How to Write Maxwell's Equations Using Only Four Characters

The Classical Form

Those of us who have studied electromagnetics extensively know that Maxwell's equations come in many forms. The vector differential form is perhaps the most widely used,

while the integral form has long seemed to me to be among the most aesthetically pleasing equations of all time [1],

More recently, I have come to understand that mathematics is often most beautiful when complex ideas are expressed in their most compact form. "Less is more," as the cliche' goes. Claude Shannon, the father of information theory, and a personal inspiration of mine, said that "Much of the power and elegance of any mathematical theory...depends on use of a suitably compact and suggestive notation, which nevertheless completely describes the concepts involved." [2]

By this measure, Maxwell's equations have been a dismal cosmetic failure throughout much of history. Maxwell's original formulation consisted of a large system of twenty different equations relating numerous quantities including the field potentials, constitutive relationships, the continuity equation, and even the Lorentz force law and Ohm's law. It was Oliver Heaviside who eventually reduced twelve of these equations to the set of four most of us know today using vector calculus notation and, importantly, eliminating the potential functions altogether. Although potentials are recognized today as a valuable mathematical tool with real physical significance, Heaviside was quite blunt in his criticism of them: "I never made any progress until I threw all the potentials overboard." He further described the electrostatic potential as a "physical inanity" and that "it was best to murder the whole lot." [3]

While our stance need not be quite as strong as Heaviside's, there is some value in exploring just how simple Maxwell's equations can be, given the right mathematical tools.

The Right Tool for the Right Job

As I explain in my latest book, Relativistic Field Theory for Microwave Engineers [4], electromagnetic theory is intimately linked to special relativity, so it should come as no surprise that the simplest (most compact) forms of Maxwell's equations are those that express the relevant physical quantities in the context of four-dimensional spacetime. One of the most fascinating things I learned about while researching this new book was a mathematical formalism known as spacetime algebra.

The core mathematical element of spacetime algebra is the four-vector, an array of numbers that identify spacetime coordinates not only by their spatial position or direction, but by their temporal location or duration as well,

Note that the temporal coordinate has been scaled by c (the speed of light in a vacuum) to make the units work out. Electromagnetic sources are similarly grouped into a four-vector known as the four-current density,

where the last three elements represent the conventional current density, and the first element is the density of electric charge (once again scaled by c).

We may also define a four-dimensional differential operator, analogous to the nabla or del operator used in the three-dimensional form of Maxwell's equations in the first section,

The four-sided box symbol used here recognizes the four spacetime dimensions over which differentiation occurs, in the same way that the three-sided del operator differentiates over three spatial dimensions.

The Faraday Field

The electric and magnetic fields, in contrast are not four-vectors — they cannot be, for reasons explained in my book [4]. Instead, they jointly form another kind of mathematical object known as a bivector — an array of six numbers describing a directed plane segment in four dimensions in the same way that a vector (or four-vector) describes a directed line segment. The electric and magnetic fields, E and H, have different values in different inertial reference frames, but the bivector Faraday field, F, is a global spacetime quantity that unifies them both. What we perceive as separate electric and magnetic fields are merely geometric projections of the Faraday field on the spatial and temporal planes associated with the worldline of our particular reference frame.

Say, for example, we are observing an experiment in an inertial reference frame defined by w, the temporal axis, or more accurately to the unit normal vector to the worldline of the reference frame. The electric field is given by the projection of the Faraday field onto that vector, written as a subscript,

where we have scaled the result according to the wave impedance of free space. The magnetic field, in contrast, is the spatial residual or rejection of the field from that worldline, which I like to write as a struck-out subscript,

where i is the unit pseudoscalar, which behaves much like the imaginary unit but is defined more concretely as a four-volume element in spacetime. (Explicit formulas for the operations of projection and rejection in arbitrary dimensions are given in the reference [4].)

The spacetime divergence of the Faraday field is given by its dot product with the four-dimensional differential operator defined above. While a rigorous exposition of the rules for the products of four-vectors and bivectors in spacetime algebra is beyond the scope of this article, suffice it to say that the spacetime divergence of the Faraday field returns the source of that field, namely the four-current density (much like the divergence of the displacement field in Gauss's law returns the source of the electrostatic field, or charge density),

In reality, this equation encompasses both Gauss's law and Ampere's law, collectively known as the inhomogeneous Maxwell's equations, as they are both differential equations that incorporate an undifferentiated source term.

