In order to understand the following, the reader should first review the Hypercomplex Math page. It describes a system of commutative 4-D numbers of the form Z=1x+iy+jz+kct [not quaternions] that behaves in all ways like the 2-D classical complex variables. It is based upon elementary group, ring, and matrix principles.
We shall reformulate Maxwell's equations of electromagnetism in terms of commutative hypercomplex mathematics. The motivation is that we will obtain a more-concise, illuminating, and more easily solved formulation, using a form of mathematics that intrinsically embodies much of electromagnetic theory.
First, some background. Engineers and scientists ordinarily work with Maxwell's equations in the form (Gaussian units)
These may not look like quaternion expressions, but they are, as we showed on the Hypercomplex Math page. We are not so much concerned about what the various symbols represent in the physical world; refer to any good introductory electromagnetics text for the details. We merely wish to reformulate them with hypercomplex mathematics.
It is possible to recast Maxwell's equations into the form of wave equations [Jackson, 1962]. The result, in Gaussian units, is
The first equation is known as the Lorentz condition,
and is necessary
to assure continuity of solutions in free space. The quantities
and J are the "sources" or "causes"
of the scalar and vector fields, respectively, and the quantities A and
are the vector and scalar potentials,
respectively, such that
A is a 3-D vector function and
is a scalar function.
Additionally, physicists describe electromagnetic fields in free space as being "conservative," i.e., having both the divergence and curl equal to zero. Such fields are conformal: The field flux lines of a static field are always perpendicular to the equipotential surfaces. Similarly, analytic functions of a complex variable are automatically conformal in two dimensions. For many years, engineers have extensively used complex variable theory to solve 2-D electromagnetic field problems [ Gibbs, 1958, Binns & Lawrenson, 1963]. They construct analytic functions that model the field. In fact, the theory of analytic functions of a complex variable is the same as the theory of 2-D electromagnetic fields in free space. We wish to extend this analogy to four dimensions.
With these observations and the wave-equation formulation,
above, we can begin
to reformulate the equations of electromagnetism in the commutative hypercomplex
notation. First, the Lorentz condition equation has an exact equivalent in the
D space, and suggests how to construct
the 4-D potential function 
:
The second equation is the reformulated Lorentz condition
which, in conjunction with the vectorial Maxwell's equations, assures continuity.
We must similarly set some condition in the hypercomplex formulation that assures
that (Z) is continuous
(and analytic). The above Lorentz condition is not sufficient. However, we
do know a sufficient condition, which we developed on the
Hypercomplex Math page (use browser BACK button
to come back). It is the full four-gradient equal to zero:
Moreover, by setting this requirement, we get that every component of
(i.e., every component of A, hence
A itself, and ) obeys a 4-D
Laplace's equation:
These can be transformed into wave equations with the
unitary transformation
x'=x, y'=y, z'=z, ct'=ict, where "i" is the classical imaginary. With the one
equation, ,
we have all the behavior of electromagnetism in free space, in terms of the vector
and scalar potentials from the classical treatments. It is just the theory of
analytic functions of a 4-D variable!
We are not quite through, however. The Maxwell's
equations in a non-vacuum
medium are inhomogeneous wave equations. We reconcile those as follows:
Multiply both sides of the A wave equation by the hypercomplex 1 and
the equation by k, then
combine the equations by addition to get:
Next, we use the relations
that we developed on the Hypercomplex Math page (use browser BACK button to come back) to reduce the partial derivatives in the composite wave equation to ordinary derivatives with respect to one 4-D variable Z. The effect of doing this is merely to impose continuity conditions on the wave equation, because the above conversion relations follow directly from the 4-D Cauchy-Riemann conditions, which themselves derive directly from continuity conditions for analytic functions. The result, after collection of like terms, is
Therefore, the full statement of Maxwell's equations in commutative hypercomplex notation is
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In typical practical applications, one is given the current and/or
the charge distribution (J and ), and first
computes the potential field . From that, the
electric and magnetic fields are calculated by use of the formulas given earlier. The
advantage, here, is that now we can use the full range of classical complex
analysis techniques on 4-D (3-D plus time) problems. That includes conformal mapping,
contour integration, 4-D ordinary integration and differentiation, Schwarz-Christoffel
mapping, Laplace and Fourier transforms, eigenfunction expansions, orthogonal series
approximations, and all the rest. Any solution that we develop for the 4-D potential
will automatically yield wave equation
behavior for the scalar and vector potentials
,A (i.e., they will obey Maxwell's
equations).
Lastly, just as for 2-D analytic functions, any 4-D analytic function or combination of functions represents a possible electromagnetic field configuration in free space. Branch cuts, if any, of the analytic potential function represent equipotential surfaces (conducting objects), isolated singularities represent point or line charges, and any equipotential surface of the analytic function can represent an equipotential conducting object. If, a priori, we know an analytic function whose singularities and/or equipotential surfaces fit our problem boundaries, then we have an immediate solution.
For free-space problems with complicated boundaries that might preclude analytic integration, the equation setting the four-gradient to zero provides a new, simpler numerical calculation tool. As we mentioned earlier, it is just a compact statement of the 4-D Cauchy-Riemann conditions, and they are a set of first-order partial differential equations of such form as to be ideally suited for parallel computation. This approach is doubly attractive because the boundary conditions are often given as gradients (first-order partial differentials) that are normal to the boundary. The 4-D solution automatically includes any wave propagation effects. We need only remember that we are solving Laplace's equation, hence need to apply a final unitary transformation to get the wave-equation solution.
As mentioned earlier, vector analysis was developed as a kind of shorthand for quaternion analysis, keeping all expressions to three dimensions or less. In recent years, there has been significant work in going back and recasting the equations of physics in full quaternion notation. For a list of references on quaternion analysis, go to Weisstein's Mathworld page. For quaternion formulations of electromagnetism, see Sweetser. These pages facilitate a comparison of the quaternion and commutative hypercomplex formulations. For a comprehensive list of electromagnetism theory and research sites, go to http://www.ieeeaps.org/emlib/emlib.html. Also see Chris Bishop's page on education and theory resources. For practical application in communications, see the resource list at The RF Cafe
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© 2000
Clyde M. Davenport
cmdaven@comcast.net