The Scalar Field

In the previous post, we saw that a homogeneous paravector wave provided an exact description of the Maxwell's equations, except there is an added scalar field, which is on the same footing as the Electric and Magnetic fields. This scalar field does not appear in standard electromagnetic theory, and we are going to find out why.

To begin, we will define the electro-magnetic bi-paravector F in terms of the potential A

A = ( V/c, A )

F = c∂A

F = ( ∂V/∂t + c∇•A, - ∂A/∂t - ∇V + ic (∇ × A) )

We can make the following definitions for the fields

E = -∂A/∂t - ∇V

B = ∇ × A

φ = (1/c) ∂V/∂t + ∇•A

In terms of these fields the electro-magnetic bi-paravector is given by

F = ( cφ, E + icB )

If you want to recover the orthodox version of the electro-magnetic field tensor, then just take the vector portion of this, ignoring the scalar portion. For now, we are not going to ignore the scalar portion, so we can see what physical implications it has.

To begin with, we see that the scalar field is a Lorentz invariant. It is the same in all reference frames.

The vector fields are gauge invariant. We can express the classical concept of a gauge transformation as follows

A' → A + ∂Χ

F' → F + c∂∂Χ

Now, if we want to get full gauge invariance, we should expect the second term to vanish. You can show that since Χ is a scalar the vector portion of the second term vanishes identically, therefore the E and B fields are trivially gauge invariant.

We can achieve gauge invariance of the scalar field, only in the case where the scalar Χ belongs to a restricted set of functions which obey the condition

∂∂Χ = 0

This expression reduces to the scalar homogeneous wave equation. The idea of a restricted gauge invariance provides for some non-trivial gauge transformations, which end up being valid, even in the context of the traditional, unrestricted, gauge principle.

For instance, consider a paravector Ψ, which is not the gradient of a scalar, but which does satisfy the expression

∂Ψ = 0

Such a paravector can be used as a gauge function, in the sense that

A' → A + Ψ

F' → F

Such a gauge function ends up being non-trivially gauge invariant for the vector fields. This means that the vector fields are gauge invariant, but it is due to the form of the gauge function, not because of mathematical identities.

We will escalate the concept of gauge invariance to include the non-trivial gauge functions as well. The higher principle that we used to introduce the non-trivial gauge functions requires that we restrict the traditional scalar gauge functions to only include solutions to the homogeneous wave equation. This refinement resides within the limitations of traditional electro-magnetic theory, though it may require us to rethink the physical meaning behind various gauge transformations.

For instance, the Lorenz gauge

(1/c) ∂V/∂t + ∇•A = 0

If you remember the definition of the scalar field, you see that applying the Lorenz gauge is the same as making the statement

φ = 0

Since the derivatives of the scalar field are equated with the source terms, we see that setting this field to zero, or any constant for that matter, has the physical meaning of having no source terms. Luckily, this is the case where the Lorentz gauge is employed.

We will continue discussing the scalar field in the next post, where we will ask the question, "what force does the scalar field apply to a particle?"