The Strangest Man,

The Hidden Life of Paul Dirac,

Mystic of the Atom

by Graham Farmelo Basic Books, 2009

Out of the blue, it occurred to Dirac that he had come across a special mathematical construction, known as a Poisson bracket, that looked vaguely like AB - BA. He had only a faint visual recollection of the construction, but he knew it was somehow related to the Hamiltonian method of describing motion... Sure enough, as Dirac had surmised, the Poisson bracket, which first appeared over a century before in the writings of French mathematician Siméon-Denis Poisson, had the form of two mathematical quantities multiplied together minus two related quantities multiplied together, the multiplication and minus signs making it appear similar to the expression AB - BA. In one of his great insights, Dirac saw that he could weave an entire carpet from this thread -- within a few weeks of uninterupted work he had set out the mathematical basis of quantum theory in analogy to classical theory. Like Heisenberg, he believed that mental pictures of the tiniest particles of matter were bound to be misleading. Such particles cannot be visualized, nor is it possible to describe them using quantities that behave like ordinary numbers, such as position, speed and momentum. The solution is to use abtract mathematical quantities that correspond to the familiar classical quantities: it was these relationships that Dirac pictured, not the particles that they described. Using the analogy with the Poisson bracket, together with the correspondence principle, Dirac found connections between the abstract mathematical quantities in his theory, including the crucial equation connecting the symbols associated with the position and momentum of a particle of matter: position symbol x momentum symbol - momentum symbol x position

symbol = h x (square root of -1)/(2 x ) where h is Planck's constant and is the ratio of the circumference to the diameter of every circle (its value is about 3.142)... The most mysterious part of the equation was on the right-hand side, especially for those unwise enough to think of the position and momentum symbols as anything other than abstractions: they are not numbers of measurable quantities but symbols, purely mathematical objects. Graham Farmelo, The Strangest Man, The Hidden Life of Paul Dirac, Mystic of the Atom, pp.86-87, color added; do Farmelo's readers really need pi explained to them?

The book by Graham Farmelo on great physicist Paul Dirac gives us some excellent examples of the Sin of Galileo, such as we see in the passage just quoted, and on some other points in the philosophy of science. The most striking thing here is that Dirac himself seems to like the idea that the "symbols" in his equations don't even refer to anything. But, in fact, as "purely mathematical objects" they are already more than just "symbols." If they have no meaning or reference at all, then we have the kind of mathematics described in the formalistic project of David Hilbert, where the symbols could mean "beer steins" rather than numbers, and all that matters are the rules that manipulate them. This formalistic view of mathematics was refuted by Kurt Gödel, who proved that any formal system of mathematics must contain propositions that must be true but cannot be proven within the system. Thus, they can only be true because they refer to some external ground, and they must have meaning and reference for that to happen. So, in mathematics the symbols in equations cannot be just "symbols."

The idea that in physics the terms are only "symbols" and do not mean "measurable quantities" is even more remarkable, especially when we actually identify those quantities as things like "position" and "momentum." We could use the notion that our quantities aren't really these things in any way that we can understand as part of a modest philosophy, consistent with a rejection of the Sin of Galileo, that we don't know what is really going on. This is a natural reflection given the curious and mysterious characteristics of quantum mechanics. But it also leads us to a crossroads. Either we can endeavor to make sense of the reality of the objects we are studying, or we can adopt a Positivist approach that we don't care what the reality is and will make no effort to figure it out. Thus, all that matters to the Positivist is that the "black box" of a physical theory produces predictions that can be confirmed. As Roger Penrose himself has commented, it hardly even makes sense to study or do science if you don't what to know what is going on. To the extent that Dirac himself thought that the pure mathematics was the only object of his study, he was either a Positivist or embraced the Sin of Galileo itself. At least in the Sin of Galileo you think that something -- pure mathematics -- provides meaning for your theory. In Positivism, you have no ambition to actually understand anything. But the expressions used by Farmelo, like "momentum symbol," in which we have somehow simply named a meaningless hieroglyph, are deeply meaningless and pointless, or evasive.

Farmelo says:

The philosophers who least offended Dirac and other theoretical physicists were the logical positivists, who held that a statement had meaning only if it could be verified by observation. There are traces of this philosophy in three pages of notes Dirac wrote out by hand in mid-January 1933... [p.220]

Since Dirac was not otherwise interested in philosophy, he never had the perspective necessary to see the problems with Logical Positivism. Briefly, (1) to know whether or not a theory could be verified, one would need to understand it, which means that even theories that could not be empirically verified must have meaning, or their character could not be understood; and (2) the Positivists venerated Hume, and they were aware of Hume's treatment of the fatal Problem of Induction, but they never seemed to understand that this meant that scientific theories could never be "verified," which means they would necessarily all be meaningless. In these terms, and more, Logical Positivism was incoherent and self-refuting.

