作用表示波动方程中与面有关项

这篇短文的内容是:(i)对於量子场论中的i(δψσ])/(δσ(x))=V(x,σ)ψσ] 如何由寻常的“曲面上的薛定谔方程”导出,作一个较严格的讨论,以及 (ii)讨论上式中的V(x,σ)在什么条件下不包含有σ。我们证明了所需的条件是 (?L^I)/(?φ_μ) (?L^I)/(?φ_ν)=(?²L)/(?φ_μ?φ_ν)F(φ,φ_ρ),式中L,L^I代表总拉格朗日及作用拉格朗日,φ代表场量,φ_μ代表φ/x^μ,F(φ,φ_ρ)代表φ及φ_μ的一个任意函数。

NORMAL-DEPENDENT TERMS IN WAVE EQUATIONS IN INTERACTION REPRESENTATION

The aim of this short paper is:( i ) To provide a rigorous proof for the deduction of the well-known equation i(δψσ])/(δσ(x))=V(x,σ)ψσ] (1) from the formulation of ordinary Schroedinger wave equations for wave functions on arbitrary space-like surfaces as given by Weiss and the author, and (ii) To study under what conditions the operator V in the above equation does not contain σ explicitly.To prove (1) from Weiss's theory, all that is necessary is to transform away the free Hamiltonian in the usual way and to prove that, according to the resulting wave equation, the difference of ψ on two adjacent surfaces which are different only in a small neighbourhood of σ certain point on the surfaces is proportional to the volume included between the surfaces. This is actually achieved. The terms in V which depend explicitly on a are worked out in terms of the total Lagrangian L and the interaction Lagrangian L^I, i.e. 1/2N_μ(?L^I)/(?φ_μ^α) G^αβN_ν(?L^I)/(?φ_ν^β), (2) where φ¹,φ², …are the various field quantities, φ_μ^α denotes (?φ^α)/?x^μ,N_μ denotes the surface normal and G^αβ are quantities defined by N_μN_ν(?²L)/(?φ_μ^α?φ_ν^β)G^βγ=δ_α^γ. For the special case in which there is only one φ, the condition for V not to contain σ explicitly reduces to (?L^I)/(?φ_μ)(?L^I)/(?φ_ν)/(?L^I)/(?φ_ρ)(?L^I)/(?φ^ρ)=(?²L)/(?φ_μ?φ_ν)/(?²L)/(?φ^ρ?φ_ρ).