The Propagator

The propagator is defined as the probability amplitude that a particle is observed at the space-time point, when it is known that the particles is at the point. The propagator , is the integral over all possible paths from the point to the point . It is a very difficult integral since there are an infinite number of paths between any two points. Let the function be the probability amplitude that the particle actually takes the path between and . It is referred to as a phase factor. If the point is held fixed, then the propagator is only a function of the location , and can simply be written as the probability wave solution of the Dirac equation for a free electron.

The wave equation for a free electron is,

(1)

where is the momentum four-vector, are the coordinate functions and is the space-time metric tensor. The electron traveling along an arbitrary path connecting the points adds to the propagator a phase factor,

(2)

This is simply the contribution of time evolution to the propagator. [1]

Probability Wave Dispersion