by Lee Smolin
There is no Rule 2 in this framework. But there is a new law that directs the particle to follow the wave, which is called the guidance equation. The system defined by the wave function together with the particle evolves deterministically, which suggests it is complete.
In other approaches to quantum mechanics, it is simply posited that the particle will be found where the wave is large. More precisely, the probability of finding the particle at some particular place is proportional to the square of the wave function there. This is what we earlier called the Born rule.
In pilot wave theory it is also the case that the particle is more likely to be found where the wave is high. But this is not posited. Rather, it turns out to be a consequence of the law that drives the particle to follow the wave.
Place a ball on a hillside and watch it roll down from there. You may observe that the ball tends to follow the steepest path downward. This is called the law of steepest descent. Roughly speaking, de Broglie’s guidance equation does the opposite, guiding the particle on the steepest path to climb the wave function.* We can call it the law of steepest ascent. A mountain climber following this law would at each moment of her climb choose to go in the direction of the steepest slope of the mountain.
De Broglie was able to demonstrate that the probability law posited by Max Born is a consequence of the particle following the steepest ascent. To illustrate this important point, imagine that you throw a bunch of particles down on a hillside representing the wave function. Wherever you throw them, the particles will quickly arrange themselves so that they are more likely to be found where the square of the wave function is largest, which reproduces Born’s law.
FIGURE 8. SQUARING THE WAVE FUNCTION The dashed line denotes a wave traveling along the horizontal line to the right. Note that it spends as much time with negative values as it does with positive. The solid line is the square of the wave, which is always above zero.
The pilot wave theory predicts everything quantum mechanics does, but explains a good deal more. The mysterious way in which the ensemble seems to influence the individual is cleared up and explained straightforwardly as the influence of the wave on the particle. Both are real, and both exist for every individual atom. Everything that was puzzling and mysterious about quantum mechanics is revealed to be a consequence of that theory leaving out half of every story.
Despite what Bohr and Heisenberg say, the electron always has a position and it follows a definite trajectory, which is perfectly predictable if you know the right law. No need for operationalism, and no sense wasting time trying to make sense of Bohr’s obscure pronouncements on complementarity. Waves and particles don’t contradict each other; instead, both are always present and they work together to explain atomic physics. What is, simply is.
There is an alternative history in which all the bright, ambitious students flocked to Paris in the 1930s to follow de Broglie, and wrote textbooks on pilot wave theory, while Bohr became a footnote, disparaged for the obscurity of his unnecessary philosophy. It was, alas, not to be. But why the convoluted philosophy of complementarity triumphed, while it was de Broglie’s pilot wave theory which became the forgotten footnote, is a question to be pondered.
The pilot wave theory overlaps with quantum mechanics, but it also differs on several points. Rule 1 is common to quantum mechanics and pilot wave theory. But pilot wave theory differs from quantum mechanics in having no Rule 2. Instead there is a law to guide the particle. The laws of pilot wave theory are deterministic.
With no Rule 2, the quantum state in pilot wave theory never collapses. This has some strange consequences which took its adherents some time to appreciate, and to which we will return in the next chapter.
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AT THE SOLVAY CONFERENCE the talks were followed by discussions, and these were transcribed and published in a book with the talks. There is not much evidence that de Broglie’s presentation changed minds, although it was discussed. One person who did get it and did comment was Einstein.
Although he doesn’t say so in the transcribed discussion, Einstein had himself thought of the idea of pilot wave theory. In May 1927 Einstein gave a talk to the Prussian Academy of Sciences in which he presented a rather complicated version of the pilot wave idea. He discussed the idea in correspondence with Heisenberg and others and submitted a paper based on the talk for publication. But just before it was to appear, Einstein withdrew his paper.* He had apparently realized his version of pilot wave theory had several problems, some of which prevented the theory from reproducing all the predictions of quantum mechanics. So far as is known, he never mentioned it again.
Einstein had been scheduled to give a talk at the Solvay conference, probably about that paper. He backed out of that talk at the last minute, writing to the conference organizer, “I kept hoping to be able to contribute something of value in Brussels; I have now given up that hope. . . . I did not take this lightly but tried with all my strength.”1
Einstein nonetheless did attend, and, of course, he contributed to the discussions that took place about the new quantum theory. Among them were the first discussions between him and Bohr in which Einstein tried to find inconsistencies in the new quantum mechanics. These intense discussions were informal and unfortunately were not transcribed. But much later, Bohr published his reminiscences of those discussions, in a paper that is both one of the most compelling reads in the history of physics and a masterpiece in scholarly propaganda.
During the meals and breaks of the conference, Einstein presented Bohr with several arguments that quantum mechanics is inconsistent. He posited that to give a complete description would require additional variables, which are hidden in the quantum mechanical description. Bohr doesn’t mention that this is what de Broglie had achieved with his pilot wave theory. On the contrary, in Bohr’s telling he was able, after a sleepless night, to refute Einstein’s objection, leaving in place his view as to the consistency and even the inevitability of complementarity.
