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Goldstein has an acronym for the Bohmian view of the quantum world (a “terrible acronym,” he acknowledged): OOEOW. It stands for the “obvious ontology evolving the obvious way.” Since Bohm came up with his formulation of quantum theory in 1952, there have been many tweaks to the theory, most notably by Bohm himself in collaboration with Basil Hiley, who worked with Bohm during the final phase of Bohm’s career at Birkbeck College in London (Bohm moved from Israel to the UK and stayed put). Others who have contributed to the growing understanding of Bohm’s ideas include English physicist Peter Holland and the team of Goldstein, Detlef Dürr, and Nino Zanghi. While the details differ (and contentiously so), the essence is captured by Goldstein’s acronym. Dürr coined the term Bohmian mechanics . It’s a term that Hiley does not like, but we’ll stick to it in this chapter, to avoid getting tangled up in the subtleties of the different points of view. So, according to Bohmian mechanics, the quantum world consists of particles with definite positions and a wavefunction that guides these particles. If you have a system of N particles, each particle has a position. There is, however, only one wavefunction, and each particle is being influenced by it. The particles have coordinates in three-dimensional space, but the wavefunction doesn’t operate in the same 3-D space. Instead, it does so in something physicists call configuration space. Take just two particles. Each particle has a coordinate in 3-D space that describes its position (x1,y1,z1 for particle 1 and x2,y2,z2 for particle 2). The wavefunction for the two particles, however, needs all six numbers (x1,x2, y1,y2, z1,z2) to describe the state of the system, and thus has to contend with six mathematical dimensions. The actual positions of the two particles in 3-D space correspond to one point in the 6-D configuration space.
It’s easy to see how the number of dimensions of the configuration space quickly mushrooms as the number of particles increases—making it impossible to visualize. Nonetheless, no matter how many particles there are in the system, their individual positions in 3-D space ultimately correspond to just one point in the 3N-dimensional configuration space, where N is the number of particles.
Despite this mathematical abstraction, the wavefunction is real in Bohmian mechanics. It propagates in configuration space and evolves according to the Schrödinger equation. And it simultaneously exerts an influence on each and every particle. The wavefunction determines the trajectory of every particle. And because this interaction between the wavefunction and the particles is not taking place in 3-D space but within the confines of configuration space, the interaction is instantaneous. This is how nonlocality is built into the very fabric of Bohmian mechanics. Many have argued that this profound nonlocality (a particle in some distant galaxy, in principle, is instantly influencing a particle here on Earth) makes Bohm’s ideas untenable. Even though Bohm was aware of the nonlocality built into his theory, it was John Bell who was the first to give it serious thought. He wondered if he could get rid of it. “He proved you couldn’t,” Goldstein said.
The tests of Bell’s inequality carried out by Clauser, Aspect, Zeilinger, and others have ruled out local hidden variable theories. The correlations in the outcomes of measurements carried out on, say, two entangled particles cannot be explained by theories that posit local hidden variables, meaning variables that are not present in standard quantum theory and whose values evolve in a local manner, unaffected by what’s happening at a distance. These tests, however, do not rule out nonlocal hidden variable theories, of which Bohmian mechanics is an example. The position of a particle, the hidden variable in the theory, is nonlocally influenced by the positions of all other particles, mediated by the wavefunction. Bell was keenly aware that his theorem did not address nonlocal theories such as Bohm’s. “In fact, Bell repeatedly stressed that any serious version of quantum mechanics, and even orthodox quantum mechanics, must be nonlocal,” Goldstein told me.
So, even though orthodox quantum mechanics wins out when it’s pitted against local hidden variable theories, the outcome of a tussle between the orthodoxy and Bohmian mechanics is far from resolved. It has proven impossible to experimentally disprove Bohm’s ideas, because the theory makes exactly the same predictions as orthodox quantum theory.
