The use of the word picturing here is significant, because Feynman’s method allows a beautiful pictorial way of thinking about quantum mechanics. Developing this new way took awhile, even for Feynman, who did not explicitly talk about a “sum over paths” in his thesis. By the time he wrote up his thesis as an article six years later in Reviews of Modern Physics, the notion was central. That 1948 paper, titled “The Space-Time Approach to Non-Relativistic Quantum Mechanics,” begins with a probability argument along the lines that I have given here, and then immediately starts discussing space-time paths. Surprisingly, drawings are conspicuously absent from the paper. Maybe it was too expensive in those days to get an artist to draw them. No matter, they would come.
WHILE FEYNMAN WAS writing up these results to form the basis of his thesis, the world in 1942 was in a state of turmoil, embroiled in the second world war of the century. Amid all of his other concerns—completing his thesis, getting married, finding a job—one morning he was suddenly interrupted in his office by Robert Wilson, then an instructor in experimental physics at Princeton. He sat Feynman down and revealed what should have been top-secret information, though the specific information was so new as to not yet have been thoroughly classified as such.
The United States was about to embark on a project to build an atomic bomb, and a group at Princeton was going to work on one of the possible methods for making the raw material for the bomb, a light isotope of uranium called uranium 235 (with the number 235 representing the atomic mass—the number of protons plus neutrons in the nucleus). Nuclear physics calculations had shown that the dominant naturally occurring isotope of uranium, uranium 238, could not produce a bomb with practicalamounts of material. The question was, How can the rare isotope uranium 235, which could produce a bomb, be separated from the far more abundant uranium 238? Because isotopes of an element differ only in the number of neutrons in the nucleus, but otherwise are chemically identical since they contain the same number of protons and electrons, chemical separation techniques wouldn’t work. Physics had to be employed. Wilson was revealing this secret because he wanted to recruit Feynman to help with the theoretical work needed to see if his own proposed experimental methods would work.
This presented Feynman with a terrible dilemma. He desperately wanted to finish his thesis. He was enjoying the problem he was working on, and he wanted to continue to do the science he loved. He also wanted to graduate, as this was one of his own preconditions for marriage. Moreover, Wilson would want him to focus on problems that Feynman viewed as engineering, a field he had explored but left for physics while still an undergraduate.
His first response was to turn down the offer. At the same time, how could he turn down a possibility to help win the war? He had earlier considered enlisting in the army if he could work in the Signal Corps, but he was told there were no guarantees. Here was a possibility of doing something far more significant. Moreover, he realized that the nuclear physics involved was not a secret. As he later said, “The knowledge of science is universal, an international thing. . . . There was no monopoly of knowledge or skill at that time . . . so there was no reason why if we thought it was possible that they [the Germans] wouldn’t also think it was possible. They were just humans, with the same information. . . . The only way that I knew how to prevent that was to get there earlier so that we could prevent them from doing it, or defeat them.” He did think for a moment about whether making such a frightening weapon was the right thing to do, but in the end he put his thesis work in a desk drawer and went to the meeting Wilson had told him about.
From that moment on he became occupied not in the abstract world of quantum mechanics and electrons, but in the minutiae of electronics and materials science. He was well prepared, as always, by what he learned on his own and in some excellent courses in nuclear physics from Wheeler and in the properties of materials from Wigner. Still, it took some getting used to. He and another research assistant of Wheeler’s, Paul Olum, a Harvard graduate in mathematics, set to work as fast as they could doing calculations they were not certain about even as the experimentalists around them were building the device that the two of them had to determine was or was not workable.
This was Feynman’s first experience with the realization, which would reoccur many times over his subsequent career, that while he loved theoretical calculations, he didn’t really trust them until they were put to experimental tests. As daunting as it was to try to understand nature at the edges of knowledge, it was equally daunting to be responsible for decisions based on his calculations that would ultimately have an immediate impact on the largest industrial project ever carried out by one country.
In the end, Wilson’s proposed device for isotope separation was not chosen. What was selected for major production of U235 were the processes known as electromagnetic separation and gaseous diffusion.
Feynman’s thesis supervisor, John Wheeler, did not abandon him during this period. Wheeler had left Princeton to work with Enrico Fermi on building the first nuclear reactor at the University of Chicago—where they would test the principle of controlled chain reactions as a first step toward the uncontrolled chain reactions that would be required to build a nuclear bomb. But Wheeler was aware of what Feynman was working on, and in the spring of 1942 he decided enough was enough. He and Wigner felt that Feynman’s thesis work was close enough to completion to be written up, and he told him so, in no uncertain terms.
Feynman proceeded to do just that. He was aware of what he had achieved. He had re-derived quantum mechanics in terms of an action principle involving a sum (or rather, in the language of mathematics, an integral) over different paths. This allowed a generalization to situations where the standard Schrödinger approach would not work—in particular the absorber theory that he and Wheeler had worked out for electromagnetism. This is what interested him—his real step forward, he thought—and the new method he had invented for deriving quantum mechanics was primarily a means toward this end.
