Null surfaces, initial values, and evolution operators for spinor fields

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. We study, in particular, the intersecting pair of characteristics t=x and t=−x (and supress the y and z dependence). In this case the values of separate components of the spinor function on the intersecting pair of null surfaces comprise the necessary initial value data. We present an expression for the construction of a solution from the null surface data; two analogs of the quantum mechanical Hamiltonian operator determine the evolution of the system.