Similarly, the spacetime curl of the Faraday field, utilizing an operation known as the wedge product, is identically zero, in a manner analogous to Faraday's law,

In fact, this equation embodies both of the homogeneous Maxwell's equations, those having no undifferentiated source terms.

It turns out that the dot product of any four-vector a and bivector B in spacetime algebra is antisymmetric in its two arguments,

while the wedge product is symmetric,

Moreover, it can be shown that these two products are in fact the symmetric and antisymmetric parts of a more general operation known as the geometric product or Clifford product,

Therefore,

As we have already stated, the spacetime differential operator behaves like a four-vector in many respects, so we may apply this template to the homogeneous and inhomogeneous equations above,

Therefore,

That's it! By expressing the electric and magnetic fields and their associated sources in the language of spacetime algebra, we reduce all of Maxwell's laws into a single equation which is remarkably compact, having only four characters (including the equal sign). In fact it is the simplest possible differential equation for the Faraday field that can be written in this mathematical system — except perhaps when J = 0, as it is in a source-free region [5,6].

Further Reading

Clearly, I have glossed over some details above for the sake of brevity. The reader is referred to [4] for a more thorough exposition. I hasten to point out that this is more than a mere trick, a concoction of arbitrary rules having no practical value other than to compress the rich mathematical elegance of Maxwell's equations into a densely coded form. On the contrary, the Clifford product on which this equation is based is a powerful mathematical tool with extraordinarily useful properties. For one, as a product it is both distributive and associative, so that multiple contributing terms can be factored and combined in order to solve problems. It is also algebraically invertible, unlike the dot and cross products of three-dimensional vector calculus. That is, if

then

where the inverse of a and other objects in spacetime algebra have well-defined forms. More pointedly, a differential equation such as Maxwell's can be directly inverted [7] such that

where the inverse of the differential operator is shorthand for a spacetime integral over the boundary conditions that apply to the Faraday field.

I hope it goes without saying at this point that the formulation of Maxwell's equation (singular!) presented in this article is a powerful way to look at the laws of electromagnetics that has real value for solving engineering problems. I work through several such examples in my book [4], including the interaction of propagating waves at material boundaries and the modes of propagation in closed metallic waveguide. While the solutions to these problems are no doubt familiar to readers educated in electromagnetic theory, I am willing to bet that even the most seasoned professionals in microwave engineering will find the new perspective that this approach affords them is both refreshing and enlightening, and, if I have succeeded in my goals, it will enable them to conceive of new and elegant solutions to practical engineering problems as a consequence.

Relativistic Field Theory for Microwave Engineers is now available at BookBaby and is available for pre-order from Amazon.com.

References

[1] M. A. Morgan, Principles of RF and Microwave Design. Norwood, MA: Artech House, November 2019.

[2] J. Soni and R. Goodman, A Mind at Play: How Claude Shannon Invented the Information Age. New York: Simon and Schuster, 2017.

[3] P. J. Nahin, Oliver Heaviside: The Life, Works, and Times of an Electrical Genius of the Victorian Age. Baltimore, MD: Johns Hopkins University Press, 2002.

[4] M. A. Morgan, Relativistic Field Theory for Microwave Engineers. Pennsauken, NJ: BookBaby, November 2020.

[5] J. Dressel, K. Y. Bliokh, and F. Nori, “Spacetime algebra as a powerful tool for electromagnetism,” Physics Reports, vol. 589, pp. 1–71, 2015.

[6] J. M. Chappell, S. P. Drake, C. L. Seidel, L. J. Gunn, A. Iqbal, A. Allison, and D. Abbott, “Geometric algebra for electrical and electronic engineers,” Proceedings of the IEEE, vol. 102, no. 9, pp. 1340–1363, September 2014.

[7] D. Hestenes. (1998) Spacetime calculus. https://geocalc.clas.asu.edu/pdf/SpaceTimeCalc.pdf