Kiefer C. Quantum gravity

130 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
This yields for H E the expression
H E = − 1
4πβ ɛabc tr(F ab {A c ,V}) . (4.136)
Thiemann (1996) also considered the integrated trace of the extrinsic curvature,
∫
T ≡ d 3 x √ ∫
hK = d 3 xKaE i i a ,
Σ
for which one gets
{A i a (x),T} =8πβKi a (x) . (4.137)
For H E ≡ ∫ d 3 x H E , one finds, using (4.114),
Σ
{H E ,V} =8πβ 2 GT . (4.138)
One now considers the following sum (written here for general β),
˜H ⊥ + 1 − β2 (σ +1)
β 2 H E = β2 σ − 1 (
F
i
2β 2 |detEi a| ab − Rab
i )
[E a ,E b ] i . (4.139)
The reason for performing this combination is to get rid of the curvature term.
From (4.116) and using (4.119), one can write
R i ab = F i ab + β2 ɛ i jk Kj a Kk b +2βD [bK i a] .
With the help of (4.137) and (4.133), one then finds after some straightforward
manipulations,
˜H ⊥ = − 1 − β2 (σ +1)
β 2 H E + β2 σ − 1
2(4πβ) 3 ɛabc tr ({A a ,T}{A b ,T}{A c ,V}) . (4.140)
This will serve as the starting point for the discussion of the quantum Hamiltonian
constraint in Section 6.3. The advantage of this formulation is that ˜H ⊥ is
fully expressed through Poisson brackets with geometric operators.
4.3.3 Loop variables
An alternative formulation that is closely related to the variables discussed in
the last subsections employs so-called ‘loop variables’ introduced by Rovelli and
Smolin (1990). This is presently the most frequently used formulation in the
quantum theory (Chapter 6). Consider for this purpose a closed loop on Σ, that
is, a continuous piecewise analytic map from the interval [0, 1] to Σ,
α :[0, 1] → Σ , s ↦→ {α a (s)} . (4.141)
The holonomy U[A, α] corresponding to A a = A i a τ i along the curve α is given by