But Dirac did not have the sensibility of a Positivist. It was more important to Dirac that a theory be mathematically elegant than that it actually passed empirical tests. The mathematical "renormalization" technique of Richard Feynman, which defused the infinities that had plagued quantum electrodynamics, were offensive to Dirac; and he hated them the rest of his life, regardless of the success of the practice.

Although [Pascual] Jordan and [Eugene] Wigner's mathematics [of a field theory of the electron] was similar to Dirac's, their theory did not appeal to Dirac, who could not see how their symbols corresponded to things going on in nature . Their work looked to him like an exercise in algebra, though later he realised he was wrong; his mistake stemmed from his approach to theoretical physics, which was 'essentially a geometrical one and not an algebraic one' -- if he could not visualize a theory, he tended to ignore it. [p.139, color added]

This remarkable statement seems to contradict what Farmelo says, as we have seen, elsewhere. If Dirac was worried about the symbols corresponding to "things going on in nature," then obviously they are not just self-referential symbols. At the same time, we get the view, expressed elsewhere in the book, that Dirac's sensiblity was, not just for mathematical elegance and beauty, but for a particularly geometrical version of this. If he realized that it was a "mistake" to use geometry to dismiss algebra, this still leaves his sensibility as essentially mathematical, with an open question how any of it is going to correspond to things going on in nature. So there is a bit of a muddle here that needs clarifying, something that I'm not sure ever happened.

With astonishing boldness, Heisenberg had abandoned the assumption that electrons can be visualized in orbit around a nucleus -- an assumption no one had previously thought to question [!] -- and replaced it by an purely mathematical description of the electron. [p.84]

First of all, it does not seem like "astonishing boldness" for Heisenberg to abandon the "assumption" that electrons must be conceived, or "visualized," as in orbit around a nucleus. I think it was already obvious that they could not be in orbits, like planets around the sun, since they would be accelerated in such orbits and would, as accelerated charges, radiate away their energy. This was a profound problem for all theories of Rutherford's atom, where the nucleus is small, dense, and positively charged, while electrons somehow fill the rest of the volume of the atom. The model of orbiting electrons is so irresistable that we still constantly see images of it (as on the popular The Big Bang Theory television show); but it never could have been correct. If "no one had previously thought to question" this, then every living physicist was out to lunch -- which strikes me as unlikely. There may have been nothing to do about it -- a situation that often happens in science -- but that did not mean it could be accepted as correct.

Heisenberg wrote that some of the quantities in the theory have a peculiar property: if one quantity is multiplied by another, the result is sometimes different from the one obtained if the sequence of multiplication is reserved. This was exemplified by the quantities he used to represent position and momentum of a piece of matter (its mass multiplied by its velocity): position multiplied by momentum was, strangely, not the same as momentum multiplied by position... Unlike Heisenberg, who had never come across non-commuting quantities before, Dirac was well acquainted with them -- from his studies of quaternions, from the Grassman algebra he had heard about at [Henry] Baker's tea parties, and from his extensive studies of projective geometry, which also features such relationships. [pp.84-85]

What Heisenberg was developing was his "Matrix Mechanics" version of quantum mechanics. As Farmelo mentions elsewhere, Heisenberg actually wasn't even familiar with mathematical matrices, which is why he was surprised that his own mathematics violated the algebraic principle of commutation (i.e. xy = yx). In effect, he was rediscovering a property of matrices. This is actually a nice example of a truth that Farmelo says was appreciated by Dirac, that unusual forms of mathematics may turn out to be the key to something in physics. At the same time, when this happens, it vindicates the Pythagorean and Platonic understanding of the world.

Dirac probably knew about matrices, but he also already knew about William Hamilton and quaternions, where Hamilton was forced to accept that commutation was violated with his strange multiples of imaginary numbers (i, j, and k; with ijk = -1), so that ij = -ji. Heisenberg, of course, was dealing with the sort of autistic mathematical abstraction whose physical meaning was entirely concealed -- and, with "boldness," left unexplained by Heisenberg. Before long, Heisenberg's Matrix Mechanics was matched by Schrödinger's Wave Mechanics, which turned out to be mathematically equivalent, but which also had a physical meaning. Nevertheless, Farmelo, like many other scientists, philosophers of science, and historians of science, has difficulty understanding or expressing the wave nature of matter, despite Niels Bohr's Principle of Complementarity, which is that the "wave/particle duality" means that in each physical event we can understand the operation of a wave, or of a particle, but not both at the same time. I don't think that Farmelo does a good job of explaining this principle, or even of showing that he understands, or accepts, it very well.

