As with Bohm, Goldstein too began his career advocating Bohr’s views. He was studying at Yeshiva University in New York in the late 1960s and early ’70s. “I was a fairly strong defender of the Copenhagen interpretation, to the extent that I understood it,” Goldstein told me when we met at Rutgers University in New Jersey on a miserably rainy day. He invited me into his long, narrow office—one side of which was lined with bookshelves filled with books on quantum mechanics. Through the large windows at the far end of the office, I could see the gray sky and the occasional skein of geese flying past. Stuck to the bookshelves were clippings of newspaper articles about Alan Sokal, a New York University professor who in 1996 duped a social studies journal into publishing what turned out to be gibberish, to prove a point that such journals would publish nonsense. A white T-shirt hung from one of the bookshelves; it had an image of Bohm, with the words David Bohm, Keepin’ it real . Goldstein sat down on his swivel chair, leaned back, and put his legs up on the table, clasped his hands behind his head, and proceeded to talk for two hours, getting up only to scribble equations on the blackboard or to grab a much-thumbed copy of Speakable and Unspeakable in Quantum Mechanics by John Bell, its dust jacket in tatters, and quote entire paragraphs to me.
“I wanted Bohr and Heisenberg and orthodox quantum theory to be right, and Einstein to be wrong,” he said.
“You wanted that?” I asked.
“Yeah, I wanted Einstein to be wrong,” said Goldstein. “I’m not too proud of that, by the way. I was excited about the quantum revolution, and Einstein was presented as somebody who wanted to go back to old-fashioned classical ways of thought. He just couldn’t get with the new modes of thinking; he was too old.”
Goldstein then expressed some remorse for those thoughts about Einstein. “I think that was very unfair, but anyway, that’s what I thought then,” he said. “You could say I wasn’t smart enough to see what a bunch of crap that was, so I swallowed it. I thought if I learned the mathematics better and looked into it carefully, I would really understand it all one day. [But] the more I learned, the more clear it became that we were all hoodwinked.”
Strong words, but not unusual from those who have developed a distaste for the orthodoxy.
As Goldstein probed further into the mathematics of standard quantum theory, he was unable to make sense of what it’s about. What are the fundamental entities of reality? Is it a theory about particles? Is it about waves? Is it a theory of measurements and observations? Is it a theory of wavefunctions? Is the wavefunction ontic (meaning it is something ) or is it epistemic (in that the wavefunction represents our knowledge about something); is the wavefunction objective or subjective?
Goldstein wasn’t done expressing his concerns about orthodox quantum mechanics. “Are there particles before you look? Do they have positions before you look? According to textbook quantum mechanics, presumably not. Then what do you have before you look? Or does looking create reality? Is that clear from the usual theory, textbook theory? No, it’s not.”
Goldstein used the double-slit experiment to further make his point about the “reality” of the wavefunction. “I don’t see how you can understand the interference unless you take seriously that you have a wavefunction, an objective thing in the world, which has these two pieces, one going through the upper slit, and one going through the lower slit, and they interfere with each other,” he said.
Disenchanted with the Copenhagen interpretation, Goldstein turned to work by a mathematical physicist at Princeton named Edward Nelson, who had proposed a theory called stochastic mechanics to arrive at a realistic theory of the quantum realm. The theory had actual particles in it, with positions and momenta, and these particles were being randomly buffeted by the wavefunction—resulting in a sort of Brownian motion. It wasn’t deterministic and it reproduced the results of standard quantum theory, albeit after many mathematical contortions. Goldstein found it enticing but soon realized that it was too complicated and that there was something simpler hiding in Nelson’s proposal.
And even as he started figuring out the simpler idea, he had a vague notion that “there was this guy David Bohm” who had proposed a deterministic formulation of quantum theory, one with hidden variables. Goldstein discovered that the idea that he was playing with—making Nelson’s stochastic mechanics simpler and deterministic—was exactly what Bohm had already clearly elucidated. Here was an alternative to the Copenhagen view of things: a deterministic theory of particles that move around because of interactions with the wavefunction, which in turn is a “real” thing and evolves according to the rules of the Schrödinger equation.
Bohm’s theory has a definite ontology: the world is made of particles and wavefunctions, even if wavefunctions are not “physical” in the sense that particles are physical, but nonetheless are real, objective aspects of nature. A particle has a definite position at all times, which means it has a trajectory—in direct contravention of the Copenhagen view of reality. The particle is “guided” by the wavefunction, and thus influenced not just by the usual forces (such as electromagnetism), but by a “quantum potential,” a new force felt by the particle because of its interactions with its wavefunction. Moreover, the theory is deterministic: given a particle’s position and its wavefunction, you can predict the particle’s position at some later time. And even more emphatically, the particle’s trajectory is objective reality—it exists independent of an observer.
And what of hidden variables? In Bohm’s theory, the much-maligned hidden variables are nothing other than the positions of particles. To those who think Bohm is right, it’s an irony that this rather obvious property has to be called “hidden”: it’s called so because it doesn’t appear in the standard formalism of quantum mechanics, unless “observed.”
