Kiefer C. Quantum gravity

CANONICAL GRAVITY WITH CONNECTIONS AND LOOPS 129
chosen β = −1 for the Lorentzian case (which does not seem to have a special
geometrical significance), so the Hamiltonian constraint reads
˜H ⊥ = ɛijk Ei aEb j
2 √ (F abk − 2R abk ) ≈ 0 . (4.129)
h
An alternative form using (4.124) is
˜H ⊥ = 1 √
h
tr ( (F ab − 2R ab )[E a ,E b ] ) . (4.130)
The constraint algebra (3.90)–(3.92) remains practically unchanged, but one
should keep in mind that the constraints have been modified by a term proportional
to the Gauss constraints G i ; cf. also (4.28). In addition, one has, of
course, the relation for the generators of SO(3),
{G i (x), G j (y)} = ɛ k
ij G k(x)δ(x, y) . (4.131)
Following Thiemann (1996), we shall now rewrite the Hamiltonian constraint in
a way that will turn out to be very useful in the quantum theory (Section 6.3).
This will be achieved by expressing the Hamiltonian through Poisson brackets
involving geometric quantities (area and volume). Consider for this purpose first
the ‘Euclidean part’ 24 of ˜H ⊥ ,
H E = tr(F ab[E a ,E b ])
√
h
. (4.132)
(As we have discussed above, for β = i only this term remains.) Recalling (4.124),
one finds
[E a ,E b ] i = − √ hɛ abc e i c . (4.133)
Here, use of the ‘determinant formula’
(dete d i )ɛabc = e a i eb j ec k ɛijk
has been made. From the expression for the volume,
∫
V = d 3 x √ ∫ √
h = d 3 x |detEi a| , (4.134)
one gets 2δV/δEi c(x) =ei c (x) and therefore
Σ
Σ
[E a ,E b ] i
√
h
abc δV
= −2ɛ
δEi
c
= −2 ɛabc
8πβ {Ai c ,V} . (4.135)
24 The name stems from the fact that for β = 1 and the Euclidean signature σ =1,thisis
already the full Hamiltonian ˜H ⊥ .