There is another important difference that is related to the mathematical formulation of SFT and that is the complete absence of gauge. To see this we must first look at the basics of gauge theory. And to look at the basics of gauge theory we must first define some terms we have already used. First let’s define Maxwell’s equations.

(1a)

(1b)

(1c)

(1d)

If we just look at the equations above (1a-d), Maxwell’s equations, then we have the basis for examining gauge in various systems. The reason being, as they stand these equations are incomplete and cannot be solved uniquely. There is no definitive form of the equations as ordinary variables. In other words, there is a family of solutions, all isomorphs of each other; we do not have a unique solution. We can add any constant we like to them and this will be part of the overall family. Now this is the basis of gauge theory and its symmetries. In terms of SFT, two spinors could be a solution as long as they obey the Maxwell equations in (1a-d) above; the solutions float about a constant of integration. There is a freedom to choose the constant anyway we like.

So what is Maxwellian? The Lorentz equation for the field forces acting on the particles is written as

(1.1e)

The constitutive equations in free space are

(1.1f)

. (1.1g)

The relationship between the speed of light[i] and the ratio of the fields

*c* = (1.1h)

The atomic energy density per volume is

, (1.1i)

which depends upon the E- and H-fields in the atomic region. Equations (1.1a–i) are termed the Maxwellian, or sometimes the EM field equations.[ii] In these equations, *v *is the particle velocity, *m* is its mass. It is assumed that the volume of integration *v _{n}* over which the charge density is evaluated, and the area the charge circulates normal to its motion

*S*, are calculated during successive periods over which the internal motions of the atom take place (Fig. 2.2). At this stage, the periodic motions are not assumed circular, regular, or constant. The area parameters need retrospective examination to check any solutions that are uncovered for a particle’s motion. In distinction to quantum theory, the charge- and mass-points are non-singular, as a particle does not to reside at the charge-point due to its assumed bi-spinorial motions.

_{n}In these equations there is no gauge because now the equations are completely defined. Now we have a unique equation. There is no family of isomorphs because we have tied the solution down; it is no longer a floating family of similar shaped solutions but a single solution.

[i] In SFT, the speed of light is not proscribed from being variable. Depending on the energy density of the region being studied, and the photon state, *c* can vary.

[ii] Where a nebular current density is used in (2.1d), the factor 4p comes about from an application of Green’s theorem leading to a surface over the volume enclosed by the charge density. For the case of discrete charges, the factor p represents the area enclosed by the moving charge point.