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(δψ[σ])/(δσ(x))=V(x,σ)ψ[σ] (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 LI, i.e. 1/2Nμ(?LI)/(?φμα) GαβNν(?LI)/(?φνβ), (2) where φ1,φ2, …are the various field quantities, φμα denotes (?φα)/?xμ,Nμ denotes the surface normal and Gαβ are quantities defined by NμNν(?2L)/(?φμα?φνβ)Gβγ=δαγ. For the special case in which there is only one φ, the condition for V not to contain σ explicitly reduces to (?LI)/(?φμ)(?LI)/(?φν)/(?LI)/(?φρ)(?LI)/(?φρ)=(?2L)/(?φμ?φν)/(?2L)/(?φρ?φρ).