Approaches to Quantum Gravity

296 L. Freidel
is now constrained to be in the conjugacy class of h me , where m e = 4πG ˜m e is the
deficit angle created by the particle of mass ˜m e and h me is the element of the Cartan
subgroup corresponding to the rotation of angle 2m e :
( )
e
im
0
h m ≡
0 e −im .
The deficit angle is related to the parameter M e in the discrete action by
κ M e = sin m e .
Then the corresponding quantum amplitude
∫ ∏ ∏ ∏ ∏ d 3 P e
I (Ɣ) = dX e dg f dλ e
4π e 1 ∑
2
2 e tr(X eG e )−iS PƔ (X e ,P e ,λ e )
e
f
e∈Ɣ
e∈Ɣ
is given by
∫ ∏
I (Ɣ) =
f
dg f
∏
e∈Ɣ
˜K me (G e ) ∏ e /∈Ɣ
δ(G e ). (16.15)
˜K m (g) is a function on SO(3) which is invariant under conjugation and defined in
terms of the momenta 2iκ ⃗P(g) ≡ tr(g⃗σ) given by the projection of g on Pauli
matrices.
˜K m (g) =
iκ 2
(κ 2 P 2 (g) − sin 2 m − iɛ) . (16.16)
Since this is a class function we can expand it in terms of characters. We have the
identity
˜K m (g) = ∑ K m ( j)χ j (g), (16.17)
j∈N
where χ j (h m ) is the character of h m in the j-representation:
and
χ j (h m ) =
sin(2 j + 1)m
,
sin m
K m ( j) = 2iκ2 e id j (m+iɛ)
.
cos m
It is interesting to note that this is essentially the usual Feynman propagator evaluated
on a discrete lattice, that is if k m (x) is the Feynamn propagator solution of
(⊔⊓+m 2 )k m (x) =−iδ(x), then K m( j)
d j
= κ3 k m (κd j )
cos m
4π
.