by Lee Smolin
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ONE REASON QUANTUM MECHANICS captured the interest of the younger generation of physicists was that it could be approached from several points of view. I have so far told the story of one way the quantum theory was invented, centering on the wave-particle duality, but there was another route, which had been discovered shortly before Schrödinger took his Christmas holiday. This was pioneered by Werner Heisenberg, a young and very confident German theorist, who completed his education in Max Born’s group in Göttingen and then in 1925 went on a research fellowship to work in Copenhagen with Bohr. He spent the next several years bouncing between Göttingen and Copenhagen, which is to say he was in close touch with the two most dynamic scientific personalities of that moment, Born and Bohr. Max Born and several of his students and assistants also played important roles in the story; indeed, the full story of how quantum mechanics was invented involves at least half a dozen theorists, in frequent communication.
Heisenberg worked from a particular idea about physics, an idea that was anti-realist to begin with. He asserted that physics does not give a description of what exists, as realists suppose, but is only a way to keep track of what is observable. For large-scale objects, we have gotten used to confusing the two. But if we want to make sense of atomic physics, we must adhere strictly to the dictum that science can only refer to what can be observed.
Hence, Heisenberg asserts that it is meaningless to talk about how the electron moves in the atom, unless that motion has consequences which can affect large-scale measuring devices. According to Bohr’s model, an atomic electron spends most of its time in stationary states, during which it has no interaction with anything outside the atom. It is then meaningless to ask how the electron moves while it is in a stationary state. It is only when it jumps between stationary states that the atom can interact with the world outside, because the jump is accompanied by the absorption or creation of a photon, and that photon’s energy can be measured by a spectrograph.
Heisenberg’s admonition not to try to model the trajectories of electrons in stationary states must have come as a breath of fresh air to others of his generation who were spending much of their time in frustrating and ultimately fruitless attempts to do just that.
Heisenberg was inspired by this thinking to invent a new way of representing the energy of the electron. Not by a single number, because to do so would be to claim that the energy is a property of the atom alone. What is relevant for physics is only what aspect of energy affects a measuring device. These are the energies carried by the photons that the atoms absorb or emit when the electrons jump between energy levels. These are the differences between the energies in the different stationary states.
Heisenberg arranged these energy differences as a table of numbers. He then imagined that such tables could represent observable aspects of other quantities, such as the electron’s position and momentum. To make a theory he had to do more, which was to find a way to write equations involving these tables of numbers. In the equations of physics we often find ourselves adding or multiplying numbers. He needed to do the same with tables of numbers. So he had to invent rules for how to do this.
As a member of both Bohr’s institute and Max Born’s research group, Heisenberg was under the influence of two masters with very different styles of work, and the contrast between them undoubtedly stimulated his thinking. But to realize his ideas in detail, he needed isolation, no less than Einstein, de Broglie, and Schrödinger had. Like Schrödinger, he took off on a holiday, in his case to a small island called Helgoland.
Once there, it took him only a few days to take himself on the journey I’ve just sketched, and to invent ways to write and solve equations with his tables of observable quantities.
He tested his ideas on a simple toy model of an atom, in which the electron is bound by a constantly increasing force, as if on a spring. This was not meant to be realistic, but it was a simple test, because the answer was known, and his method passed. There was only one hitch: he discovered that the order in which he multiplied two tables together matters. In the language I proposed earlier, Heisenberg’s tables of numbers don’t commute. This is of course not the case for ordinary numbers, and at first this discovery dismayed Heisenberg.
Nonetheless, he wrote up his findings in a paper published at the end of 1925. It was in the introduction to that paper that he announced his program of constructing laws of physics that dispensed with mechanical models describing the trajectories of the electrons and involved only relationships between observable quantities, namely the spectra of light the atoms emit and absorb.
This was a big step, but it was not yet the complete theory. He then returned to Göttingen and worked with Max Born and a brilliant student of his, Pascual Jordan. Born and Jordan were already partway to a new theory, and explained to Heisenberg that his tables of numbers were known to mathematicians as matrices; and they were able to reassure him that the failure to commute was a feature and not a bug. Heisenberg then understood that since the tables/matrices represent a process of measurement, the order does matter—because it matters in which order we make measurements. Together the three theorists then worked out the rest of the new theory, which they named quantum mechanics. A joint paper by the three of them was the first complete statement of the new theory.
Austrian wunderkind Wolfgang Pauli quickly followed up and applied the new theory to find the spectrum of the hydrogen atom, and it came out exactly right. Thus was quantum mechanics born by a second route, and in a way that was directly inspired by the anti-realist principles Heisenberg had expressed in his 1925 paper. The new theory of Born, Heisenberg, and Jordan is expressed in terms of quantities that describe how an atom responds to being probed by an external measurement device; there are no quantities that describe the exact trajectories of the electrons, independent of our interactions with them.
