Quantum Gravity

82 PARAMETRIZED AND RELATIONAL SYSTEMSµ =0,...,D− 1. Denoting the derivative with respect to τ ≡ σ 0 by a dot andthe derivative with respect to σ ≡ σ 1 by a prime, one has∂X µ ∂X νG 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. Thecanonical momenta conjugated to them read1[]P µ = −2πα ′√ −detG (ẊX′ )X µ ′ − (X′ ) 2 Ẋ µ . (3.41)αβFrom this one gets the conditionsP µ X µ′ 1[]= −2πα ′√ −detG (ẊX′ )(X ′ ) 2 − (X ′ ) 2 (ẊX′ ) = 0 (3.42)αβas well asP µ P µ = − (X ′ ) 24π 2 (α ′ ) 2 . (3.43)In fact, the last two conditions are just constraints—a consequence of the reparametrizationinvarianceτ ↦→ 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 Hamiltonianis constrained to vanish. For the Hamiltonian density H, one finds thatH = NH ⊥ + N 1 H 1 , (3.44)where N and N 1 are Lagrange multipliers, andH ⊥ = 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 commutationrelations[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 allowedwave functionals,Ĥ ⊥ Ψ[X µ (σ)] ≡ 1 (− 2 δ2 Ψ2 δX 2 + (X ′ ) 2 )Ψ4π 2 (α ′ ) 2 =0, (3.48)