.2) proposed for 3D Quantum Gravity in [2]. For the plane waves, we usethe canonical formψk ⃗ = eık·x , |⃗k| < π λin terms of the local ‘logarithmic’ coordinates as in Section 24.2. The compositionlaw for plane waves is the SU 2 product in these coordinates (given by the CBHformula) and we have a quantum Fourier transform (24.5) with e i = x i in thepresent application. We also have [2]:dx i = λσ i ,x i − x i = ı λ2μ dx i,(dx i )x j − x j dx i = ıλɛ ij k dx k + ıμδ ij ,where is the 2 × 2 identity matrix which, together with the Pauli matrices σ icompletes the basis of left-invariant 1-forms. The 1-form provides a natural time