Feynman Path Integral Formulation

112 4 Hamiltonian and Wheeler-DeWitt Equation
while Eq. (4.50) gives the evolution equation for the intrinsic metric,
∂ t g ij (x,t) = { g ij (x,t),H N + H N
}
. (4.53)
Here {A,B} ≡∑ r (∂A/∂q r ∂B/∂ p r − ∂A/∂ p r ∂B/∂q r ) is the classical Poisson
bracket. Its use is motivated here by the fact that the transition from classical to
quantum mechanics can be affected easily by promoting the Poisson bracket to a
quantum commutator,
{H,O} →
i¯h 1 [Ĥ,Ô] , (4.54)
thus leading in a natural and simple way from Eqs. (4.52) and (4.53) to the Heisenberg
equations of motions for the canonically conjugate quantum operators ĝ ij and
ˆπ ij . The quantities H N and H N are given by
∫
∫
H N ≡ d 3 xN(x)H(x) , H N ≡ d 3 xN i (x)H i (x) . (4.55)
In this notation, the Einstein field equations in the absence of sources G μν = 0are
equivalent to the initial value constraint
H(x) =H i (x) =0 , (4.56)
supplemented by the canonical evolution equations of Eqs. (4.52) and (4.53), with
G ij = 0. The quantity
∫
H = d 3 x [ N(x)H(x)+N i (x)H i (x) ] , (4.57)
should then be regarded as the Hamiltonian for classical general relativity. Then
Eq. (4.56) is equivalent to
H = 0 , (4.58)
for all N and N i .
4.6 Matter Source Terms
When matter is added to the Einstein-Hilbert Lagrangian,
I[g,φ] = 1 ∫
d 4 x √ g 4 R[g μν (x)] + I φ [g μν ,φ] , (4.59)
16πG
where φ(x) are some matter fields, it is easy to see that the action within the ADM
parametrization of the metric coordinates needs to be modified to
∫
I[g,π,φ,π φ ,N] = dt d 3 x ( 1
16πG πij ∂ t g ij + π φ ∂ t φ − NT− N i )
T i . (4.60)