Approaches to Quantum Gravity

478 S. Majidwhere we introduce p 0 , p 1 as coordinates on the group M and θ as the coordinateof SO 1,1 .Hereλ is a fixed but arbitrary normalisation constant and we have θ/2because we are working with the double cover of SO 2,1 . According to the group lawof matrix multiplication, the p i viewed abstractly as functions enjoy the coproduct()e λ 2 p0 0=λp 1 e λ 2 p0 e − λ 2 p0()e λ 2 p0 0⊗λp 1 e λ 2 p0 e − λ 2 p0()e λ 2 p0 0λp 1 e λ 2 p0 e − λ 2 p0where matrix multiplication is understood. Thus in summary we have[p 0 , p 1 ]=0, p 0 = p 0 ⊗ 1 + 1 ⊗ p 0 , p 1 = p 1 ⊗ 1 + e −λp0 ⊗ p 1 (24.18)S(p 0 , p 1 ) = (−p 0 , −e λp0 p 1 ) (24.19)as the Hopf algebra C[R>⊳R] corresponding to our nonAbelian momentum groupand its group inversion.We now take group elements in the wrong order and refactorise:( cosh(θ2 ) sinh( θ 2 ) ) ( )e λ 2 p0 0sinh( θ 2 ) cosh( θ 2 ) =λp 1 e λ 2 p0 e − λ 2 p0()(C + Sλp 1 )e λ 2 p0 Se − λ 2 p0(S + Cλp 1 )e λ 2 p0 Ce − λ 2 p0() (cosh(e λ 2=p0′ θ0′λp 1′ e λ 2 p0′ e − λ 2 ) sinh( θ ′2 ) )2 p0′ sinh( θ ′2 ) cosh( θ ′2 )where S = sinh(θ/2), C = cosh(θ/2), which gives according to (24.6):p 0′ = θ⊲p 0 = p 0 + 1 )((Cλ ln + Sλp 1 ) 2 − S 2 e −2λp0 )(24.20)p 1′ = θ⊲p 1 = (C + Sλp1 )(S + Cλp 1 ) − SCe −2λp0λ ( (C + Sλp 1 ) 2 − S 2 e−2λp0) (24.21)()θ ′ = θ⊳(p 0 , p 1 Se −λp0) = 2arcsinh √(C + Sλp1 ) 2 − S 2 e −2λp0 (24.22)where we have written formulae in the domain where C + Sλp 1 > 0. Therefactorisation is possible (so the actions ⊲, ⊳ are well-defined) only when()()C + S(λp 1 − e −λp0 ) C + S(λp 1 + e −λp0 ) > 0. (24.23)This can be analysed in terms of the regions in Figure 24.2, which shows orbitsunder ⊲ in (p 0 , p 1 ) space. One can check from the expressions above that theseorbits are lines of constant values of||p|| 2 λ = (p1 ) 2 e λp0 − 2 (cosh(λp 0 ) − 1 ) (= eλp0λ 2 (p 1 ) 2 − (1 − e −λp0 ) 2)λ 2 λ 2(24.24)