Quantum
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He had been in Brazil only a matter of weeks when the American embassy, fearing that his final destination might be the Soviet Union, confiscated Bohm’s passport and reissued it as valid only for travel to the United States. Worried that his South American exile would cut him off from the international physics community, Bohm acquired Brazilian nationality to circumvent the travel ban imposed by the Americans. Back in the United States, Oppenheimer faced a hearing. Pressure on him intensified the moment it emerged that Klaus Fuchs, a physicist he had selected to work on the atomic bomb, was a Soviet spy. Einstein advised Oppenheimer to turn up, tell the committee they were fools, and return home. He did no such thing, but another hearing in the spring of 1954 revoked Oppenheimer’s security clearance.
Bohm left Brazil in 1955 and spent two years at the Technion Institute in Haifa, Israel before moving to England. After four years at Bristol University, in 1961 Bohm settled once and for all in London after being appointed professor of theoretical physics at Birkbeck College. During his troubled time in Princeton, Bohm had largely devoted himself to studying the structure and interpretation of quantum mechanics. In February 1951 he published quantum Theory, one of the first textbooks to examine in some detail the interpretation of the theory and the EPR thought experiment.
Einstein, Podolsky and Rosen had conjured up an imaginary experiment that involved a pair of correlated particles, A and B, so far apart that it should be impossible for them to physically interact with one another. EPR argued that a measurement carried out on particle A could not physically disturb particle B. Since any measurement is performed on only one of the particles, EPR believed they could cut off Bohr’s counter-attack – an act of measurement causes a ‘physical disturbance’. Since the properties of the two particles are correlated, they argued that by measuring a property of particle A, such as its position, it is possible to know the corresponding property of B without disturbing it. EPR’s aim was to demonstrate that particle B possessed the property independently of being measured, and since this was something that quantum mechanics failed to describe, it was therefore incomplete. Bohr countered, never so succinctly, that the pair of particles were entangled and formed a single system no matter how far apart they were. Therefore, if you measured one, then you also measured the other.
‘If their [EPR] contention could be proved,’ wrote Bohm, ‘then one would be led to search for a more complete theory, perhaps containing something like hidden variables, in terms of which the present quantum theory would be a limiting case.’6 But he concluded ‘that quantum theory is inconsistent with the assumption of hidden causal variables’.7 Bohm looked at quantum theory from the prevailing Copenhagen viewpoint. However, in the process of writing his book he became dissatisfied with Bohr’s interpretation, even as he agreed with the dismissal by others of the EPR argument as ‘unjustified, and based on assumptions concerning the nature of matter which implicitly contradict the quantum theory at the outset’.8
It was the subtlety of the EPR thought experiment, and what he came to regard as the reasonable assumptions on which it was constructed, that led Bohm to question the Copenhagen interpretation. It was a brave step for a young physicist whose contemporaries were busy using quantum theory to make their reputations rather than risking career suicide by raking over the embers of a dying fire. But Bohm was already a marked man after his appearance before the House Un-American Activities Committee, and, suspended by Princeton, he had little left to lose.
Bohm presented Einstein with a copy of quantum Theory and discussed his reservations with Princeton’s most famous resident. Encouraged to examine the Copenhagen interpretation more closely, Bohm produced two papers that appeared in January 1952. In the first of these he publicly thanked Einstein ‘for several interesting and stimulating discussions’.9 By then Bohm was in Brazil, but the papers had been written and sent to the Physical Review in July 1951, just four months after the publication of his book. Bohm appeared to have had a Paul-like conversion on the road not to Damascus, but Copenhagen.
In his papers Bohm outlined an alternative interpretation of quantum theory and argued that ‘the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy’.10 Reproducing the predictions of quantum mechanics, it was a mathematically more sophisticated and coherent version of Louis de Broglie’s pilot wave model, which the French prince had abandoned after it was severely criticised at the 1927 Solvay conference.
Whereas the wave function in quantum mechanics is an abstract wave of probability, in the pilot wave theory it is a real, physical wave that guides particles. Just as an ocean current carries along a swimmer or a ship, the pilot wave produces a current that is responsible for the motion of a particle. The particle has a well-defined trajectory determined by the precise values of position and velocity that it possesses at any given time but which the uncertainty principle ‘hides’ by preventing an experimenter from measuring them.
On reading Bohm’s two papers, Bell said that he ‘saw the impossible done’.11 Like almost everyone else, he thought that Bohm’s alternative to the Copenhagen interpretation had been ruled out as impossible. He asked why no one had told him about the pilot wave theory: ‘Why is the pilot wave picture ignored in textbooks? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?’12 A part of the answer was the legendary Hungarian-born mathematician John von Neumann.
The eldest of three brothers, the Jewish banker’s son was a mathematical prodigy. When his first paper was published at eighteen, von Neumann was a student at Budapest University but spent most of his time in Germany at the universities of Berlin and Göttingen, returning only to take his exams. In 1923 he enrolled at the ETH in Zurich to study chemical engineering after his father insisted that he have something more practical to fall back on than mathematics. After graduating from the ETH and gaining a doctorate from Budapest in double-quick time, von Neumann became at 23 the youngest-ever privatdozent appointed by Berlin University in 1927. Three years later he began teaching at Princeton University and in 1933 joined Einstein as a professor at the Institute for Advanced Study, remaining there for the rest of his life.