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Just as Goldstein discovered that his nascent ideas had already been worked out by Bohm, Bohm would discover that his theory wasn’t entirely novel either. Louis de Broglie, the young French prince whom we encountered earlier, had made the first clear attempt at a theory that incorporated both realism and determinism, back in the 1920s. Recall that in 1924, de Broglie came up with the theory that particles of matter such as electrons had wavelike properties. Then in 1927, de Broglie presented another radical idea at the Fifth Solvay Conference in Brussels—that reality is made of particles and that these particles are being guided by a “pilot wave,” which behaves like a wavefunction and evolves according to a form of the Schrödinger equation. So de Broglie was proposing that reality isn’t wave or particle, as Bohr was arguing, but rather it’s wave and particle. At the Solvay meeting, Wolfgang Pauli—who sided with Bohr—ripped into de Broglie’s theory, claiming to point out certain experimental situations that it couldn’t explain. A disheartened de Broglie gave up on the pilot-wave theory, and actually became a supporter of the Copenhagen interpretation.
Until, that is, Bohm entered the picture. Bohm, unaware of de Broglie’s work, had reinvented the theory, but with far greater conceptual and mathematical clarity. Einstein and Pauli both alerted him to de Broglie’s work. Pauli, in particular, raised some of the same issues that he had brought up after de Broglie’s presentation in Brussels. But Bohm, unlike de Broglie, did not back down. He revised his draft to address Pauli’s concerns and sent it to Pauli, who apparently did not read it because it was too long. Bohm wasn’t amused. He sent Pauli a rather stern note: “ If I write a paper so ‘short’ that you will read it, then I cannot answer all of your objections. If I answer all of your objections, then the paper will be too ‘long’ for you to read. I really think that it is your duty to read these papers carefully.”
As for giving de Broglie his due, Bohm did so somewhat reluctantly. In his 1952 paper, he acknowledged that he had been alerted to de Broglie’s work after he had completed his paper, and that de Broglie had abandoned his approach following criticism from Pauli and after de Broglie had himself realized what he took to be some of the theory’s shortcomings. “ All of the obj
ections of de Broglie and Pauli could have been met if only de Broglie had carried his ideas to their logical conclusion,” wrote Bohm.
He put this argument rather more colorfully in a letter to Pauli: “ If one man finds a diamond and then throws it away because he falsely concludes that it is a valueless stone, and if this stone is later found by another man who recognizes its true value, would you not say that the stone belongs to the second man? I think the same applies to this interpretation of the quantum theory.”
To Bohm’s credit, he did push the ideas to their logical conclusion and the result was the first deterministic, realistic, hidden variable quantum theory. As Bell subsequently said, Bohm had done the impossible.
Today, the pilot-wave theory is often referred to as the de Broglie-Bohm theory. De Broglie, once he became acquainted with Bohm’s work, left the Copenhagen camp and started working on a variant of his own idea called the double-wave solution, something he had started on in 1926 but had given up as being too difficult.
After decades in exile, both the de Broglie-Bohm pilot-wave theory (which Goldstein favors) and de Broglie’s double-wave solution are getting some attention and even support. The latter from an unlikely group of researchers studying how droplets of silicone oil bounce on a vibrating surface of the same oil. What, you might ask, has that got to do with quantum physics?
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As a graduate student, John Bush felt the same annoyance with quantum mechanics as he did with Sunday school growing up in London, Ontario, Canada. There were some questions that were off-limits when it came to religion. Unfortunately for him, he encountered similar attitudes when learning quantum mechanics and the Copenhagen interpretation. “You are telling me that the particle doesn’t exist unless you observe it?” he’d ask. The instructor would go, “You can’t ask that question.” Bush felt he was back in Sunday school.
It was galling to Bush that human observers could somehow be held responsible for creating quantum reality. It still galls him. “This is the latest in the long line of epic human intellectual follies that have resulted from man putting himself at the center of the universe,” he said when we met at his office at MIT. “It strikes me as nonsense.”
Disillusioned with quantum mechanics, Bush ended up studying fluid mechanics. Little did he know that his chosen field would lead him back to quantum mechanics. The impetus came from the 2006 work of two French researchers, Yves Couder and Emmanuel Fort. They had conjured a curious setup. Imagine a petri dish filled with silicone oil that’s being vibrated up and down. These vertical vibrations of the bath of oil are kept below what’s called the Faraday threshold for the fluid. Above this threshold, waves form on the surface, but below the threshold, the surface remains smooth, even though there is vibrational energy in the fluid. The researchers discovered that if they let a millimeter-size droplet of the same oil fall onto the vibrating surface, the droplet would keep bouncing and begin wandering across the surface.
Here’s why. A thin cushion of air between the droplet and the surface prevents the droplet from coalescing into the oil bath. Upon first impact, the vibrating surface gives the droplet a vertical kick, causing it to bounce up. The impact also creates a small wave on the bath surface. When the droplet falls back onto the surface, it encounters this wave. This time, the droplet gets both a horizontal and a vertical kick, and the process now keeps repeating. The droplet starts “walking” over the surface, guided by the very wave it creates and sustains with each bounce. The wave dictates the droplet’s speed and direction.
