A little while ago I was asked to give a lecture at the very elegant Weisman Art Museum in Minneapolis on an assigned title: “Elegance in Physics.” As I get older, the things I’m asked to do get stranger, so I wasn’t surprised. Alarmingly, the older I get, the stronger is my inclination to do the peculiar ones. So I accepted the invitation, and soon found myself brooding, not, as I had imagined, about the glory of the eternal verities, but about the highly contentious nature of elegance in physics.

Here is the first such difference of opinion I came upon. In a lecture at Fermilab with a title similar to the one I was given, Subrahmanyan Chandrasekhar talked about “harmoniously organizing a domain of science with order, pattern, and coherence.” (See his article in Physics Today, July 1979, page 25.) He cited five examples of such pinnacles of exposition, one of them being Paul Dirac’s celebrated book, Principles of Quantum Mechanics. “The translucence of the eternal splendor through material phenomena,” Chandrasekhar remarked, “[is] made iridescent in these books.”

Keeping that iridescent translucence firmly in mind, consider the following remarks of the eminent mathematician Jean Dieudonné: “When one arrives at the mathematical theories on which quantum mechanics is based, one realizes that the attitude of certain physicists in the handling of these theories truly borders on delirium . . . . One has to ask oneself what remains in the mind of a student who has absorbed this unbelievable accumulation of nonsense, real hogwash! It would appear that today’s physicists are only at ease in the vague, the obscure, and the contradictory.” ¹

What is Dieudonné talking about? He is addressing the approach to quantum mechanics laid out in Dirac’s book.

Elegance in physics is as much in the eye of the beholder as it is in any other field of human endeavor. Dirac’s formulation appeals to physicists because, by being a little vague and ambiguous about its precise mathematical structure, it enables them to grasp and manipulate the physical content of the theory with a clarity and power that would be greatly diminished if one were distracted by certain complicating but fundamentally uninteresting mathematical technicalities. But for mathematicians, those minor technical matters lie at the heart of the subject. Quantum mechanics becomes ill-formulated and grotesque if it does not properly rest on impeccable mathematical foundations.

Chandrasekhar and Dieudonné having thus sensitized me to what one might call interdisciplinary aesthetic dissonance, I realized that the same mechanism had been at work in a difference of opinion I had had a few years ago with crystallographers. For over a century, the hallmark of crystallinity had been taken to be periodicity at the atomic scale. all crystals were thought to be built out of primitive units made from a comparatively small number of atoms, repeated over and over again. But then new crystalline materials were discovered. Some of them, dubbed quasicrystals, had symmetries that no periodic structure could possibly have. One of the important features all these aperiodic crystals shared with the ordinary periodic ones was a characteristic display of sharp Bragg peaks in their x-ray photographs.

The crystallographers realized early in the game that all these aperiodic crystals could be described as three-dimensional cross sections of structures periodic in spaces of four, five, six, or even more dimensions. Motivated by this insight, the International Union of Crystallography has had for many years a subcommittee that struggles over the appropriate nomenclature to describe the kinds of symmetries one can encounter in four or more dimensions. Thinking about four-dimensional crystals can be highly entertaining, and is it not elegant that nature has actually produced materials that require us to think that way?

No, it is not. The new materials don’t require us to think that way. Some of us noticed that if we shifted the defining feature of crystallinity from the periodicity of the atomic structure to the presence of Bragg peaks in the diffraction diagram, then the only relevant symmetries continued to be those associated with patterns of reflected x rays in boring old three-dimensional space. The patterns made by aperiodic crystals can be more complex, but the geometrical description of the symmetry of those patterns remained familiarly three-dimensional.

We thought that this triumphant return to three-dimensional geometry, at the price of shifting the emphasis from the crystal itself to its x-ray photographs, was an elegant step forward. We expected the crystallographers to throw out their irrelevant books on higher-dimensional geometry, dissolve or at least redirect their commissions on higher-dimensional nomenclature, and glory in the elegance of rethinking crystal symmetry in terms of diffraction patterns.

Did that happen? No, of course not! Most crystallographers found our beautiful and illuminating shift from the structure of the crystal to the structure of its diffraction pattern to be unnatural, ungainly, and unintelligible. They were not impressed that our approach provided a direct link between crystal symmetry and electronic properties, because, being crystallographers but not physicists, they were not interested in electronic properties. I won’t abuse my privilege as a “Reference Frame” columnist to elaborate on why our way is better, but I report the sad tale here as another, less lofty example of the relativity of elegance.