A year earlier, in 1932, the then 28-year-old von Neumann wrote a book that became the quantum physicist’s bible, Mathematical Foundations of quantum Mechanics.13 In it he asked whether quantum mechanics could be reformulated as a deterministic theory by the introduction of hidden variables, which, unlike ordinary variables, are inaccessible to measurement and therefore not subject to the restrictions imposed by the uncertainty principle. Von Neumann argued that ‘the present system of quantum mechanics would have to be objectively false in order that another description of the elementary processes than the statistical one may be possible’.14 In other words, the answer was ‘No’, and he offered a mathematical proof that outlawed the ‘hidden variables’ approach that Bohm would adopt twenty years later.
It was an approach with a history. Ever since the seventeenth century, men like Robert Boyle had studied the various properties of gases as their pressure, volume and temperature were varied, and had discovered the gas laws. Boyle found the law describing the relationship between the volume of a gas and its pressure. He established that if a certain quantity of a gas was kept at a fixed temperature and its pressure was doubled, its volume was halved. If the pressure was increased threefold, then its volume was reduced to a third. At constant temperature, the volume of a gas is inversely proportional to the pressure.
The correct physical explanation of the gas laws had to wait until Ludwig Boltzmann and James Clerk Maxwell developed the kinetic theory of gases in the nineteenth century. ‘So many of the properties of matter, especially when in gaseous form, can be deduced from the hypothesis that their m
inute parts are in rapid motion, the velocity increasing with temperature,’ wrote Maxwell in 1860, ‘that the precise nature of this motion becomes the subject of rational curiosity.’15 It led him to conclude that ‘the relations between pressure, temperature, and density in a perfect gas can be explained by supposing the particles to move with uniform velocity in straight lines, striking against the sides of the containing vessel and thus producing pressure’.16 Molecules in a continual state of motion, haphazardly colliding into one another and the walls of the container holding the gas, produced the relationships between pressure, temperature and volume expressed in the gas laws. Molecules could be regarded as the unobserved microscopic ‘hidden variable’ that explained the observed macroscopic properties of gases.
Einstein’s explanation of Brownian motion in 1905 is an example where the ‘hidden variable’ is the molecules of the fluid in which the pollen grains are suspended. The reason behind the erratic movement of the grains that had so perplexed everyone was suddenly clear after Einstein pointed out that it was due to the bombardment by invisible, but very real, molecules.
The appeal of hidden variables in quantum mechanics had its roots in Einstein’s claim that the theory is incomplete. Maybe that incompleteness was due to the failure to capture the existence of an underlying layer of reality. This untapped seam in the form of hidden variables – possibly hidden particles, forces, or something completely new – would restore an independent, objective reality. Phenomena that at one level appear probabilistic would with the help of hidden variables be revealed as deterministic, and particles would possess a definite velocity and position at all times.
As von Neumann was acknowledged as one of the great mathematicians of the day, most physicists simply accepted, without bothering to check, that he had proscribed hidden variables when it came to quantum mechanics. For them the mere mention of ‘von Neumann’ and ‘proof’ was enough. However, von Neumann admitted that there remained the possibility, though small, that quantum mechanics might be wrong. ‘In spite of the fact that quantum mechanics agrees well with experiment, and that it has opened up for us a qualitatively new side of the world, one can never say of the theory that it has been proved by experience, but only that it is the best known summarization of experience’,17 he wrote. Yet despite these words of caution, von Neumann’s proof was held to be sacrosanct. Virtually everyone misinterpreted it as proving that no theory of hidden variables could reproduce the same experimental results as quantum mechanics.
When he analysed von Neumann’s argument, Bohm believed that it was wrong but could not clearly pinpoint the weakness. Nevertheless, encouraged by his discussions with Einstein, Bohm attempted to construct the hidden variables theory that was deemed to be impossible. It would be Bell who demonstrated that one of the assumptions used by von Neumann was unwarranted, and therefore that his ‘impossibility’ proof was incorrect.
Born in July 1928 in Belfast, John Stewart Bell was descended from a family of carpenters, blacksmiths, farm workers, labourers and horse dealers. ‘My parents were poor but honest’, he once said.18 ‘Both of them came from large families of eight or nine that were traditional of the working class people of Ireland at that time.’ With a father who was in and out of work, Bell’s childhood was far removed from the comfortable middle-class upbringing of the quantum pioneers. Nevertheless, before he reached his teens, the bookish Bell had earned the nickname ‘The Prof’, even before he told his family that he wanted to become a scientist.
There was an older sister and two younger brothers, and though their mother believed that a good education was the route to future prosperity for her children, John was the only one who went on to secondary school aged eleven. It was not a lack of ability that denied his siblings the same opportunity, only a shortage of money for a family always struggling to make ends meet. Luckily the family came into a small sum of money that enabled Bell to enrol at the Belfast Technical High School. Not as prestigious as some of the other schools in the city, it offered a curriculum that combined the academic and the practical that suited him. In 1944, aged sixteen, Bell gained the qualifications necessary to study at Queen’s University in his home town.
