Quantum Gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 149where one additional renormalization constant is needed in comparison to theFock-space formulation; see Symanzik (1981). 9The simplest example for the Schrödinger picture is the free bosonic field.The implementation of the commutation relationsleads to[ ˆφ(x), ˆp φ (y)] = iδ(x − y) (5.35)ˆφ(x)Ψ[φ(x)] = φ(x)Ψ[φ(x)] , (5.36)ˆp φ Ψ[φ(x)] = δΨ[φ(x)] ,i δφ(x)(5.37)where Ψ[φ(x)] is a wave functional on the space of all fields φ(x), which includesnot only smooth classical configurations, but also distributional ones. The Hamiltonoperator for a free massive scalar field reads (from now on again =1)∫ (Ĥ =1/2 d 3 x ˆp 2 φ (x)+ ˆφ(x)(−∇ 2 + m 2 ) ˆφ(x))∫∫≡ 1/2 d 3 x ˆp 2 φ (x)+1 d 3 xd 3 x ′ ˆφ(x)ω 2 (x, x ′ )2ˆφ(x ′ ) , (5.38)whereω 2 (x, x ′ ) ≡ (−∇ 2 + m 2 )δ(x − x ′ ) (5.39)is not diagonal in three-dimensional space, but is diagonal in momentum space(due to translation invariance),∫ω 2 (p, p ′ ) ≡ d 3 p ′′ ω(p, p ′′ )ω(p ′′ , p ′ )= 1 ∫(2π) 3 d 3 xd 3 x ′ e ipx ω 2 (x, x ′ )e −ip′ x ′with p ≡|p|. Therefore,=(p 2 + m 2 )δ(p − p ′ ) , (5.40)ω(p, p ′ )= √ p 2 + m 2 δ(p − p ′ ) ≡ ω(p)δ(p − p ′ ) . (5.41)The stationary Schrödinger equation then reads (we set =1)ĤΨ n [φ] ≡(− 1 ∫2d 3 xδ2δφ 2 + 1 ∫2)d 3 xd 3 x ′ φω 2 φ Ψ n [φ] =E n Ψ n [φ] . (5.42)9 This analysis was generalized by McAvity and Osborn (1993) to quantum field theoryon manifolds with arbitrarily smoothly curved boundaries. Non-Abelian fields are treated, forexample, in Lüscher et al. (1992).