Bohmian mechanics makes a case for itself in other ways. Some aspects of quantum theory, which are axiomatic in standard quantum mechanics, can be derived in Bohmian mechanics. For example, the uncertainty principle can be shown to be an outcome of not knowing enough about the exact initial conditions of a system. So, despite the fact that Bohmian mechanics is deterministic, inadequate knowledge can make precise predictions impossible. This is not unlike chaos theory in classical mechanics: a small initial perturbation can lead to wildly different eventual outcomes in the evolution of a chaotic system (such as weather), making the system appear nondeterministic. In fact, even with all the challenges to accurate weather prediction, it is nonetheless a deterministic system.
Even the collapse of the wavefunction, which is such a bugbear in standard quantum theory—in that we don’t know what it really means or whether a physical collapse actually happens—can be deduced in Bohmian mechanics. Consider Schrödinger’s cat. It can be described by the positions of the “N” particles that make up the cat and its wavefunction. In Schrödinger’s thought experiment, when the cat ends up in a superposition of being dead and alive, orthodox quantum mechanics requires a measurement (or observation) to collapse the wavefunction from its state of superposition to one in which the cat is either dead or alive. In Bohmian mechanics, the overall system ends up in either a cat-dead or cat-alive state regardless of measurement. The N particles that make up the cat will be in one configuration or the other. The observer merely discovers the state. There is no collapse of the wavefunction. The part of the wavefunction that captures the cat-dead state and the part of the wavefunction that captures the cat-alive state diverge in configuration space and no longer influence each other. There is an effective collapse—but nothing that is mysterious. The physical process is clearly mapped out by what happens to the particles that make up the cat. In addition, if one focuses on what should be regarded as the wavefunction of the cat after the experiment, one finds an actual collapse and not merely an effective one.
“So Bohmian mechanics is not a repudiation of the rules of quantum mechanics,” said Goldstein. “It’s simply a clarification of them. You understand where they come from, you understand more clearly what they say.”
Despite such claims, support for Bohm’s ideas was hard to come by. Those who favored the orthodox interpretation even pointed out that the very strengths of Bohmian mechanics—its clear ontology and the fact that particles have trajectories—were leading to problems. These problems were laid bare using—what else?—the double-slit experiment. Theorists claimed that the trajectories predicted by Bohmian mechanics to describe particles passing through the double-slit apparatus made no sense. They dubbed them surreal, a term they used to deride Bohmian mechanics.
For experimentalists, proving the existence of such surreal trajectories was going to be a challenge. First they had to find a way to determine trajectories in general, before they could focus on surreal ones. It required a whole new way of thinking about measuring trajectories. After all, traditional quantum mechanics eschewed the very notion of trajectories.
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Bohmian mechanics, however, says that particles have well-defined trajectories. Chris Dewdney can recall the first time he saw the trajectories of particles going through a double slit. It was the late 1970s. He had just applied to do a PhD with David Bohm at Birkbeck College. Basil Hiley replied instead and said he’d take on Dewdney as his student. “Okay, close enough,” Dewdney recalled thinking.
While he was looking around for a thesis topic, Dewdney stumbled on a book at a local bookstore on quantum mechanics by Frederik Belinfante, which had a chapter on Bohm’s hidden variable theory. Belinfante suggested the possibility of calculating the Bohmian trajectories of particles in a two-slit experiment
. Dewdney recalled being puzzled. “I thought, ‘This is very strange.’ At Birkbeck, nobody was talking about this,” he said, despite the fact that Bohm himself was at Birkbeck. Dewdney talked it over with Hiley and Chris Philippidis in a coffee lounge at Birkbeck, and they decided to plot the trajectories. Today, our smartphones would make quick work of such calculations. But four decades ago, they needed a supercomputer, especially for the graphics. The programming was done using punch cards. You had to submit the cards and wait. The results came back in a small film canister. “You had to get them printed or hold them up to the light,” said Dewdney, who is now at the University of Portsmouth in the UK. And when they did, the particle trajectories were clearly visible. “It was amazing, totally amazing,” Dewdney told me. A particle went through one slit or the other, and then zigzagged its way to the screen on the far side. Taken together, trajectories bunched up in ways that eventually mimicked an interference pattern.