I must now display my own ignorance. My understanding is that the introduction of a negation as the result of commutation, as with Hamilton's quaternions, is the key to the Pauli Exclusion Principle of Wolfgang Pauli. I saw this clearly explained just once, when I was watching physics documentaries shown on "Thursday Night at the Physics Movies," put on by the Physics Department at the University of Texas in the late 1970's. Unfortunately, I wasn't taking notes, supposing that the explanation was something otherwise readily available. It wasn't; and in fact I have never seen anything of the sort since in popular expositions of science.

Indeed, we get a similar oversight in Farmelo's book. Thus, even though Dirac is immortal precisely for his equation of the electron, this equation is never given in its full form in the book and its features are never explained in the least way. All we are given is a "succinct version of the Dirac Equation," , such as is carved on the monument to Dirac in Westminster Abbey [p.142]. Not even the elements of this "succinct version" are identified or explained in the book. "When set out in full, in the form he originally used, the equation looked intimidating even to many theoreticians simply because it was so unusual..." [pp.142-143]; but we are not even vouchsafed a peek at so "intimidating" and "unusual" an equation, let alone have any of it explained.

Perhaps this is the Sin of Galileo in action. Farmelo does not explain the equation because it cannot be explained except in its own esoteric mathematical terms, and those of us without the background will just have to leave it to the experts to know what is going on. This is rather different from saying that the mathematics is "just symbols" and a meaningless device for making Positivistic predictions. If it is really so meaningless, how can its formalism not be explained? Or at least shown? Indeed, the obscurity of these modern equations in physics mainly lies in their mathematical detail largely being concealed. Many of the symbols are mathematical operators that, far from being mere symbols, stand in for a great many other equations that unpack the function. These are just never given at the popular level, let alone explained to the ignorant.

Nevertheless, Farmelo gives some explanations that I have not seen elsewhere. The prediction of anti-matter from Dirac's Equation is because energy is expressed in the equation (although not in the forms shown here) as a square, which will have both positive and negative roots:

Dirac's problem was that his equation predicted that, in addition to perfectly sensible positive energy levels, a free electron has negative energy levels, too. This arose because his theory agreed with Einstein's special theory of relativity, which said that the most general equation for a particle's energy specifies the square of the energy, E2. So if one knows that E2 is, say, 25 (using some chosen unit of energy), then it follows that the energy E could be either +5 or -5 (each of them, multiplied by itself, equals 25). So, Dirac's formula for the energy of a free electron predicted that there were two sets of energy values -- one positive, the other negative. [pp.144-145]

This was a troubling feature of Dirac's theory. No one believed there could be negative energy. It could well mean that his equation was gravely flawed, or wrong. And it all depended on the basic rule of algebra that positive numbers have positive and negative roots -- it never occurred to physicists, apparently, that we could just change the rules of algebra (as suggested by Alberto Martínez).

Dirac and others developed different ways of dealing with the required negative energy electrons. Dirac's own strange idea was that space had become filled with negative energy electrons, the equivalent, in a way, of the ether, which generally would be undetectable but occasionally would have a "hole," a missing electron, which would then appear as a positive particle in interactions with other matter. Unfortunately, Dirac at first thought that these positive "holes" in space were actually protons. This bizarre theory, however, was soon vindicated, in part, by observation. Cosmic rays, first observed in cloud chambers, seemed to contain, or to create, particles with the mass of the electron but with a positive charge -- "positrons." If these were the "holes," they were not protons, which were much more massive. But they were something quite as bizarre as was required by Dirac's theory. It was anti-matter, like ordinary matter in mass (and energy) but opposite in charge and all other quantum numbers (the esoteric baryon number, lepton number, strangeness, charm, etc.). This is now a commonplace of science and science fiction. The engines of the starship Enterprise somehow operate by using anti-matter, which releases pure energy when brought into contact with ordinary matter.

Meanwhile, we might still wonder what it would mean if anti-matter is ordinary matter with negative energy -- let alone why the unverse seems to consist of matter rather than anti-matter (a cosmological problem). All we really need is a place to move the negative sign. As it happens, we can just move the negation down to time. Anti-matter is ordinary matter going back in time. Now, in its own way, this is as strange as negative energy. However, the fundamental equations of physics (like those of either Newton or Einstein) are symmetrical in time, so there is nothing particularly outrageous about particles going back in time -- the problem of physics in time is order and entropy, which a lot of physicists and philosophers would rather not worry about. And we don't need to worry about them here, especially when the particles going back in time are not easily and conventionally represented in Feynman Diagrams, which Feynman himself orginated and used to represent the intereactions of particles. In the diagram here we have an electron going forward in time, emitting a photon of energy, and then going back in time as a positron. The photon itself decays into a quark and an anti-quark, with the idea that the anti-quark is coming back in time and being scattered by the photon into a quark, which goes forward in time.

