Kiefer C. Quantum gravity

where here
PARTICLE SYSTEMS 79
H S ≡ η µν p µ p ν + m 2 ≈ 0 (3.27)
plays the role of the Super-Hamiltonian which is constrained to vanish. The interpretation
of the Lagrange multiplier N can be gained from Hamilton’s equations,
ẋ 0 = ∂(NH S)
∂p 0
= −2Np 0 ,
to give
ẋ 0
N =
2 √ p 2 + m = ẋ0
2 2mγ = 1 ds
2m dτ , (3.28)
where γ is the standard relativistic factor. In contrast to (3.11), the lapse function
N here is proportional to the rate of change of proper time (not x 0 ) with respect
to parameter time. 2
If we apply Dirac’s quantization rule on the classical constraint (3.27), we get
Ĥ S ψ(x µ ) ≡ ( − 2 ✷ + m 2) ψ(x µ )=0. (3.29)
This is the Klein–Gordon equation for relativistic one-particle quantum mechanics
(spinless particles). We emphasize that the classical parameter τ has completely
disappeared since particle trajectories do not exist in quantum theory.
With regard to the Hamiltonian action (3.26), the question arises how x, p,
and N must transform under time reparametrizations in order to leave the action
invariant; cf. Teitelboim (1982). Since the first-class constraint H S generates
gauge transformations in the sense of (3.20), one has (neglecting the space–time
indices)
δx(τ) =ɛ(τ){x, H S } = ɛ ∂H S
∂p , (3.30)
δp(τ) =ɛ(τ){p, H S } = −ɛ ∂H S
∂x . (3.31)
(In the present case this yields δx =2ɛp, δp = 0, but we shall keep the formalism
general for the moment.) But how does the Lagrange multiplier N transform?
We calculate for this purpose
τ 1
∫ τ2
δS = dτ (ẋδp + pδẋ − H S δN − NδH S ) .
τ 1
The last term is zero, and partial integration of the second term leads to
∫ τ2
(
δS = dτ −ɛ ∂H ) [
S
∂x ẋ − ɛ∂H S
∂p ṗ − H SδN + pɛ ∂H ] τ2
S
∂p
2 If we had chosen (3.25) instead of (3.27), we would have found N =ẋ 0 .
τ 1
.