Approaches to Quantum Gravity

478 S. Majid
where we introduce p 0 , p 1 as coordinates on the group M and θ as the coordinate
of SO 1,1 .Hereλ is a fixed but arbitrary normalisation constant and we have θ/2
because we are working with the double cover of SO 2,1 . According to the group law
of 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 0
λp 1 e λ 2 p0 e − λ 2 p0
where 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 group
and its group inversion.
We now take group elements in the wrong order and refactorise:
( cosh(
θ
2 ) sinh( θ 2 ) ) ( )
e λ 2 p0 0
sinh( 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′ θ
0
′
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 )
((C
λ ln + Sλp 1 ) 2 − S 2 e −2λp0 )
(24.20)
p 1′ = θ⊲p 1 = (C + Sλp1 )(S + Cλp 1 ) − SCe −2λp0
λ ( (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. The
refactorisation 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 orbits
under ⊲ in (p 0 , p 1 ) space. One can check from the expressions above that these
orbits are lines of constant values of
||p|| 2 λ = (p1 ) 2 e λp0 − 2 (
cosh(λp 0 ) − 1 ) (
= eλp0
λ 2 (p 1 ) 2 − (1 − e −λp0 ) 2)
λ 2 λ 2
(24.24)