With two such examples to think about, it occurred to me that a curious episode, early in my professional career, was yet another manifestation of the same phenomenon.

Over 25 years ago, I became interested in the physics of the newly discovered superfluid phases of helium-3. I realized that one of the phases of this unique fluid bore a striking similarity to a type of structure known to mechanical engineers as a Cosserat continuum. So when I noticed one day that there was a seminar on Cosserat continua on the Cornell Engineering quad, I wandered over.

I didn’t learn anything useful about helium-3 from the engineers, but in the discussion period after the lecture, rather to my surprise, a heated debate broke out over whether a point particle could have an angular momentum—the terms of the argument were whether a particle with no internal structure at all could nevertheless spin like a top. I found this dispute remarkable for two reasons. First, because I hadn’t thought that hard-headed engineers could become so passionate about so fundamentally metaphysical an issue. And second, because the question, insofar as it had empirical content, had an elegant answer whose relevance to their argument the disputants seemed not to have noticed. So I rose to my feet and made a remark, elegantly stated in four words, that I was sure would settle the whole debate: “What about an electron?” There followed a sickening silence. It was as if someone in the crowd had shouted an obscenity. (This was the early 1970s, when somebody in any crowd was quite likely to shout an obscenity.) A senior professor of theoretical and applied mechanics rose slowly from his seat, fixed me with a baleful gaze, and delivered this crushing rejoinder: “Have you ever seen an electron?” His riposte elicited nods and murmurs of approval throughout the auditorium. Then they returned to their deliberations with undiminished vigor.

My elegant invocation of physical reality to cut through a metaphysical argument was viewed as a clumsy introduction of speculative metaphysics into a tough-mindedly practical debate about—about what? To this day, I do not know what the debate was about. So I slunk back to the physics corner of the campus, where the elegance and relevance of spinning electrons remained unchallenged.

Having thus become well attuned to the highly contentious nature of elegance in physics, I now realize that an excellent example of debatable elegance is provided by the new field of quantum computation, which I offer as a final illustration of the contingency of scientific aesthetics.

You can set up a quantum computer to act on a superposition of all possible inputs. Because a system in a superposition of inputs evolves into the superposition of outputs it would have evolved into for each of the superposed possibilities, in no more time than it takes to do a single calculation, a quantum computer with an n-bit input register can produce a state whose structure encodes the outcome of 2n separate calculations! If you have 100 bits—hardly anything for a classical computer—that amounts to doing 2100 = 1030 calculations in a single pass.

But have you really done that astronomical number of calculations? How much of that vast output of information can actually be extracted? Not much! Indeed, the most obvious approach gives you only a single output, randomly chosen from the enormous range of possibilities, so you do no better than you would have done by feeding a randomly chosen input into a classical computer. But one of the funny things about quantum mechanics is that it offers you trade-offs. If you’re willing to renounce the possibility of getting any information about any individual computation, you can get certain kinds of partial information about the results of all the computations. In particular, if you know that the output is a periodic function of the input but you don’t know the value of the period, then a quantum computer turns out to offer you clues that permit you to determine the value of the period spectacularly more efficiently than you can with a classical computer.

Is this elegant? It is for computer scientists, because it offers a striking demonstration that computational complexity theory—the study of how the time it takes to do a computation scales with the size of the input—cannot be divorced from assumptions about the physical nature of your computer. It is elegant for the National Security Agency, because finding the period of an unknown periodic function is the key step in cracking a widely used coding scheme. For me the elegance lies in the entirely new perspective quantum computation provides on the exquisitely intricate ways in which information can be encoded in quantum states.

On the other hand, many people find it altogether inelegant, because of the enormous, quite possibly insuperable (but please don’t tell the NSA!) obstacles in the way of building a quantum computer capable of performing these wonderful tricks. Quantum computation raises the question of whether feasibility is or is not an essential ingredient in determining the elegance of a proposed technology. Ordinarily it surely is, but quantum computation seems to me a case where the conceptual charm of the idea is so very powerful that the strong possibility that it will never prove feasible fails to undermine its elegance. Long may it flourish, if only as a gedanken technology!

Reference
1. J. Dieudonné, in La Pensée physique contemporaine, S. Diner, D. Fargue, G. Lochak, eds., Editions Augustin Fresnel, Paris (1982).   

  © 2000 American Institute of Physics