Kiefer C. Quantum gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 149
where one additional renormalization constant is needed in comparison to the
Fock-space formulation; see Symanzik (1981). 9
The simplest example for the Schrödinger picture is the free bosonic field.
The implementation of the commutation relations
leads 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)] = δ
Ψ[φ(x)] ,
i δφ(x)
(5.37)
where Ψ[φ(x)] is a wave functional on the space of all fields φ(x), which includes
not only smooth classical configurations, but also distributional ones. The Hamilton
operator 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 ′ )
2
ˆφ(x ′ ) , (5.38)
where
ω 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 ∫
2
d 3 x
δ2
δφ 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 theory
on manifolds with arbitrarily smoothly curved boundaries. Non-Abelian fields are treated, for
example, in Lüscher et al. (1992).