Einstein responded positively during the discussion of de Broglie’s talk. After describing an objection to the Copenhagen version, he said, “In my opinion, one can remove this objection only in the following way, that one does not describe this process solely by the Schrödinger wave, but that at the same time one localizes the particle during the propagation. I think that Mr. de Broglie is right to search in this direction.”2
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FEW QUANTUM PHYSICISTS MENTIONED de Broglie’s theory in the years after its presentation in 1927. In spite of de Broglie being justly admired for his insight of extending the wave-particle duality to matter, and in spite of his having presented the pilot wave theory at the most important conference on quantum physics, with an audience of virtually everyone who mattered in atomic physics, it was as if de Broglie had never published or presented his theory. So far as I know, no textbooks mentioned it for decades after. It is not that there were Copenhagen textbooks and pilot wave textbooks. There were only Copenhagen textbooks. These either ignored the foundational issues with the theory or presented a confident assertion that all questions that were meaningful had already been answered by Bohr and Heisenberg.
One important reason anti-realism triumphed was that the mathematician John von Neumann published a proof he claimed showed there could not be a consistent alternative to quantum mechanics. This was published a few years after the Solvay conference in a book on the mathematical structure of quantum mechanics. This claim had to be wrong, as it implied de Broglie’s pilot wave theory had to be inconsistent, which it wasn’t. You might have thought that someone would have mentioned this.
Von Neumann’s incorrect proof seems to have been one of those cases that happens far too often in the history of science, where a result is as influential as it is wrong. Von Neumann had a formidable reputation, and in the face of his theorem, opposition to the view that quantum mechanics was th
e most complete theory possible caved. In particular, de Broglie himself capitulated to the combined criticisms of von Neumann and other theorists, including Wolfgang Pauli.
It is not quite true that nobody noticed that von Neumann’s theorem contained a mistake. A young mathematician called Grete Hermann took an interest in quantum mechanics and was naturally drawn to study von Neumann’s book. A good mathematician in her own right, Hermann was a PhD student of Emmy Noether,* and among her accomplishments are several results which anticipated the modern study of algorithms in computer science. She also had a keen interest in philosophy and was concerned with the implications of quantum mechanics for the neo-Kantian philosophy then popular in the German-speaking academy.
Grete Hermann quickly noticed a mistake in the proof of the theorem on the impossibility of hidden variables in von Neumann’s book. One of the assumptions of the theorem was already equivalent to the basic structure of quantum mechanics. So all the theorem proved was that any theory equivalent to quantum mechanics would turn out to be equivalent to quantum mechanics.
Very unfortunately, the paper she wrote exposing the fault in von Neumann’s proof had no impact.3 Part of the reason may have been that she published it in an obscure journal, but it is hard to avoid the thought that she wasn’t taken as seriously as she might have been due to her gender, as well as to the fact that her paper punctured one of the main arguments used to establish the inevitability of quantum mechanics.
It took two long decades for someone else to notice that von Neumann’s proof had to be wrong, because it disagreed with the manifest existence of pilot wave theory. This was David Bohm, who will be the protagonist of the next chapter. Ten years after that, John Bell isolated the error as an erroneous assumption. Here is how he put it:
[T]he von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It’s not just flawed, it’s silly. . . . When you translate [his assumptions] into terms of physical disposition, they’re nonsense. You may quote me on that: The proof of von Neumann is not merely false but foolish!4
David Mermin, in a lucid review of various impossibility theorems, regretted the “many generations of graduate students who might have been tempted to try to construct hidden variables theories [who] were beaten into submission by the claim that von Neumann . . . had proved that it could not be done.” Mermin “wonder[ed] whether the proof was ever studied by either the students or those who appealed to it to rescue them from speculative adventures.”5
It is hard now, looking back from our present vantage point, in which several competing views about how to understand quantum theory flourish, to appreciate the state of mind of the first several generations of quantum physicists. In spite of the persistent and powerful dissents of Einstein, de Broglie, and Schrödinger, for at least the first half century following the invention of quantum mechanics in 1925, the anti-realist philosophy initiated by Bohr dominated all discussions of quantum theory.
Through all those years, if someone raised the possibility of a realist version of quantum mechanics, the response, I was told, was a good dose of Copenhagen-speak which, if one persisted, was capped off with “Von Neumann proved there is no alternative.” One can imagine it would have changed things at least a little if Grete Hermann’s paper showing that no, von Neumann hadn’t proved anything, had been known. But it simply wasn’t.
EIGHT
Bohm: Realism Tries Again
In 1952, David Bohm solved the biggest of all problems in quantum mechanics, which is to provide an explanation of quantum mechanics. . . . Unfortunately, it is widely under-appreciated. It achieves something that was often (before and even after 1952) claimed impossible: To explain the rules of quantum mechanics through a coherent picture of microscopic reality.
