Einstein always loved his violin and played it wherever he traveled. Musicians were welcomed into the Einstein apartment to play. He sometimes performed in public as well. Einstein occasionally played during dusk services at Berlin’s New Synagogue. There was room for 3,000 people. The men filled the main floor; the women sat silently in the balcony while the organist played Bach. Einstein accompanied on his violin and the city’s fading daylight crept through stained-glass windows. Those sounds in the darkness were Einstein’s escape. “Whenever he felt that he had come to the end of the road …,” his son Hans Albert recalled, “he would take refuge in music and that would usually resolve all his difficulties.” It is too happy an interpretation. In the morning, the difficulties, especially the meaning of quanta, persisted. They could rebound with unanticipated strength.
The surprise challenge of May 1926 was the publication of Schrödinger’s proof that wave mechanics and matrix algebra were mathematically equivalent. This idea was much in the air. Schrödinger had submitted his article in mid-March. At March’s end, an American named Carl Eckart sent a paper to the National Academy of Sciences making the same argument, and in April, Wolfgang Pauli outlined a proof in a private letter to Born’s assistant, Pasqual Jordan.
Schrödinger’s proof took the most peculiar feature of matrix alge-
bra—the fact that p times q does not equal q times p—and showed that he found the same peculiarity when using his wave equation’s differential calculus. Schrödinger then showed that any matrix equation could be translated into his wave mechanics and, conversely, any wave function could be translated into matrix math.
This formal unity served the function of a scandalous moment at a dinner party. It revealed something serious and different about each person who was witness to the incident. Heisenberg, Schrödinger, Bohr, Einstein, and Born each responded separately. The immediate question was what this mathematical equivalence meant. Were the equations physically equivalent too? Heisenberg became much more ardent in insisting that his system was the correct one, and that Schrödinger’s wave mechanics was “disgusting.” Schrödinger and some other wave enthusiasts were equally determined to defend their system.
For Einstein, the situation showed that the revolution still had a long way to go. The mathematical equivalence reflected the fact that neither one got down to nature’s deeper truth.
Max Born’s reaction was the most distinctive. He usually focused on keeping the math rigorous and clear, a taste that made him seem a bit odd to many other physicists. Even his assistant, Wolfgang Pauli, once complained to Born about his “tedious and complicated formalisms” and how he spoiled good physical ideas by his “futile mathematics.” Presumably Pauli meant that physical meaning was not the same thing as mathematical meaning, yet Born looked to physics’ mathematical side. Born’s approach, however, became a great strength during the birth of quantum mechanics. Although nobody was having any luck finding physical meanings for the discoveries, mathematical meanings were there for the taking. A grand example of that success lay in the birth of quantum mechanics. Heisenberg had been looking for a math system that could predict changes in a particle’s quantum state. He finally succeeded in drafting a paper and giving it to Born to look at. Heisenberg then immediately left Göttingen for vacation. When Born read the paper, he saw the peculiarity that p times q did not equal q times p and he knew that this oddity must mean something important.
Most physicists presented with such an oddity would look for some insight into physical reality. What property might nature have that would give such a result? But Born looked for a mathematical meaning. He remembered that he had already encountered the same thing in his student days. So for him, Heisenberg’s paper meant that quantum mechanics rests on matrix algebra.
Einstein would have pressed on to ask what was found in nature that required matrix algebra; however, for Born, matrix algebra was the whole solution. He translated Heisenberg’s confusing notation into matrix form, and was well pleased because the mathematics now made sense. In matrix algebra, the multipliers (p and q) are not numbers but systems of numbers and multiplication is a method of combining number systems.
Suppose we have two sets of letters, IE and TM, which we can symbolize by p and q. We can combine the sets by alternating letters from each group. We take the first letter from one group and then the first letter from the next group; next we take the second letter from the first group and then the second letter from the second group. If we begin with the p group, pq = ITEM, but if we begin with the q group, qp = TIME. There is nothing inherently puzzling in this case about pq giving us a different result from qp, but while this mathematical understanding might relieve the feeling that the situation is impossible, it does nothing to explain the reality of what lies behind quantum mechanics. If you take the matrix algebra to be the whole story, normal physical explanations go out the window. There is only a mathematical explanation, just as there is only a mathematical reason for why three oranges priced at 30 cents apiece cost 90 cents. There is no natural cause at work here, nothing about the properties of oranges or pennies. There is just the mathematical fact that 3 times 30 equals 90. Similarly, the changes in quantum states have no underlying physical explanation. They are what they are because of the mathematics of how matrices work.
It was this kind of abracadabra that led Einstein in the summer of 1926 to write Ehrenfest, “I look upon quantum mechanics with admiration and suspicion.” Who could not help but admire the multiplication table that allows us to find a price for any number of oranges?
Yet how can you not be suspicious that this system misses much about an orange’s quality, size, and freshness?
Schrödinger’s proposal to explain quantum states in terms of waves and wave packets had suggested a method that could look beyond the mysterious matrices to reveal what was really going on. But in June 1926, Max Born submitted a paper that reinterpreted Schrödinger’s equation, using it to consider what happens when atomic particles collide. The matrix approach to this issue was to define a quantum state at the instant of collision and a second state after the collision, with no causal process between the input and the outcome. Schrödinger’s method viewed the collision as the interaction of two wave packets and calculated their behavior through space and time. In this view, every part of the process from start to finish is connected. But Max Born proposed a new way of thinking, one he credited to Einstein’s influence.
