Kiefer C. Quantum gravity

82 PARAMETRIZED AND RELATIONAL SYSTEMS
µ =0,...,D− 1. Denoting the derivative with respect to τ ≡ σ 0 by a dot and
the derivative with respect to σ ≡ σ 1 by a prime, one has
∂X µ ∂X ν
G αβ = η µν
∂σ α ∂σ β , (3.39)
|detG αβ | = −detG αβ =(ẊX′ ) 2 − Ẋ2 (X ′ ) 2 . (3.40)
The embeddings X µ (σ, τ) will play here the role of the dynamical variables. The
canonical momenta conjugated to them read
1
[
]
P µ = −
2πα ′√ −detG (ẊX′ )X µ ′ − (X′ ) 2 Ẋ µ . (3.41)
αβ
From this one gets the conditions
P µ X µ′ 1
[
]
= −
2πα ′√ −detG (ẊX′ )(X ′ ) 2 − (X ′ ) 2 (ẊX′ ) = 0 (3.42)
αβ
as well as
P µ P µ = − (X ′ ) 2
4π 2 (α ′ ) 2 . (3.43)
In fact, the last two conditions are just constraints—a consequence of the reparametrization
invariance
τ ↦→ τ ′ (τ,σ) , σ ↦→ σ ′ (τ,σ) .
The constraint (3.43), in particular, is a direct analogue of (3.24).
As expected from the general considerations in Section 3.1.1, the Hamiltonian
is constrained to vanish. For the Hamiltonian density H, one finds that
H = NH ⊥ + N 1 H 1 , (3.44)
where N and N 1 are Lagrange multipliers, and
H ⊥ = 1 (P 2 + (X′ ) 2 )
2 4π 2 (α ′ ) 2 ≈ 0 , (3.45)
H 1 = P µ X µ′ ≈ 0 . (3.46)
Quantization of these constraints is formally achieved by imposing the commutation
relations
[X µ (σ),P ν (σ ′ )]| τ =τ ′ =i δ ν µ δ(σ − σ ′ ) (3.47)
and implementing the constraints à la Dirac as restrictions on physically allowed
wave functionals,
Ĥ ⊥ Ψ[X µ (σ)] ≡ 1 (
− 2 δ2 Ψ
2 δX 2 + (X ′ ) 2 )
Ψ
4π 2 (α ′ ) 2 =0, (3.48)