IVE1. Example of an Underlying Theory. 

The Single-Particle Equation.

It is useful to have an example of an underlying theory so that one can see the possibilities.  We will give one here, originally published in Journal of Mathematical Physic, 29, 1389, (1988), in which the independent variables are complex variables (unlike space and time which are real variables).  Complex variables actually seem more appropriate than real variables because it is easier to define unitary (SU(n)) internal symmetries.

We start off with the equation for a single “particle.”  It is a linear, second order partial differential equation that has the same general structure as the harmonic oscillator problem in quantum mechanics, only in this case the harmonic oscillator is relativistic.  This form was chosen because one can fairly easily find explicit solutions.  The equation is

(IVE1-1a)  

(IVE1-1b)  

There is a sum on i from 1 to n, and the bar denotes complex conjugate.  The 1 and 2 subscripts will be explained presently.