Born was unwilling to abandon material reality quite as thoroughly as Heisenberg did and he appreciated the way Schrödinger’s method kept space in physics’ story. Yet Born refused to believe that particles were illusions, wave packets that passed themselves off as something more. Still, Schrödinger’s equation described something; if it did not describe a wave, what was it about? Born was back in the same blind alley that quantum work kept entering. How do you reconcile waves and particles? But Max Born’s instinct was to look for a mathematical reconciliation rather than a physical one.
One possibility was that instead of defining a wave, Schrödinger’s equation defined a field. After all, electromagnetic fields played a large role in the behavior of quantum radiation and particles. Born recalled the “ghost field” idea that had bothered Einstein for so many years. The same old difficulty showed itself. What was this field? How did it work?
Faced with the question that never seemed to go away, Max Born took what many physicists consider one of the grandest strides in the history of science. At the time, however, the step was not universally appreciated and Einstein always considered it a terrible move backward. Born stripped the field of any physical reality whatsoever and gave it a purely mathematical meaning. The wave or field in
Schrödinger’s equation—the thing that was represented by ψ—was a statement of probability. As Born put it, “One obtains the answer to the question, not ‘what is the state after the collision,’ but ‘how probable is a specified outcome of the collisions.’”
The radicality of Born’s idea was that there was no physical explanation anywhere behind these probabilities. It was not like a weather forecast in which the chances refer to physical events. If there is a 50-50 chance of rain, the forecaster is saying, in effect, “Well, the clouds might go this way or they might go that way.” Whatever way they do move is determined by meteorological forces too subtle for the forecaster to anticipate, but once it is all done the forecaster can say, “We didn’t get that rain after all because that high pressure system came in and pushed the rain clouds away.” The forecast was given in terms of probabilities, but the events were all physical.
Max Born’s probability field had no such physical meaning. Born then made an even more radical assertion, saying that in “quantum mechanics there exists no quantity which in an individual case causally determines the effect of a collision…. I myself tend to give up determinism in the atomic world.”
Born was taking the position Einstein had always refused to take. Einstein had added two extra years to his struggle with general relativity because he insisted on finding a causal explanation for his equation, and, after general relativity, he again had a chance to abandon causality when he published a theory of spontaneous emission of light quanta. The trouble with the theory, as Einstein noted at the time, was that it could not predict where the light would go. Ever since then, Einstein kept an eye open for some new idea that would complete his theory by predicting the light’s direction. Now Born was saying there was no need to spend years looking for an explanation. At the quantum level, things do as they damn well please.
This was the return of BKS with a vengeance. Niels Bohr had proposed that quanta behave statistically, not on the basis of what had come before. Einstein had said then that he would prefer to work in a casino than to be a physicist in such a world. BKS had failed, but Max Born had restored its croupier spirit. Or almost restored it. The triumph of Einstein’s photons in the Compton effect had vanquished
the notion of an absolutely statistical universe in which nothing had to be conserved. Particles, in Born’s account, moved statistically, just as they did in the BKS paper, but the probabilities of those motions were absolutely fixed in accordance with Schrödinger’s wave field.
Viewed from that perspective, Max Born’s paper on particle collisions marked the conservative cooling point, the Thermidor, of the quantum revolution. The most radical elements—Niels Bohr’s assault on conservation and Heisenberg’s attempt to abolish space from quantum considerations—failed. But Born was no counterrevolutionary and he did not restore the old regime. Statistics, rather than reasons, governed the outcomes of collisions; space was filled with probabilities rather than motion. The particle was there; now it is here. As the philosopher David Albert put it much later, “Electrons [in an experiment] do not take [one] route and do not take [the other] route and do not take both of those routes and do not take neither of those routes.” Baffling, wouldn’t you say?
If we return for a moment to the metaphor of the bullwhip, in Max Born’s interpretation the whip is gone and the wave packet is gone, but sometimes the cigarette still flies out of the lovely assistant’s lips.
Whenever a revolution cools there are people who resist its halt. Like America’s whiskey rebels after the establishment of the federal constitution, they grumble that this outcome was not what they were looking for when they enlisted in the sacred cause. Rarely, however, does a living symbol of the revolution resist its settlement quite so ferociously and famously as Einstein resisted this one. He did see much to admire. The new quantum mechanics was a technical achievement of enormous imagination. Whatever finally emerged would have to be just as accurate as the existing mathematics. Quite possibly, the existing mathematics would survive but be understood in a new way. This kind of transformation had happened before. Einstein’s theory of relativity had found a new way to derive and explain Lorentz’s equations. Bose’s paper had provided a new way to derive and explain Planck’s Law. Einstein was confident that, now that it was making progress, the quantum revolution had to heat up still more. After decades of struggle, physicists had a new toolkit to work with, but Einstein never thought
tools alone led anywhere. Technique by itself is not enough to advance a civilization from Babylonian practicality to Greek understanding.
In the summer of 1926, it was still not clear where the quantum revolution would end. Colleagues assumed they would remain colleagues. Max Born did not expect to break with Einstein, and in his second paper on wave mechanics he publicized his debt to his great friend, “I start from a remark by Einstein on the relation between wave field and light quanta. He said approximately that the waves are only there to show the way to the corpuscular light-quanta, and talked in this sense of a ‘ghost field’ that determines the probability for a light-quantum … to take a different path.” Born’s recollection puts Einstein suspiciously in tune with the probabilistic interpretation, but the gist is plain. Einstein had long toyed with the idea, yet he never published it. In his 1917 paper on light quanta, Einstein had objected that his work did not get physics “any closer to making the connection with wave theory.” He could—and did—have the same doubts about probabilistic waves. It gets us no closer to understanding how and why particles move through space. As he later wrote, he feared that the emerging quantum mechanics offered “no useful departure for future development.” If quantum mechanics was a step, Einstein feared it was a stride into a box canyon and could never lead to an understanding of what lay at the heart of the universe.