With seventeen the minimum age for admission and his parents unable to finance his university studies, Bell looked for work and fortuitously found it as an assistant technician in the laboratory of the physics department at Queen’s University. Before long, the two senior physicists recognised Bell’s abilities and allowed him to attend the first-year lectures whenever his duties permitted. His enthusiasm and obvious talent were rewarded with a small scholarship, and this, together with the money he was able to set aside, meant that he returned after his year as a technician as a fully-fledged physics student. With the sacrifices that he and his parents had made, Bell was focused and driven. He proved to be an exceptional student and in 1948 obtained a degree in experimental physics. A year later he gained another in mathematical physics.
Bell admitted that he ‘had a very bad conscience about having lived off my parents for so long, and thought I should get a job’.19 With his two degrees and glowing references, he went to England to work for the United Kingdom Atomic Energy Research Establishment. In 1954 Bell married a fellow physicist, Mary Ross. In 1960, having gained a PhD from Birmingham University, he and his wife moved to CERN, the Conseil Européen pour la Recherche Nucléaire, near Geneva, Switzerland. For a man who would make his name as a quantum theorist, Bell’s job was designing particle accelerators. He was proud to call himself a quantum engineer.
Bell first came across von Neumann’s proof in 1949, his last year as a student in Belfast, when he read Max Born’s new book, Natural Philosophy of Cause and Chance. ‘I was very impressed that somebody – von Neumann – had actually proved that you couldn’t interpret quantum mechanics as some sort of statistical mechanics’, he later recalled.20 But Bell did not read von Neumann’s book as it was written in German and he did not know the language. Instead he accepted Born’s word for the soundness of von Neumann’s proof. According to Born, von Neumann had put quantum mechanics on an axiomatic basis by deriving it from a few postulates of a ‘very plausible and general character’, such that the ‘formalism of quantum mechanics is uniquely determined by these axioms’.21 In particular, Born said, it meant that ‘no concealed parameters can be introduced with the help of which the indeterministic description could be transformed into a deterministic one’.22 Implicitly, Born was arguing in favour of the Copenhagen interpretation, because ‘if a future theory should be deterministic, it cannot be a modification of the present one but must be essentially different’.23 Born’s message was that quantum mechanics is complete, therefore it cannot be modified.
It was 1955 before von Neumann’s book was published in English, but by then Bell had read Bohm’s papers on hidden variables. ‘I saw that von Neumann must have been just wrong’, he said later.24 Yet Pauli and Heisenberg branded Bohm’s hidden variables alternative as ‘metaphysical’ and ‘ideological’.25 The ready acceptance of von Neumann’s impossibility proof proved only one thing to Bell, a ‘lack of imagination’.26 Nevertheless, it had allowed Bohr and the advocates of the Copenhagen interpretation to consolidate their position even while some of them suspected that von Neumann might be wrong. Even though he later dismissed Bohm’s work, Pauli in his published lectures on wave mechanics wrote that ‘no proof of the impossibility of extending [i.e. completing quantum theory by hidden variables] has been given’.27
For 25 years, hidden variable theories had been ruled impossible by the authority of von Neumann. However, if such a theory could be constructed to yield the same predictions as quantum mechanics, then there would be no reason for physicists to simply accept the Copenhagen interpretation. When Bohm demonstrated that such an alternative was possible, the Copenhagen interpretation was so well entrenched as the only interpretation of quantum mechanics that he was either ignored or attacked. Einstein, who had initially encouraged him, dismissed Bohm’s
hidden variables as ‘too cheap’.28
‘I think he was looking for a much more profound rediscovery of quantum phenomena’, Bell said as he tried to understand Einstein’s reaction.29 ‘The idea that you could just add a few variables and the whole thing would remain unchanged apart from the interpretation, which was a kind of trivial addition to ordinary quantum mechanics, must have been a disappointment to him.’ Bell was convinced that Einstein wanted to see some grand new principle emerge on a par with the conservation of energy. Instead, what Bohm offered Einstein was an interpretation that was ‘non-local’, requiring the instantaneous transmission of so-called ‘quantum mechanical forces’. There were other horrors lurking in Bohm’s alternative. ‘For example,’ clarified Bell, ‘the trajectories that were assigned to the elementary particles were instantaneously changed when anyone moved a magnet anywhere in the universe.’30
It was in 1964, during a year-long sabbatical from CERN and his day job designing particle accelerators, that Bell found the time to enter the Einstein-Bohr debate. Bell decided to find out if non-locality was a peculiar feature of Bohm’s model or if it was a characteristic of any hidden variable theory that aimed to reproduce the results of quantum mechanics. ‘I knew, of course, that the Einstein-Podolsky-Rosen setup was the critical one, because it led to distant correlations’, he explained. ‘They ended their paper by stating that if you somehow completed the quantum mechanical description, non-locality would only be apparent. The underlying theory would be local.’31