The analogy with the theories of de Broglie and Bohm is hard to ignore. The droplet is a particle being guided by its pilot wave. What else can one do at this stage but carry out a version of the double-slit experiment?
Couder and Fort did just that. They made a barrier with two openings and submerged it fractions of a millimeter below the oil surface, such that anything moving on the surface would be influenced by the barrier. The subsurface barrier made for a double slit. When the walking droplet approached the barrier, it went over one or the other opening (like a particle going through one slit or the other). The attendant pilot wave, however, spanned both openings and thus went over both. When the wave emerged on the other side of the submerged barrier, it was now the outcome of the interaction between two diffracted waves, each influenced by one subsurface slit. This more complex wave now guided the bouncing droplet away from the barrier. For each run of the experiment, the droplet went to a different location on the far side. The researchers collected seventy-five such trajectories, and their initial analysis showed that the droplets were going to some places and not to others—suggestive of an interference pattern. Despite there only ever being one particle-like droplet in the apparatus at any one time, its accompanying pilot wave was causing the droplet to behave as a wave. If you didn’t know about the pilot wave, you’d think the droplet had gone through both slits and was interfering with itself.
Had Couder and Fort done an actual double-slit experiment using a bouncing droplet of silicone oil? Had they found a classical analogue of what happens in the quantum world? Other teams raced to duplicate the results, but failed. One team was led by Tomas Bohr, Niels Bohr’s grandson, at the Technical University of Denmark near Copenhagen. Another was led by John Bush and his team at MIT. Their results revealed inadequacies in the experiment done by Couder and Fort. Bohr and colleagues showed that the French team’s statistics were inadequate—seventy-five trajectories were just too few to make strong claims about what the droplets were doing. And Bush’s team pointed out that the French experiment hadn’t been adequately sealed off from environmental influences, so the droplet’s patterns on the surface, for example, may have been affected by ambient air currents.
When the MIT researchers did a more rigorous version of the experiment, they did not see the double-slit interference pattern. Nor did they see the kind of diffraction patterns expected when a particle goes through a single slit. They attribute this to “boundary conditions”: the interaction of the droplets and the waves with the walls of the petri dish, for example, making it difficult to reproduce the conditions that would be experienced by, say, a photon going through a double slit, where there are no such boundary effects. Maybe future experimentalists can come up with walking-droplet setups that negate any effects of physical boundaries. “ Our results do not close the door on the quest for diffraction and interference of walking droplets,” Bush’s team concluded in one of its papers.
But Bush told me that they do see the kind of mystery highlighted by Richard Feynman’s analysis of the double-slit experiment. In the quantum mechanical version, when both slits are open, the particle goes to certain places on the far side and not to others. Close one of the slits, and the behavior of the particle changes, as if the particle senses the closing of one slit. The classical walking droplet does the same, even though it does not exactly replicate an interference pattern in where it goes and does not go. It’s fair to say that the droplet, which is going over only one opening in the barrier or the other, nonetheless can “sense” whether both slits are open or not. “Our system has that feature, if that’s the mystery,” said Bush.
Bush thinks the walking-droplet setup is an important classical analogue of a quantum mechanical system. They may not have replicated the double-slit experiment yet, but they are seeing phenomena that are too suggestive to ignore. For example, when they follow the seemingly chaotic movements of a droplet in the circular bath, over time its statistics resemble those of an electron moving inside a quantum mechanical corral of atoms.
Bush explains this result using de Broglie’s double-wave solution. De Broglie revisited this idea, which he had abandoned after the 1927 Solvay Conference, when Bohm revitalized the single pilot-wave theory in 1952. The double-wave solution is what it says: there are two waves involved in guiding a particle. One is a localized wave, and the particle is centered on that wave and is guided by it. This particle-wave combination gives rise to another wave that be
haves like the wavefunction in orthodox quantum mechanics.
According to Bush, the vibrating oil bath and the walking droplet physically replicate this two-wave system. The bouncing droplet creates and sustains the pilot wave. This wave is localized and the droplet is centered on the wave. The interaction of the droplet and pilot wave with the geometry of the vibrating surface also creates another wave pattern whose properties emerge over time and mimic those of a wavefunction. “Now we have a macroscopic realization of the physical picture suggested by de Broglie, and it exhibits many of the allegedly inscrutable features of quantum mechanics,” Bush told me. “That’s a hell of a coincidence.”
There’s a chance that that’s all it might be: a coincidence. Bush isn’t overly concerned. The main thrust of his argument is that physicists have to challenge the Copenhagen interpretation, and anything that gets them to do that is worth the effort. “That’s why I’m a believer in this venture, even if its sole result is to get young people to question their views on quantum mechanics,” Bush said.
To other quantum physicists, even those who are anti-Copenhagen, the idea that a classical system can replicate all the features of quantum mechanics is a hard sell. Goldstein, for one, thinks that the walking droplets can never replicate the key feature that distinguishes the quantum world from the classical: nonlocality, which depends on the wavefunction that, for a system of two or more particles, doesn’t live in the familiar three-dimensional space of physical things.
Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality Page 13