Quantum mechanics on a roll

The July issue of the Skeptical Inquirer, a magazine published by the Committee for the Scientific Investigation of Claims of the Paranormal (CSICOP), carries an article by Paul Quincey (PQ), a physicist at the National Physical Laboratory at Teddington, Middlesex, UK. This article purports to explain Why Quantum Mechanics Is Not So Weird after All.

It’s not hard to see why CSICOP wants to downgrade the weirdness level of quantum mechanics. As Quincey rightly points out, “[i]t seemed [and still seems] that any weird idea could gain respectability by finding similarities with some of the weird features of quantum mechanics.” If your aim is to undermine the respectability of weird claims, deflating the weirdness of quantum mechanics seems like a good place to start.

Quincey sets out “to see if either quantum mechanics is less weird than we usually think it is or the world around us is more so.” What he ends up demonstrating is definitely not that Quantum Mechanics Is Not So Weird after All but that the world around us is weirder than we are wont to think.

The first step in Quincey’s argument is to translate classical physics from the language of Newton into the language of Hamilton, which replaces the concept of force by the principle of least action (a.k.a. Hamilton’s principle, although it was originally formulated by Maupertuis).

PQ: Hamilton’s Principle is fundamentally equivalent to Newton’s Laws, and comes into its own when solving more advanced types of classical problems. But as an explanation, it has a major flaw — it seems to mean that things need to know where they are going before they work out how to get there.

Let’s compare the respective “flaws” of stories about path-changing forces and stories about path-selecting actions. You are probably familiar with the following story, which at one time was offered to explain how charges act on charges:

• The electromagnetic field is a physical entity in its own right;
• it is locally generated by charges here;
• it locally acts on charges there.

Did you notice the absence of any explanation of how a charge locally acts on the electromagnetic field, or how the electromagnetic field locally acts on itself, or how the electromagnetic field locally acts on a charge? Comment by B. S. DeWitt and R. N. Graham:

Physicists are, at bottom, a naive breed, forever trying to come to terms with the “world out there” by methods which, however imaginative and refined, involve in essence the same element of contact as a well-placed kick.

Fact is that the calculation of electromagnetic effects can be carried out in two steps:

• given the distribution and motion of charges, we calculate six real-valued functions of time and position using Maxwell’s equations,
• given these six functions, we calculate the effects that these charges have on a test charge, using the Lorentz force law.

The rest — the transmogrification of an algorithm for calculating the effects of charges on charges into a mechanism or process by which charges act on charges — is embroidery. It’s a sleight-of-hand useful only for shutting up students who ask too many questions.

One advantage of the Hamiltonian formalism is that if you want to similarly embroider it with a regaling story, or if you want to transmogrify its mathematical formulas into physical mechanisms or processes, you end up with something nonsensical. That’s not a flaw! As Quincey correctly points out, it’s what brings us closer to the correct theory, quantum mechanics, which is essentially a tool for calculating the probabilities of possible measurement outcomes on the basis of actual outcomes. Try to transmogrify the quantum-mechanical probability algorithms into physical mechanisms or processes, and you end up with horrendous nonsense.

By the way, in passing, quantum mechanics also comes in two fundamentally equivalent versions, Schrödinger’s version based on wave functions and Feynman’s propagator-based version. (Heisenberg’s third version isn’t much used these days.) Schrödinger’s version is analogous to Newton’s version (of classical mechanics) in that it invites attempts to embroider it, and Feynman’s version is analogous to Hamilton’s in that it resists such attempts. All attempts to transmogify the mathematical formalism of quantum mechanics into a physical mechanism or process are focused on Schrödinger’s version, even though in the real world (which is not only quantum but also relativistic) Feynman’s version is vastly superior.

Unfortunately Quincey’s otherwise commendable article gives a wrong account of Hamilton’s principle.

Correct is that “given its starting points and motion [momentum], an object will end up at locations that are connected to its starting point by a path whose action is a minimum compared to neighboring paths.”

