In light of the unknown nature of the independent variables, how do our familiar space and time enter the theory?  They are introduced through the translation operators (which exist by hypothesis—point 2, Sec. IVD1).  Each generator  is a first order differential operator, , and as such, it defines a direction, , at each point in the independent variable space.  If the total number of independent variables is ,  then each space-time “point,” x, y, z, t, corresponds to an  dimensional surface.  So if the surface corresponding to x, y, z, t is , then the surface corresponding to  is .  Each point in the independent variable space will be associated with one and only one value of x,y,z,t.

Time Evolution.

The solutions  of Eq. (IVD1-1) will be defined throughout the space of the independent variables.  Since different ’s correspond to different times, a solution will contain the state at all times.  To obtain a mental picture of this, imagine we are dealing with a solution corresponding to a single particle and consider the set of points  for fixed time t.  Then as the are varied so that (but not t) changes,  will be nonzero for some values of  and zero for others.  As time changes, the set of values of the where is nonzero will change.  That is, the nonzero region will sweep out a “trajectory” in the independent variable space, with each “point” on the trajectory corresponding to a specific time.

Equations of Motion.

The equations of motion of the solutions can in principle be obtained by using the time translation operator  to define surfaces of constant t and then examining how the solution changes from one surface to another.  To correspond to the current theory, however, the equations of motion will be obtained in a different, representation-based way.  First we note that if a solution corresponds to a spin ½ representation that includes both a particle and its antiparticle, the basis vectors can always be chosen so the representatives of the solution obey the Dirac equation.  (That is, the Dirac equation follows from group representation theory.¹)  Then, instead of coming directly from Eq. (IVD!-1), the equation of motion for the representation of a fermion comes from the representation-theory-based free particle Dirac equation.  To include interactions, the free particle equation is  modified in the way specified by gauge theory.  The same holds for vector bosons and the Proca equation.

Notes:

1. Weinberg, S., The Quantum Theory of Fields, Vol. I, pg. 225. (Cambridge Univ. Press, Cambridge, 1996).