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A new unification of the Maxwell equations is given in the domain of Clifford algebras with in a fashion similar to those obtained with Pauli and Dirac algebras. It is shown that the new electromagnetic field multivector can be obtained from a potential function that is closely related to the scalar and the vector potentials of classical electromagnetics. Additionally it is shown that the gauge transformations of the new multivector and its potential function and the Lagrangian density of the electromagnetic field are in agreement with the transformation rules of the second-rank antisymmetric electromagnetic field tensor, in contrast to the results obtained by applying other versions of Clifford algebras.

Clifford algebras provide a unifying structure for Euclidean, Minkowski, and multivector spaces of all dimensions. Vectors and differential operators expressed in terms of Clifford algebras provide a natural language for physics which has some advantages over the standard techniques [

It has been shown that when the electromagnetic field is defined as the sum of an electric field vector and a magnetic field bivector, the four Maxwell equations reduce into a single equivalent equation in the domain of Pauli and Dirac algebras [

The second is the reconstruction of the combined electric and magnetic fields by a single transformation of the four-potential

Our investigation differs in approach from those in Hestenes and Chisholm- Common in its simplicity and ability to use a single potential function to cor- rectly derive Maxwell’s equations in a vacuum.

In what follows, we first lay out the theory of the Clifford algebra employed in this work. We then discuss its applications to electromagnetism and obtain a new electromagnetic field multivector, which is closely related to the scalar and vector potentials of the classical electromagnetics. We show that the gauge transformations of the new multivector and its potential function and the La- grangian density of the electromagnetic field are all in agreement with the transformation rules of the rank-2 antisymmetric electromagnetic field tensor. Finally, we give the matrix representation of the electromagnetic field multive- ctor and its Lorentz transformation.

Consider the Clifford algebra

and no others [

An important subspace of

which is isomorphic to the generalized Minkowski space

A product

If

The inner and outer products of blades are defined as follows [

The outer product of

By linearity, these definitions extend to

Some examples of inner and outer products are:

There are three important involutions on

1) inversion:

2) reversion:

3) conjugation:

Then it follows that

Let

It is straightforward to show that the following identities hold:

The Clifford algebra

The geometric product on

If

Theorem 1. Suppose

Proof. A vector field

In Gaussian units, the differential form of the Maxwell equations for sources in vacuum are [

where

We recast the Maxwell equations in the language of Clifford algebras by keeping the electric field as a vector, but replacing the magnetic field vector by the magnetic field bivector

The electromagnetic field multivector is then defined as

In terms of

Theorem 2. The Maxwell equations are equivalent to the single equation

Proof. Since

Using the Maxwell equations, we obtain

But

Therefore, we obtain Equation (32). Conversely, assuming Equation (32), the Maxwell equations follow by setting the real parts, the vector parts, the bivector parts, and the trivector parts of each side equal. This completes the proof. □

From classical electrodynamics [

where

and the continuity equation

We can formulate this as follows: Let

where

Note that

The derivative of

Using Equations (42) and (43) and noting that

we obtain

Theorem 3. The electromagnetic field

Proof. From Equation (46) we have

□

Note that Equation (47) may also be written as

A Lorentz transformation is an isometry

where

In the general case, writing

and

Thus the operator

Suppose now that

and

Theorem 4. Under the Lorentz transformation

Again, associativity does not hold in this equation.

Before we get to the mathematics of this section, let us note the difference in Lorentz and Lorenz. These names, in fact, do belong to different scientists and thus we consider both types of gauge invariance here.

The common gauge invariant from classical electrodynamics is to consider

In our formalism this leads us to

Examining this a little more fully, we know that the electric and magnetic fields do not change under Lorenz or Coulomb gauges and thus we obtain

Following through we see

As the multivector field must remain unchanged we obtain the gauge invariant condition

Recall that in classical electromagnetism the Lagrangian density in a vacuum is given by

By expanding this a bit, we find

In order to recreate this in the Clifford algebraic formulation we consider

Thus we might expect that the Lagrangian density becomes

Examining this a little we see that

Since our inner product is commutative we have a cancellation of field product terms

In higher dimensions, one may wish to restrict to the 0-blade so as to disallow higher dimensional cross terms. Thus we write

Now let’s consider the situation outside a vacuum. We have

Let us write

Then using our potential

Complex numbers can be represented by

From this we can write an

The

then the Lorentz transformation of

A quite lengthy calculation (see Appendix) shows that the two transformations given by Equations (54) and (73) are exactly identical.

We have shown that in the framework of the Clifford algebra defined in Equation (3), the Maxwell equations in vacuum reduce to a single equation in a fashion similar to that in other types of Clifford algebras. The multivector

Furthermore, we have shown that the electromagnetic field multivector can be derived from a potential function

Finally, we have discussed the Lorentz transformation of the potential function

The formulation given by other investigators [

By repeating our calculations with

Note that this matrix is not antisymmetric and the representation is not the same as that of the electromagnetic field tensor, and the transformation rule is also different. This is in contrast to the result obtained from applying a Clifford algebra with

Mohazzabi, P., Wielenberg, N.J. and Alexander, G.C. (2017) A New Formulation of Maxwell’s Equations in Clifford Algebra. Journal of Applied Mathematics and Physics, 5, 1575-1588. https://doi.org/10.4236/jamp.2017.58130

Here we show that the two transformations given in Equations (54) and (73) are identical.

We have

and

Therefore,

It follows that the matrix representation of

We also have

So the matrix representation of

The general Lorentz transformation and its inverse are given by the following matrices:

From Equation (54) we have

where

and

Let

and

Recall that from Equation (42) we have

and from Equation (37) we have

Therefore,

Then we find the following identity by carrying out the multiplication,

For example, with

and with

It is now straightforward to show that the identity in Equation (18) is identical to

where

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