Revision 1, August 2, 2008
© 2000, 2008
Clyde M. Davenport
The objective of this Revision is to take the same coordinate perspective (frame of reference) as did Hamilton, Maxwell, and their vector analysis successors. This will provide more clarity and will facilitate comparison with the standard vector analysis presentation.
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:
[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 following 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
These are classical vector expressions. (x,y,z) is a scalar function and A(x,y,z) is a 3-D vector function; consequently, E,B are also functions of (x,y,z).
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 our reformulation of the equations of electromagnetism in the commutative hypercomplex (C-H) notation. First, we must make the change to an independent variable that reflects Hamilton's and Maxwell's particular coordinate frame of reference, as we alluded at the outset. In the standard C-H notation, we use an independent variable of the form:
which was chosen because of its natural extension of the classical complex variable z=x+iy. Here, we wish to use the Hamilton-Maxwell perspective, which in our notation is:
This represents a simple change of coordinate frames (a rotation + reflection, with determinant -1). The latter is Hamilton's original form, except that he did not designate ct as the first component. It was continued by [Maxwell, 1873], [Heaviside, 1893], and [Gibbs, 1881]. Recall that is a scalar function and A is a 3-D vector function:
such that the expression 1+A is a 4-D vector function. In these terms, taking into account the change of coordinate frames, the vector quad operator is:
Now, still in these terms, consider the expression
Therefore, the Lorentz condition stated in C-H notation is:
This is very suggestive that the quantity makes up a 4-D analytic function [ is a scalar and A is a 3-D vector quantity with i,j,k components]. This makes further sense when we recall that we will be deriving the electromagnetic field quantities E,B from the A, potentials, and we know from existing theory that E,B must be analytic (conservative, in physics parlance).
Therefore, for reasons that we shall explain, we shall define the electromagnetic potential as follows:
Recall that a 3-D function (e.g., A, alone) cannot be analytic in a 4-D sense because the Cauchy-Riemann conditions require four differentiable components. With the above notation and definitions, then the Lorentz condition in C-H notation is:
This condition, 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. It is the full four-gradient equal to zero:
By setting this requirement, we assure analytic continuation of into 4-D. Although it looks arbitrary, we have not constrained any solutions of Maxwell' equations, because the above requirement is just a succinct statement of the 4-D Cauchy-Riemann conditions which hold for any and all analytic functions. Further, because of distributivity, 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!
Our definitions and methods to this point might seem arbitrary, but we have had a specific objective in mind. Consider Jackson's ,A wave equations, above. If we combine them by straight addition, we obtain:
Next, we use the relations
that we developed on the Hypercomplex Math page [but, notice the permutation (ct,x,y,z) of the coordinates in order to reflect Hamilton's and Maxwell's perspective] 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
Finally, we are at our objective. On the right-hand side, (ct,x,y,z) is a scalar and J(ct,x,y,z) is a 3-D vector with i,j,k components, meaning that their sum is a 4-D vector. The derivative on the left is always a 4-D analytic quantity. The above expression emphasizes the direct relationship between the potential function and the "sources" of the field:
The reader might notice a problem with the above. We want to be an analytical function, implying four components with each a function of x,y,z,ct. However, most practical problems provide only boundary conditions and initial conditions over only part of 4-D space, not usually in the form of a 4-D function. In fact, if and J are time-invariant quantities, then traditional treatments [Jackson(1), 1962] define the components of as:
where the integrals are taken over all infinitesimal increments of charge and directed current. This usually provides only a numerical solution.
In summary, the full statement of Maxwell's equations in commutative hypercomplex notation is
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).
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 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.
One Last Observation
Let us go back and examine the 3-D vector relations that give E,B in terms of A,:
Note that all elements are 3-D vector quantities. We can subtract the Lorentz condition form (a zero quantity) from the B definition in order to convert it back to a quaternion expression:
If we compare this with the corresponding E expression,
we see an intriguing symmetry. If we add them, we obtain
In the Hamilton-Maxwell coordinate frame in which we are working, we note that
Recall that the E,B vector functions were constrained to 3-D by Maxwell and his successors. However, by the above operations, we have analytically continued the function E+B into four dimensions to obtain:
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, search for "ieee" and "emlib". Also see Chris Bishop's page on education and theory resources. For practical application in communications, see the resource list at The RF Cafe