Farmelo does not mention or discuss this interpretation of anti-matter; but, if we don't worry too much about the metaphysics of time, it provides a simple and elegant representation of what "negative energy" could possibly mean. At the very least, it is a nice convention for use in the Feynman diagrams. At best, some like the idea because we could imagine that there is a dearth of anti-matter in the universe because all the anti-matter actually went back in time from the point of the Big Bang. This would make the Big Bang symmetrical in time, which has its own appeal.

In later life, Dirac liked to point out that quantum mechanics was the first physical theory to be discovered before anyone knew what it meant. He spent months on the problem of interpreting its symbols and came to see that the theory was mathematically less complicated than he had first thought. [Max] Born pointed out to Heisenberg that each array of numbers in his quantum theory was a matrix, which consists of numbers arranged in horizontal rows and vertical columns that behave according to simple rules spelt out in textbooks. Heisenberg had never heard of matrices when he discovered the theory, as Born often reminded his colleagues... [p.96]

However, one could easily say that Newton's theory of gravity was also a physical theory that was formulated before anyone knew what it meant. The "action at a distance" nature of Newtonian gravity was perplexing and objectionable to many, while that very feature of it has been serendipitously eliminated, as we have seen above, in both Einstein's Relativity and in quantum mechanics. At the same time, if Dirac decided that quantum mechanics was "mathematically less complicated than he first thought," because it can be arranged as mathematical matrices, as Born pointed out to Heisenberg, it is not clear how this gives us any notion of what quantum mechancis, as a "physical theory," is supposed to mean. Heisenberg's matrices were precisely how he could avoid explaining what the theory meant.

I listened to their arguments, but I did not join in them, essentially because I was not very much interested [...] It seemed to me that the foundation of the work of a mathematical physicist is to get the correct equations, that the interpretation of those equations was only of secondary importance... Yet, whereas Einstein remained interested in philosophy, for Dirac it was a waste of time. What Dirac had retained from his reading of Mill, bolstered by his studies of engineering, was a ultilitary approach to science: the salient question to ask about a theory is not 'Does it appeal to my beliefs about how the world behaves?' but "Does it work?' [p.137]

So here, beginning with Dirac's response to Bohr and Einstein arguing about the meaning of quantum mechanics in 1927, we are back to a metaphysical, or even as physical, Know-Nothing-ism. The equations seem to exist for Dirac in their own reality; and if physics is a study of the world, the actual explanation that we end up with in those terms is actually "of secondary importance." Now, if Dirac were a Platonist, like Gödel, this would make a little more sense. The mathematics of the world would be the nature of the world. But it doesn't look like he was. So the result is a Skeptical (epoché), a suspension of judgment.

However, Dirac could not even be consistent about this:

According to Heisenberg, Dirac thought religion was just 'a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.' For Dirac, 'the postulate of an Almighty God' is unhelpful and unnecessary, taught only 'because some of us want to keep the lower classes quiet'. Heisenberg wrote that he objected to Dirac's judgement of religion because 'most things in this world can be abused -- even the Communist ideology which you recently propounded'. Dirac was not to be deflected. He disliked 'religious myths on principle' and believed that the way to decide what was right was 'to deduce it by reason alone from the situation in which I find myself: I live in a society with others, to whom, on principle, I must grant the same rights I claim for myself. I must simply try to strike a fair balance.' [John Stuart] Mill would have approved. [p.138]

Now, having decided that the meaning of a physical theory is of "secondary importance," Dirac passes judgment on religion as having "no basis in reality." Didn't we just see that he wasn't interested in reality? But if he was not interested in reality, then his inability to see anything in religion would make sense. So Dirac needed to make up his mind. At the same time, while his desire to deduce "what was right" by "reason alone" would be agreeable to John Locke, the "Communist ideology" that seemed to always be tempting Dirac would not have been commensurable with Locke's liberalism. In the same way, Mill's principle of liberty that for others "I must grant the same rights I claim for myself," would not be consistent with "Communist ideology" either. In any case, it is not clear how the principles of either Locke or Mill could be "deduced" from "the situation in which I find myself," which, as a matter of fact, could not logically imply, according to Hume, any moral principle at all, let alone one that established maxims of bourgeois morality rejected by Marx.

But then, as we see also in another physicist who expressed no interest in philosophy whatsoever, namely Richard Feynman, we cannot expect Dirac to have very sophisticated ideas about either metaphysics or ethics. Like Feynman, he has trouble even maintaining consistency when venturing into the perilous territory of those matters. Yet he occasionally had some things to say about them anyway, unlike Feynman. But those forays were indeed rare; and even his trips to the Soviet Union and friendship with Russian physicists seemed to be largely non-political. So, as we might expect, we never do get a systematic account from Dirac about science, reality, or morals.

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