One hundred and fifty years ago, James Clerk Maxwell presented a set of equations that describes virtually any manifestation of electromagnetism. Is it possible to find similarly compact descriptions — and is the search even worthwhile — in every branch of physics ?
The most significant event of the nineteenth century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance...
In 1861, James Clerk Maxwell published the first instalment of his four-part paper 'On Physical Lines of Force' in the Philosophical Magazine. The paper describes the mathematical physics of electromagnetism, and was to become a major landmark in science. In the words of Richard Feynman: “From a long view of the history of mankind — seen from, say, ten thousand years from now — there can be little doubt that the most significant event of the nineteenth century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.”
Maxwell showed that electricity, magnetism and light — three phenomena that seem so vastly different in their manifestation — are different aspects of one and the same phenomenon. More than that, Maxwell provided a set of just a few equations to succinctly describe what we know today as electromagnetism. For all the wide-ranging consequences and ramifications of Maxwell's work, it may be that compact description of electromagnetism in a single set of equations that makes us physicists so very proud of Maxwell's equations.
Maxwell had distilled an entire body of work on electricity and magnetism into 20 equations originally; Oliver Heaviside is credited with having compressed them into the four partial differential equations that now typically appear. Indeed, using tensor notation (or, alternatively, that of geometric algebra, as Mark Buchanan discusses on page 442 of this issue), these four equations can be collapsed into just one, crystallizing them into a monolithic monument of iconic beauty. One equation to rule all of electromagnetism.
But Maxwell's equations also set out very practical guidelines. If an experimental discovery or a new theory describing phenomena in the realm of Maxwell's equations is at odds with them, then that finding is either revolutionary or — probably more likely — flawed. This touchstone function is all the more remarkable if you stop to consider how far removed the modern-day problems we tackle by solving Maxwell's equations are from the world in which Maxwell lived and worked.
This was captured beautifully in the words of Max Planck in 1931, at the Clerk Maxwell centenary celebrations at the University of Cambridge (Nature 128, 604–607; 1931): “[Maxwell's] theory of electrodynamics was originally founded on quite special ideas of the mechanical nature of the ether, in accordance with the fact that in his time the mechanical conception of nature was considered nearly a matter of course, having received strong support through the discovery of the principle of conservation of energy. But after, by the aid of such mechanical notions, Maxwell had found his electromagnetic differential equations and had recognised the extent of their efficiency, he did not hesitate in pushing aside as a negligible accessory the mechanical interpretations of the differential equations, in order to make them independent and to give his theory its pure and sublime shape.”
Just as we have Maxwell's monument to lean on, there is comfort in knowing that there is a compact set of laws for thermodynamics, or postulates for special relativity. These basic equations, laws or axioms may not, in themselves, give a clear notion of the wealth of phenomena that are ruled by them. Yet they provide markers for how far we can stray: once real boundaries are reached in a given field, then major revision is called for. (Famously, the fact that Maxwell's equations are not consistent with Newtonian mechanics led to the formulation of the theory of special relativity.)
However, not every field of physics has such a basis set at hand. This might be most evident in quantum mechanics, which is widely considered to be based on ad hoc axioms. In the past decade, there has been a drive from the quantum-information community to put information-theoretic constraints on quantum theory (Nature Phys. 1, 2–4; 2005) — an effort also discussed in the book review on page 443 of this issue. Although, once again, an axiomatic basis of quantum mechanics might not immediately guide us to new discoveries, it could offer a more intuitive understanding of quantum mechanics, and make some of the puzzling discoveries routinely reported from labs and in theoretical work less surprising and easier to grasp.
Similarly in the vast field of 'complexity' we are regularly amazed at how mechanisms underlying, say, a social network are also found to be at work in the functioning of the metabolism of a living being. Can we also here push aside the original context and find just a few general governing principles? Can these principles, in turn, help us understand what constitutes a complex system in the first place? There could be many other fields that lack but could benefit from a compact basis.
Or are there? Does every field really need an axiomatic basis from which everything can be derived? In a glance over at mathematics, for example, it could be argued that Hilbert's momentous axiomatization of geometry, based on 20 (originally 21) assumptions, has little significance for the everyday practitioner. So how much further can we, or should we, take the spirit that shines through Maxwell's legacy?
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Keep it simple?. Nature Phys 7, 441 (2011). https://doi.org/10.1038/nphys2024