Null Surfaces, Initial Values and Evolution Operators for Spinor Fields

TL;DR

This paper investigates the initial value problem for Dirac spinor fields on null surfaces, proposing a new formulation using null surface data and deriving evolution operators analogous to quantum Hamiltonians.

Contribution

It introduces a novel initial value formulation for Dirac fields on characteristic null surfaces, differing from the standard Cauchy problem, with explicit solution construction and evolution operators.

Findings

Solution can be constructed from null surface data

Two Hamiltonian analogues govern evolution

Addresses discontinuities on characteristic surfaces

Abstract

We analyze the initial value problem for spinor fields obeying the Dirac equation, with particular attention to the characteristic surfaces. The standard Cauchy initial value problem for first order differential equations is to construct a solution function in a neighborhood of space and time from the values of the function on a selected initial value surface. On the characteristic surfaces the solution function may be discontinuous, so the standard Cauchy construction breaks down. For the Dirac equation the characteristic surfaces are null surfaces. An alternative version of the initial value problem may be formulated using null surfaces; the initial value data needed differs from that of the standard Cauchy problem, and in the case we here discuss the values of separate components of the spinor function on an intersecting pair of null surfaces comprise the necessary initial value data. We 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.