Wrong is that “there is a path from the start to the black points whose action is a minimum compared to neighboring paths, but there is no such path from the start to the white spots.” For any two points there is at least one path whose action is a minimum compared to neighboring paths. Whereas the principle of least action applies exclusively to paths with specified initial and final positions, Quincey applies it to paths with specified initial positions and momenta. This is bound to produce such nonsense as his self-contradictory conclusion that “there will always be a path with the least action, but this is not a minimum.”

PQ: Suppose we take the action question seriously and give it a rather simple answer: Nature has to check out all possible destinations to see if they are on the right track. It must do this by trying to find out if there is a path of minimal action to each destination.

Let me formulate something analogous that has the merit of being halfway correct:

Suppose we take the action question seriously and give it a rather simple answer: Nature has to check out all possible paths to a given destination to see which path qualifies. It must do this by trying to find out which path has the minimal action for the given destination.

If this were what Quincey meant, he would have explained why classical mechanics is much weirder than we ever thought it was, rather than Why Quantum Mechanics Is Not So Weird after All.

Quincey’s analogy of the surveyor’s wheel is not half bad (after all, it’s vintage Feynman) but he uses it the wrong way.

PQ: if we want to check out destinations that are too close to the start, as gauged by the size of the wheel, the mechanism doesn’t work.

Why not? One fundamental departure of quantum mechanics from classical mechanics lies in the answer to this question, which Quincey studiously avoids. He only tells us that this mechanism “cannot say where the object should be going, and there is an intrinsic fuzziness associated with it…”

What about this “intrinsic fuzziness”? Properly understood, it revolutionizes nearly everything that classical physics took for granted. It is anything but a harmless, natural, humdrum modification or generalization of classical mechanics. “Classical mechanics with a mechanism,” my foot! The way Quincey introduces this “intrinsic fuzziness” is completely ad hoc. In fact, it is inconsistent with his mechanism: there is no reason why the wheel should be incapable of correctly measuring any distance as a fraction (however small) of its circumference.

PQ: A second feature arises from the simple circular nature of the measuring device. It cannot tell the difference between paths that differ by an amount of action that is an exact whole number of Planck’s constants. This can lead to patterns of probabilities that look just like classical waves, because the mathematics of waves is very similar to the mathematics of circular motion.

The analogy between the quantum-mechanical interference of probability amplitudes and the classical interference of wave amplitudes is limited to single objects without internal degrees of freedom. The reason this is so is that the paths to which Hamilton’s principle assigns actions are path in a system’s configuration space(-time), and this has as many dimensions as the system has degrees of freedom (plus one temporal dimension). To ignore this fact is to ignore the entanglement of composite systems, the possibility of which is another fundamental departure of quantum mechanics from the classical theory.

(Just how misleading the wheel analogy can be, may also be seen by considering paths in the 3+1 dimensional configuration space(-time) of a single object without internal degrees of freedom. Any timelike path that looks shorter than its neighbors in a spacetime diagram actually has the largest action. A lightlike path has zero action, and a spacelike path has an imaginary action! We can rescue the “least” in “the principle of least action” only by making all real-valued actions negative.)

PQ: The most important change comes when we consider objects in very small orbits, like electrons around nuclei. The mechanism gives zero probability unless the orbit (or more correctly the state) has an action that is an exact multiple of Planck’s constant. This crude mechanism explains why atoms can only shrink to a certain point…

Electrons don’t orbit. In the central-field approximation, and only there, they have orbitals. Orbitals are position probability distributions defined by a more general probability algorithm, a so-called quantum “state”. Actions can be attributed to paths such as orbits but neither to orbitals nor to quantum states. The “crude mechanism” to which Quincey refers is not part of quantum mechanics; it belongs to the first tentative steps away from classical mechanics towards quantum mechanics, taken by Bohr and de Broglie, when electrons were thought of as standing waves on a circle.

PQ: With one extra idea [spin] the mechanism seems to explain the workings of chemistry, biology, and all the other successes of quantum mechanics, without ever really stopping being classical mechanics.

Everyone has the right to define “classical mechanics” anyway he likes, but if he wants to be understood or taken seriously, he had better stick to the universal acceptation of the term. (By the way, the existence of spin is yet another fundamental departure of quantum mechanics from classical mechanics.)

To be continued!