—RODERICH TUMULKA
By 1930 de Broglie had given up. From then on, the anti-realist Copenhagen interpretation dominated the teaching and application of quantum mechanics, as well as most discussion of the new theory’s implications. The only significant exceptions were Einstein and Schrödinger, who continued to challenge the Copenhagen school and insist on the need for a realist formulation of quantum theory. But their dissent had little impact.
That was the situation in the early 1950s when the young American theorist David Bohm set out to write a textbook on quantum mechanics. Bohm was an interesting character destined to have an interesting life. At that point he was an assistant professor of physics at Princeton University, specializing in plasma physics. He had come to Princeton from Berkeley, where he had been a student of J. Robert Oppenheimer. Like many people around Oppenheimer, he had been a communist sympathizer and briefly a Communist Party member before the war. As a result, the U.S. Army had refused Oppenheimer’s request to bring Bohm along to work on the atomic bomb at Los Alamos.
There is no evidence that Bohm was ever a spy or a Soviet agent, but, like others with integrity, when called in 1950 to testify before the House Un-American Activities Committee, he asserted his Fifth Amendment rights and so avoided informing on others. He was arrested and charged with contempt of Congress, but acquitted. Princeton, to its shame, suspended, and then declined to renew, his faculty appointment.
Einstein proposed appointing him at the Institute for Advanced Study, but was unable to overcome opposition from its administration. At that very moment, when Bohm found himself unemployed and, in the United States, likely unemployable, his textbook was published to high praise.
There has been no shortage of textbooks published on quantum mechanics since the first, by Paul Dirac, one of the inventors of the theory, which appeared in 1930. Bohm’s is one of the best. And despite persistent doubts over many years, when he discussed interpretational issues he kept close to the Copenhagen orthodoxy. One section of his book was titled “Proof that quantum theory is inconsistent with hidden variables.” Another was about the “Unlikelihood of completely deterministic laws on a deeper level.”
Einstein summoned him. He expressed his admiration for the lucidity of Bohm’s defense of the Copenhagen view, but asked for a chance to explain his point of view and perhaps change Bohm’s mind.
It appears Einstein succeeded. After talking with Einstein, Bohm began to think about whether there might be a deeper theory, which was realist and deterministic. Perhaps it was the appeal of realism to a Marxist materialist; perhaps it was the clarity of Einstein’s thinking. But it didn’t take long for Bohm to invent a realist completion of quantum mechanics. What he did was, basically, to reinvent de Broglie’s forgotten pilot wave theory.
There is, it should be mentioned, a difference between de Broglie’s and Bohm’s theories, in that Bohm chose a different law for the guidance equation by which the wave guides the particle. As I explained above, de Broglie’s guidance equation has the particle taking the path of steepest ascent up the wave function. This determines the speed and direction of motion of the particle.
In Bohm’s theory, the law that guides the particle is a version of Newton’s law of motion: it describes how a particle accelerates in response to a force. What is new is that there is a force which guides the particle to move to where the wave function is largest. In addition, Bohm has to assume one more condition, which is that at the initial moment, the velocities of the particles are those given by de Broglie’s guidance equation.
Apart from this difference, de Broglie’s and Bohm’s theories are different versions of the same idea, which is that both the wave function and the particles are real, with the waves guiding the particles. As presented originally, they are equivalent in that they predict the same trajectories for the particles. As a result, both theories predict that if an ensemble of particles starts off distributed according to Born’s rule, that rule will continue to be satisfied as the wave function changes and the particles move around.
It didn’t take long for Bohm to write two papers presenting his new theory.1 He submit
ted them to the most prestigious journal at that time, Physical Review. He also sent drafts to several people, including de Broglie, who quickly published a short article explaining why Bohm’s theory, like his own previous proposal, was wrong.
Bohm added a very interesting sentence to his manuscript: “After this article was completed, the author’s attention was called to similar proposals for an alternative interpretation of the quantum theory made by de Broglie in 1926, but later given up by him.”
This sentence certainly claims that he didn’t know of de Broglie’s pilot wave theory when he invented his own version. This in itself is a little shocking, given that de Broglie was a world-famous Nobel Prize winner, universally recognized for having proposed that electrons and other particles have waves. But there it is.
Bohm also devoted a section of his second paper to explaining why von Neumann’s theorem doesn’t apply to the theory he is proposing.
Bohm’s first paper on the pilot wave theory appeared in January 1952. By then he had taken a professorship in São Paulo, Brazil. From that far remove, lonely and sick from the unfamiliar food, he waited as the responses to his revolutionary papers drifted in by letter.
One person Bohm might have hoped for support from was Einstein. The great savant had, after all, praised pilot wave theory when it was first presented by de Broglie. But, apparently, by the time Bohm published his papers, twenty-five years later, Einstein had changed his mind.
Einstein described his reaction in a letter to Max Born: “Have you noticed that Bohm believes (as de Broglie did, twenty-five years ago) that he is able to interpret the quantum theory in deterministic terms? That way seems too cheap to me.”2
He went on, “This path seems to me too easy.” It is a “physical fairy-tale for children, which has rather misled Bohm and de Broglie.”3