The trio published a paper showing the trajectories in 1979, but measuring them remained a pipe dream. “Mostly everyone in the community had believed that these are trajectories that you cannot directly measure,” experimentalist Aephraim Steinberg of the University of Toronto told me.
That’s because a traditional “strong” measurement, which ostensibly collapses the wavefunction of the thing being measured, destroys the coherent superposition of the particle. The particle is irrevocably disturbed, even destroyed, by a strong measurement.
So, finding a particle’s possible trajectory using such measurements is an impossibility. Think of how different this is from determining the path of, say, cars on a highway. If you put cameras every 100 meters to record the passing of cars, the information from these cameras can be used to reconstruct the trajectories of the cars moving past. But if you try to do this for photons or electrons, it doesn’t work. Each strong measurement that tries to find out the location of a particle gives the particle such a kick that it no longer goes where it would have gone had there been no measurement. There’s really no way to measure trajectories of particles using strong measurements without altering them.
In 1988, Yakir Aharonov (one of Bohm’s students), along with David Albert and Lev Vaidman, came up with a theory of what they called “weak measurements.” What if one doesn’t try to find out the precise value of some property of a quantum system but, rather, probes it ever so gently, so as to not disturb the particle, allowing it to continue on its trajectory as if nothing happened? It turns out that the outcome of any such individual measurement is rather useless. The uncertainty in the measurement means the result could be widely off the mark. But Aharonov and colleagues showed that if you do such measurements on a large ensemble of identically prepared particles, then although each measurement alone isn’t revelatory, taken together they are. The team argued that the average value of all measurements—say, in this case, of a particle’s position—is an indication of its average position.
There’s plenty of controversy about whether this average value is indeed giving you any relevant information about a particle’s property. But some physicists saw in weak measurements a way to measure particle trajectories. In 2007, Howard Wiseman of Griffith University in Brisbane, Australia, showed that you could use weak measurements to seemingly measure the positions and momenta of particles moving through a double-slit apparatus. The idea is simple: you take hundreds of thousands or even millions of identically prepared particles and send them through the double slit one by one, and you perform weak measurements, say, at different locations between the double slit and the screen where the interference is observed. These weak measurements can then be used to reconstruct the trajectories of particles traversing the apparatus. “ It must be emphasized,” wrote Wiseman, “that the technique of measuring weak values does not allow an experimenter . . . to follow the path of an individual particle.” That would be a violation of the rules of quantum mechanics. Nonetheless, one could, in principle, reconstruct average trajectories.
Until Wiseman’s paper, the idea of measuring trajectories had been anathema, but his work changed minds. Aephraim Steinberg’s team took on the task. As always with such experiments, it involved some very sophisticated optics. Even so, conceptually, the experiment is easy to understand. Steinberg’s team sends each photon into a beam splitter, which steers the photon into one optical fiber or another, with equal probability. The fibers are set up to hit a mirrored prism that reflects the photons at right angles (with the left fiber hitting the left prism and the right fiber hitting the right prism). The net effect of this arrangement is the two prisms act as two slits. A CCD camera placed on the far side of the prisms records the photons. For each photon that lands on the camera, there is no way to tell which prism (or slit) it came from. This indistinguishability leads to interference, which is captured by the camera.
The innovation was in a block of calcite crystal placed between the double slit and the CCD camera. The photons have to traverse the calcite crystal, which has the property of rotating the angle of polarization of the photon moving through it. By carefully aligning the crystal, it’s possible to use this change in the photon’s angle of polarization to get a sense of the direction in which the photon is moving, relative to the midline. This is a weak measurement: the change in polarization is tantamount to catching a whiff of the propagation direction without destroying the photon. It’s a measure of the angle at which the photon is traveling and hence a proxy for its momentum. Of course, then to actually measure the change in polarization requires a strong measurement, which destroys the photon.