One quantum theory of the atom is great, but two are a problem, especially since they both reproduced the right spectrum of hydrogen. The two theories could not have differed more, as reflects the philosophies of their discoverers. Einstein, de Broglie, and Schrödinger were realists. Even if there were mysteries, they believed an electron was real and somehow existed as both wave and particle. Bohr and Heisenberg were enthusiastic anti-realists who believed we have no access to reality, only to tables of numbers which represent the interactions with the atom, but not the atom directly.
The tension lasted a few months, and then had an unexpected resolution when Schrödinger showed that the two forms of quantum mechanics are completely equivalent. Like two languages, you could speak in terms of waves or talk the language of matrices, but the math problems you had to solve turned out to be just different expressions of the same logic.
Heisenberg and Bohr, together in Copenhagen, shared an anti-realist perspective. They sought a way to speak consistently about properties that could not be realized simultaneously, such as waves versus particles or position versus momentum. Bohr’s resolution of the apparent paradoxes was his principle of complementarity. Heisenberg’s was his great uncertainty principle, which we talked about in chapter 2.
The uncertainty principle is a very general principle, as it says that we cannot know exactly both where a particle is and with what momentum it is moving. It has, as Heisenberg and his mentor Bohr realized immediately, stunning consequences. One is that the determinism of Newtonian physics cannot survive in the quantum world, because to predict the future motion of a particle you must know both its present position and how fast and in what direction it is moving, and hence its momentum. If you cannot know both precisely, you cannot predict where the particle will be at later times. As a result, the best that quantum theory can do is to make probabilistic predictions about the future.
The consistency of complementarity depends on there never being a case where we are forced to use both the particle picture and the wave picture in the description
of a single experiment. The impossibility of doing so is safeguarded by Heisenberg’s uncertainty principle, which he proposed in 1927, after he had moved back to Copenhagen and was in close contact with Bohr.
Historians tell us that luck plays a big role in science. Heisenberg was doubly fortunate for, as the protégé of both Max Born and Niels Bohr, he was not just in the right place at the right time, but doubly so! From his mentor Bohr he was inspired to abandon realism and model the atom only in terms of the energies it exchanges with our measuring devices, and from his mentor Born he got the mathematical tools needed to give these ideas a precise expression.
Of course, Heisenberg knew his good fortune and was the one who pushed to frame the new theory precisely. There were perhaps half a dozen young theorists who were also in the orbits of Bohr and Born, who contributed pieces, like Pauli, or got partway there, like Jordan, or were a few months late and so got to elegantly frame the new theory, like the English theorist Paul Dirac. The full story of the invention of the matrix form of quantum mechanics is far more complex than I can tell here, as it reveals a very dynamic, collective effort of a diverse community of theorists, in close interaction.
Still, diverse as they were, the matrix mechanicians were by 1927 all framing the new theory in terms of the radically anti-realist philosophy that Bohr preached. The only holdouts were those who had come to quantum mechanics through the wave-particle duality, Einstein, de Broglie, and Schrödinger, who stubbornly remained realists. But once it was proved that Schrödinger’s wave mechanics was equivalent to Heisenberg’s matrix mechanics, the realists could be dismissed as stubbornly grasping on to old metaphysical fantasies, and ignored.
The essence of Bohr’s philosophy is the necessity of basing science on incompatible pictures and languages. Heisenberg preached a view which differed in emphasis from Bohr’s while being loosely compatible with it. Heisenberg emphasized that science concerns only measurable quantities and can’t give an intuitive picture of what is happening at atomic scales. The observable quantities relevant for interacting with an atom include the energies and lifetimes of the stationary states, but do not include the positions or motions of electrons in their orbits around the nucleus. So quantum physics only has to yield an answer to a question of where an electron is if you force it into a context where that position is measured. According to Heisenberg, observable quantities are brought into existence only by the act of measuring them. When an atom is free of a measuring apparatus, no quantity describes it.
This may be called an operationalist perspective. It is certainly anti-realist, in that Heisenberg stressed that this view is mandatory. There was, according to him, no possibility of seeing deeper into the atom to perceive how the electrons move in their orbits. His uncertainty principle precluded it.
Heisenberg explained that uncertainty and complementarity were closely connected.
We can no longer speak of the behavior of the particle independently of the process of observation. As a final consequence, the natural laws formulated mathematically in quantum theory no longer deal with the elementary particles themselves but with our knowledge of them. Nor is it any longer possible to ask whether or not these particles exist in space and time objectively. . . .
When we speak of the picture of nature in the exact science of our age, we do not mean a picture of nature so much as a picture of our relationships with nature. . . . Science no longer confronts nature as an objective observer, but sees itself as an actor in this interplay between man and nature. The scientific method of analyzing, explaining and classifying has become conscious of its limitations, which arise out of the fact that by its intervention science alters and refashions the object of investigation. In other words, method and object can no longer be separated. . . .
