Approaches to Quantum Gravity

Algebraic approach to Quantum Gravity II 47724.3.1 Nonlinear factorisation in the 2D bicrossproduct modelSuch models provide noncommutative spacetimes and Poincaré quantum groups inany dimension n based on a local factorisation of SO n,1 or SO n−1,2 . The 4D modelis known [23] but the 2D case has the same essential structure and we shall use thisnow to explore global and nonlinear issues, with full derivations.The first remark in the 2D case is that for a convenient description of the globalpicture we work not with SO 2,1 exactly but its double cover X = SL 2 (R) → SO 2,1 .The map here at the Lie algebra level isã 0 = λ 2( ) 1 0→ λ0 −1⎛ ⎞0 0 1⎝0 0 0⎠ , Ñ = 1 21 0 0ã 1 = λ( ) 0 11 0⎛ ⎞( ) 0 0→ √ λ 0 −1 0⎝1 0 1⎠1 0 20 1 0⎛ ⎞→ √ 1 0 0 0⎝0 0 1⎠20 1 0for xt, yt boosts and xy-rotations with ++−signature being generated by−ıã 0 , √ 2Ñ, ˜M = −ı √ 2(λÑ −ã 1 ) respectively. The ã i close to the Lie algebra[ã 1 , ã 0 ]=λã 1 so generate a 2-dimensional nonAbelian Lie group M = R>⊳Ralong with G = SO 1,1 = R generated by Ñ. This gives a factorisation SL 2 (R) ≈(R>⊳R).SO 1,1 as)) ( ) √1 b( a bc d( aμ 0=ac−bdaμ1aμμbaμaμ1μ; μ =1 − b2a 2 ,|b| < |a|.This is valid in the domain shown which includes the identity in the group. Itcannot be a completely global decomposition because topologically SL 2 (R) andPSL 2 (R) = SO 2,1 have a compact direction and so cannot be described globallyby 3 unbounded parameters (there is a compact SO 2 direction generated by ˜M).If one does not appreciate this and works with unbounded parameters one will atsome stage encounter coordinate singularities, which is the origin of the Planckianbound for this model as well as other new effects (see below). From an alternativeconstructive point of view, as we solve the matched-pair equations for (R>⊳R)⊲⊳Rwe must encounter a singularity due to the nonlinearity. Note that this nonAbelianfactorisation and our construction of it cf. [23] is not the KAN decomposition intothree subgroups.In the factorisation we now change variables to( )aμ = e λ θ2 p0 , ac − bd = λp 1 e λp0 , sinh2= baμ