So, to reconstruct entire trajectories, Steinberg’s team performed a series of such measurements on a whole ensemble of identical photons as the photons passed from the double slit to the camera. These were weak measurements, so they were average values computed for a large number of photons. The measurements were repeated for different locations between the double slit and the camera, by placing the calcite crystal at varying distances from the double slit, each time in a plane parallel to the plane of the slits. These gave the average momenta of the particles in each plane.
There was one more measurement of relevance: the position of the photon as it crossed the plane of the calcite crystal. Optics were used to capture the image of each photon on the CCD camera as it passed through the crystal. This was used to calculate the positions of the photons. “The important thing is that we can’t follow any given photon from plane to plane, but in each plane we can correlate position with direction,” said Steinberg. So “in each plane, we construct a map of momentum versus position, and then connect the ‘arrows’ to build up trajectories.”
The net result was that the researchers were able to reconstruct the average trajectories of the photons. The seemingly impossible had been done. The reconstructed trajectories looked very much like the simulated Bohmian trajectories. It has to be pointed out that one can arrive at the same predictions using standard quantum mechanics, so it’s impossible to use the experiment to say which interpretation is correct. But despite similar predictions, the two interpretations have entirely different things to say about the nature of reality: in Bohmian mechanics, particles and their trajectories exist independent of observation, whereas in standard quantum mechanics the act of observation creates reality.
In 2011, Physics World named Steinberg’s experiment the Breakthrough of the Year, and said, “ The team is the first to track the average paths of single photons passing through a Young’s double-slit experiment—something that Steinberg says physicists had been ‘brainwashed’ into thinking is impossible.”
Not everyone is convinced that Steinberg’s team has actually reconstructed Bohmian trajectories. One naysayer is Basil Hiley, who is now professor emeritus and still active. Hiley contends that the Bohmian trajectories shown in his 1979 paper with Dewdney and Philippidis are for nonrelativistic particles that are moving far slower than the speed of light. Photons are massless particles that move at the speed of light and so are relativis
tic. Particles of matter, such as atoms, are nonrelativistic. Hiley argues that Steinberg’s experiment with photons is not the correct one to reconstruct particle trajectories—even though he’s impressed with the experiment itself.
To verify the trajectories for nonrelativistic particles, Hiley has teamed up with Robert Flack and his team at University College London (UCL), and they are working on their own double-slit experiment. Hiley and Flack have chosen to work with argon atoms. The idea for their experiment came about a few years ago, when the two happened to meet over breakfast at a conference in Sweden. They began chatting about possibly testing Bohm’s ideas, and Flack (an experimentalist) said, “I think we could have a go.”
But it’s been a hard slog. “Sometimes I wish I hadn’t said that,” joked Flack when I met him and Hiley in their lab in the basement of the physics department at UCL.
“Come on, Rob, you know, it’s made your life worth living,” Hiley bantered back.
The experiment involves first creating a cloud of argon atoms in an excited, metastable state (meaning they are stable in this state for about 100 seconds), and beam these atoms toward a double slit. After going through the two slits, and before they hit a detector, the atoms have to pass through a magnetic field. And just as in Steinberg’s experiment, where the angle of the path of the photons through the calcite crystal is reflected in a change in the angle of polarization of the photons, the path the argon atoms take through the magnetic field is also reflected in certain internal properties of the atoms. This is a weak measurement. A series of such weak measurements can then be used to reconstruct the trajectories of the atoms. In theory.
The team is still working on getting the experiment up and running. I couldn’t help but comment on how workmanlike these quantum physics experiments look. Optical benches, vacuum pumps, lasers, mirrors, and the like strewn about everywhere (the experimentalists would point out that everything is rather precisely positioned). The whole place looks like a mechanic’s shop, albeit a very clean one with no grease, and it belies the profundity of what’s being tested: the underpinnings of reality. “As a theoretician, I didn’t think I’d live to see the day when the Bohm theory was being tested effectively,” Hiley said.
Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality Page 14