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The Man Behind the Curtain

Physics is not always the seamless subject that it pretends to be

Tony Rothman

Where the Action Is Not

One of the great moments in the lives of young physicists is their first encounter with “Lagrangian mechanics.” God parts the firmament to reveal Truth. Lagrangian mechanics finds its roots in ancient Greece and in the 17th century suggestion of Pierre de Fermat that light travels between two points in the shortest possible time. Fermat’s principle of least time allows one to derive the famous law of refraction, Snell’s law and, apparently, explains the behavior of some dogs in retrieving bones. The idea that Nature minimizes certain quantities eventually led to the “principle of least action,” which states that you take a quantity known as the action, minimize it and—Presto!—Newton’s equations for the system miraculously emerge! (For classical systems the action is basically the system’s kinetic minus potential energy, a quantity known as the Lagrangian, multiplied by the time.) The realization that Newton’s laws themselves follow from the principle of least action is genuinely awe inspiring and the young physicist is immediately convinced that, logically if not historically, the action precedes Newton. Moreover, if you recollect (with pain) how difficult it was to even write down Newton’s equations in freshman physics for those infuriating systems of ropes and pulleys, it becomes surprisingly easy in the Lagrangian formalism, as the technique is known.

But the great swindle of freshman physics is indeed the conceit that problems have exact solutions. A standard question during my sophomore year was to write down the equations for a double pendulum, which is merely one pendulum swinging from the tip of another. With the Lagrangian formalism, the task is a simple one. What went unsaid, or was perhaps unknown at the time, is that the double pendulum is a chaotic system, and so to solve the equations is strictly impossible. What then did we really learn from the exercise?

Nevertheless, in terms of its avowed purpose of deriving equations, the Lagrangian formalism works extraordinarily well for “standard systems” where the damned pulleys are connected by ideal, unstretchable ropes. Unfortunately, once the ropes are allowed to stretch with time, for example, the Lagrangian technique becomes anything but straightforward and the fixes needed to apply it to a given problem become so elaborate that virtually every textbook author either makes a misstatement or avoids those scenarios altogether. Indeed, the general situation is so delicate that one is often unsure whether one has obtained the correct result.

This is not a moot point. Einstein did not consider his theory of gravitation—general relativity—complete until he could derive his field equations from an action, a feat that the mathematician David Hilbert accomplished five days before Einstein himself. General relativity allows for many model cosmologies, most not resembling the real universe, but in any case today it is known that when deriving the field equations for certain of them from an action one gets an incorrect result. A fair amount of esoteric research has gone into understanding why the failure occurs and how to patch it up but, as far as I am aware, all fixes require assuming the correct answer to begin with—the Einstein equations. Modern physicists take the primacy of Lagrangian mechanics seriously: contemporary practitioners, be they cosmologists or string theorists, invariably begin by postulating an action for their pet theory, then derive the equations, but if one does not have a set of previously accepted field equations, how is one certain that one has obtained the correct answer, especially in this day when theory is so far removed from experiment?

It would be surprising if the strange world of subatomic and quantum physics did not lead the field in mysteries, conceptual ambiguities and paradoxes, and it does not disappoint. The standard model of particle physics, for instance (the one containing all the quarks and gluons), has no fewer than 19 adjustable parameters, about 60 years after Enrico Fermi exclaimed, “With four parameters I can fit an elephant!” Suffice to say, “beauty” is a term not frequently applied to the standard model.

One doesn’t have to go so far in quantum theory to be confused. The concept of electron “spin” is basic to any quantum mechanics course, but what exactly is spinning is never made clear. Wolfgang Pauli, one of the concept’s originators, initially rejected the idea because if the electron was to have a finite radius, as indicated by certain experiments, then the surface would be spinning faster than the speed of light. On the other hand, if one regards the electron as a point particle, as we often do, then it is truly challenging to conceive of a spinning top whose radius is zero, not to mention the aggravation of infinite forces.

Unfortunately, quantum texts habitually ignore the difficulties that infect the heart of the field. The most fundamental of these is the notorious “measurement problem.” The equation that governs the behavior of any quantum system, Schrödinger’s equation, is as deterministic as Newton’s own, but as many people know, quantum mechanics predicts only the probability of an experiment’s outcome. How a deterministic system, in which the result is preordained, abruptly becomes a probabilistic one at the instant of measurement, is the great unresolved mystery of quantum theory, and yet virtually none of the dozens of available quantum textbooks even mention it. One well-known graduate text completes the irony by including a section titled “measurements” without addressing the issue.

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