Approaches to Quantum Gravity

Algebraic approach to Quantum Gravity II 479λp 0 2–21A0BC–1D–2 –1 0 12λp 1Fig. 24.2. Deformed orbits under the Lorentz group in the bicrossproduct modelmomentum group. Increasing θ moves anticlockwise along an orbit in regionsA, D and clockwise in regions B, C.which deforms the Minkowski norm in momentum space. It is also invariant underinversion in the curved momentum group and hence under the antipode S. Note thatthis has nothing to do with the Poincaré algebra which we have not constructed yet;it is part of the nonlinear geometry arising from the factorisation.Theorem 2 (i) The actions ⊲, ⊳ are defined for all θ if and only if (p 0 , p 1 ) lies inthe upper mass shell (region A).(ii) For any other (p 0 , p 1 ) there exists a finite boost θ c that sends p 0 → −∞, afterwhich ⊲ breaks down.(iii) For any θ there exists a critical curve not in region (A) such that approachingit sends θ → ±∞, after which ⊳ breaks down.For the proof we use the shorthand q ≡ e −λp0 . We analyse the situation for thetwo cases S > 0andS < 0; if S = 0 then the condition (24.23) always holds.Doing the first case, to lie in regions A, C means λp 1 + 1 − q ≥ 0. HenceC + S(λp 1 − q) = (C − S) + S(λp 1 + 1 − q) >0which also implies that the other factor in (24.23) is also positive, so the conditionholds. But conversely, strictly inside regions B, D mean that q − λp 1 > 1andC +S(λp 1 −q) = 0 has a solution θ c > 0 according to coth( θ c2) = q −λp 1 . We alsonote that our assumption C + Sλp 1 > 0 holds here and for all smaller θ.Asθ → θ cfrom below, the denominator or argument of log in the actions (24.20)–(24.21) → 0and the transformed p 0′ → −∞. IfS < 0thenλp 1 + q − 1 ≤ 0 in regions A, B