Kiefer C. Quantum gravity

112 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
p N ≡ ∂Lg
∂Ṅ =0,
pg a ≡ ∂Lg =0. (4.62)
∂Ṅ a
Because lapse function and shift vector are only Lagrange multipliers (similar
to A 0 in electrodynamics), these are constraints (called ‘primary constraints’
according to Dirac (1964), since they do not involve the dynamical equations).
Second,
p ab ≡ ∂Lg
∂ḣab
= 1
16πG Gabcd K cd =
√
h (
K ab − Kh ab) . (4.63)
16πG
Note that the gravitational constant G appears here explicitly, although no coupling
to matter is involved. This is the reason why it will appear in vacuum
quantum gravity; see Section 5.2. One therefore has the Poisson-bracket relation
13 {h ab (x),p cd (y)} = δ(a c δd b) δ(x, y) . (4.64)
Recalling (4.48) and taking the trace of (4.63), one can express the velocities in
terms of the momenta,
ḣ ab = 32πGN √
(p ab − 1 )
h 2 ph ab + D a N b + D b N a , (4.65)
where p ≡ p ab h ab . One can now calculate the canonical Hamiltonian density
H g = p ab ḣ ab −L g ,
for which one gets the expression 14
√
h(
H g =16πGNG abcd p ab p cd (3) R − 2Λ)
− N
− 2N b (D a p ab ) . (4.66)
16πG
The full Hamiltonian is found by integration,
∫
∫
H g ≡ d 3 x H g ≡ d 3 x (NH g ⊥ + N a Ha) g . (4.67)
The action (4.61) can be written in the form
∫ (
)
16πG S EH = dtd 3 x p ab ḣ ab − NH g ⊥ − N a Ha
g . (4.68)
Variation with respect to the Lagrange multipliers N and N a yields the constraints
15
13 This is formal at this stage since it does not take into account that √ h>0.
14 This holds modulo a total divergence which does not contribute in the integral because Σ
is compact.
15 These also follow from the preservation of the primary constraints, {p N ,H g } = 0 =
{p g a,H g }.