[T]he different intuitive pictures which we use to describe atomic systems, although fully adequate for given experiments, are nevertheless mutually exclusive. Thus, for instance, the Bohr atom can be described as a small-scale planetary system, having a central atomic nucleus about which the external electrons revolve. For other experiments, however, it might be more convenient to imagine that the atomic nucleus is surrounded by a system of stationary waves whose frequency is characteristic of the radiation emanating from the atom. Finally, we can consider the atom chemically. . . . Each picture is legitimate when used in the right place, but the different pictures are contradictory and therefore we call them mutually complementary.3
Bohr’s point was even more radical. For him,
An independent reality in the ordinary physical sense can . . . neither be ascribed to the phenomena nor to the agencies of observation. . . .
A complete elucidation of one and the same object may require diverse points of view which defy a unique description. Indeed, strictly speaking, the conscious analysis of any concept stands in a relation of exclusion to its immediate application.4
Other quantum luminaries, such as Wolfgang Pauli, a wunderkind who published a textbook on general relativity when he was twenty-one, and John von Neumann, a Hungarian mathematician who is famous for his inventions in a broad range of fields, from the architecture of computers to the mathematics of quantum theory, taught variants of these anti-realist philosophies. Their views differed in emphasis, but anything written by them was classified as part of the “Copenhagen interpretation” of quantum mechanics. This name recognized Bohr’s dominance as the oldest of the group and mentor to most, as well as the originator of nothing less than a new way of talking about science. The name also recognized Bohr’s institute as the central node in the network of quantum physicists, where they all studied, worked, or visited.
One of the hardest lessons to learn in academic life—and for me one of the most disconcerting—is the speed with which a radical insurgency can become orthodoxy. In just a few years a generation of students championing a dangerous new idea are elevated by an initial success into professorships. From these positions of influence they form a powerful network of academic power brokers, which they use to ensure the continuation of the revolution. Such was the case with the generation of quantum revolutionaries. In 1920 Heisenberg was a student, as were Dirac, Pauli, and Jordan; 1925 found them young researchers fully engaged in the invention of quantum theory; by 1930 they were senior professors, and the revolution was over. The fact that there remained a handful of defectors—Einstein and Schrödinger from the older generation, and de Broglie among their contemporaries—did nothing to diminish their triumph, for students knew which way the wind blew and followed the ascendant orthodoxy. For the next half century, the anti-realism of the Copenhagenists would be the only version of quantum theory taught.
PART 2
REALISM REBORN
SEVEN
The Challenge of Realism: de Broglie and Einstein
There was never a single Copenhagen interpretation. Bohr, Heisenberg, and von Neumann each told a somewhat different story. But they all agreed that science had crossed a threshold. There could be no retreat back to a realist version of physics. They gave diverse arguments against the possibility, all leading to the conclusion that quantum physics is inconsistent with realism. No version of atomic physics could exist if it included electrons with definite positions and trajectories.
One way all these arguments might have been defeated—one would think—was for someone to come up with an alternative quantum theory based on realist ideas.
What is really bizarre, looking back, is that from 1927 on, there had existed a realist version of quantum mechanics. This is based on a stunningly simple idea. Perhaps you have already thought of it. It is simply to posit that there are both waves and particles. What gets created and detected, what gets counted, is a particle. Meanwhile, a wave flows through the experiment. The wave guides the particle. The result of this guidance is that the particle goes to where the wave is high.
Faced with a choice of which way to go around an obstacle, such as in the double slit e
xperiment, the wave goes both ways. The particle goes through only one slit around only one side, but where it goes once it gets through is guided by the wave, and shows the influence of both paths.
This obvious solution to the challenge of the wave-particle duality was thought up by Louis de Broglie. He worked it out in detail and called it the pilot wave theory. De Broglie presented his theory at a famous conference held in Brussels in 1927. Named the Fifth Solvay Conference after its sponsor, the conference featured talks by most of the revolutionaries of the new quantum physics.
The core of pilot wave theory was de Broglie’s idea that the electron is actually two entities, one particle-like and one wavelike. The particle is always located some particular place and always follows some particular path. Meanwhile, the wave flows through space, taking simultaneously all the possible paths or routes through the experiment. The wave then directs the particle where to go, and that piloting will be based on conditions along all the paths. Even though the particle must take one route or another, which route it takes is influenced by the wave, which flows through all routes.
This influence of a wave on a particle is the new thing which is responsible for much of what is strange in the quantum world. There are two laws, one for the wave and one for the particle. The wave law is relatively familiar; it is not so different from the laws that physicists use to describe sound waves or light waves. The waves spread out, and as they travel they diffract and interfere. Like water and sound, these quantum waves will flow down every channel open to them. And when waves coming down different channels meet, they will interfere.
The wave in question is called the wave function. It propagates according to the simple equation that Schrödinger invented during his romantic ski weekend. This is Rule 1, and it is the key equation in every approach to quantum physics.