Null Surfaces, Initial Values and Evolution Operators for Scalar Fields

TL;DR

This paper investigates the initial value problem for scalar fields on null surfaces, proposing an alternative data formulation and solution construction using null surface data and evolution operators, especially for the Klein-Gordon equation.

Contribution

It introduces a novel initial data framework on null surfaces and derives solution expressions and evolution operators for scalar fields, extending traditional Cauchy problem methods.

Findings

Null surface data suffices for initial value problems on characteristic surfaces.

Derived explicit solution construction from null data for Klein-Gordon fields.

Presented evolution operators analogous to quantum Hamiltonians for scalar fields.

Abstract

We analyze the initial value problem for scalar fields obeying the Klein-Gordon equation. The standard Cauchy initial value problem for second order differential equation is to construct a solution function in a neighborhood of space and time form values of the function and its time derivative on a selected initial value surface. On the characteristic surfaces the time derivative of the solution function may be discontinuous, so the standard Cauchy construction breaks down. For the Klein-Gordon equation the characteristic surfaces are null surfaces. An alternative version of the initial data needed differs from that of the standard Cauchy problem, and in the case we discuss here the values of the function on an intersecting pair of null surfaces comprise the necessary initial value data. We also present an expression for the construction of a solution from null surface data; two analogues of the quantum mechanical Hamiltonian operator determine the evolution of the system.