But he was more concerned about what he hadn’t yet achieved, and he devoted the final section of his thesis to describing the limitations of his work thus far. First and foremost, his thesis did not contain any comparison with experiments, which he regarded as the real test of the worth of any theoretical idea. Part of the problem was that while he had reformulated purely nonrelativistic quantum mechanics, he was acutely aware that in order to address real experiments involving charges and radiation, the appropriate theory—quantum electrodynamics—was needed so as to incorporate relativity, which involved addressing a host of problems he had not yet dealt with.
Finally, Feynman was concerned with the physical interpretation of his new viewpoint for dealing with the quantum world—in particular, the issue of making a connection between the temporally spread-out paths and probability amplitudes inherent in his new formulation, and the possibility of making real physical measurements at any specific time. The problem of measurement was not new or unique to Feynman’s thesis. His work merely appeared to exacerbate it. The world of measurements lies within the classical world of our experience, where weird quantum paradoxes don’t ever seem to arise. How does a “measurement” ensure that the underlying quantum universe ends up appearing sensible to our eyes?
The first person to comprehensively attempt to quantitatively discuss this measurement problem in the context of quantum mechanics was John von Neumann, at Princeton, whom Feynman had the opportunity to interact and disagree with. Anyone who has heard anything about quantum mechanics often hears that one cannot separate the observer from that which is being observed. But in practice this is exactly what is required in order to make predictions and compare them with experimental data. Feynman was particularly concerned about this key question of how to separate the measuring apparatus and the system being observed in the context of the specific quantum mechanical calculations he wanted to perform.
The conventional verbiage goes as follo
ws: When we make a measurement, we “collapse the wave function.” In other words, we suddenly reduce the probability amplitude to be zero in every state but one. Therefore, the system has a 100 percent probability of being in only one configuration, and different possible configurations do not interfere with each other, as in the examples discussed in the last chapter. But this simply begs the questions: How does a measurement collapse the wave function, and what is so special about such a measurement? Is a human being needed to make the observation?
New-age hucksters aside, consciousness is not the key. Rather, Feynman argued that we must consider the system plus the observer together as a single quantum system (which is fundamentally true, after all). If the observing apparatus is “large”—that is, it has many internal degrees of freedom—then we can show that such a large system behaves classically—interference between different possible macroscopic quantum states of the apparatus becomes infinitesimally small, so small as to be irrelevant for all practical purposes.
By the act of measurement, we somehow produce an interaction between this “large” observing system and a “small” quantum system, and these become correlated. This correlation ultimately fixes the small quantum mechanical system to also exist in a single well-defined state, the state we then “measure” the system to be in. In this sense, we say the wave function of the small system has collapsed (meaning the probability amplitude of being in any state other than the one we measure is now zero). Humans have nothing to do with it. The observing system simply has to be large and classical and correlated with the quantum system via a measurement.
This still does not fully resolve the question, which then becomes: how do we determine what comprises the large classical observing part of the combined system and what comprises the quantum part? Feynman spent considerable time discussing this issue with von Neumann. He was not satisfied with von Neumann’s argument that someone had to decide, in some sense arbitrarily, where to make the cut between classical observer and observed. That sounded like a philosophical cop-out. Feynman believed that since quantum mechanics underlies reality, it should be consistently incorporated throughout instead of making ad hoc separations between an observer and the observed. In fact, he worked hard to define measurements purely in terms of correlations between different subsystems and letting the size of one of them go to infinity. If nonzero and finite correlations remained in this limit, Feynman labeled this a “measurement” of the smaller subsystem, which could be made arbitrarily accurate as the size of the “measuring” part of the system grew to be larger and larger. As he colorfully put it in a note he wrote to himself, regarding the example of a spot on a photographic plate that somehow recorded an event involving a single atom,
What can we expect to end with if we say we can’t see many things about one atom precisely, what in fact can we see. Proposal: Only those properties of a single atom can be measured which can be correlated (with finite probability) (by various experimental arrangements) with an unlimited number of atoms. (i.e. the photographic spot is “real” because it can be enlarged and projected on screens, or affect large vats of chemicals, or big brains etc etc—it can be made to affect ever increasing sizes of things—it can determine whether a train goes from N.Y. to Chicago—or an atom bomb explodes—etc).
Measurement theory still remains the bugaboo of quantum mechanics. While great progress has been made, it is still fair to say that a complete description of how the classical world of our experience results from an underlying quantum reality has not been developed, at least to the satisfaction of all physicists.
This example of Feynman’s focus when finishing his thesis is important because it demonstrates the sophisticated issues this mere graduate student in physics insisted on wrestling with as he worked. In addition, Feynman’s “path-integral formalism” made it possible to separate systems into pieces, which seems central to the idea of measurement in quantum mechanics, allowing one to isolate parts of a system one either does not or cannot measure and to separate these clearly from those parts one wishes to focus on. This is generally not possible in standard formulations of quantum mechanics.
The idea is really relatively straightforward. We sum all those weights corresponding to the action associated with those paths or parts of paths we wish to ignore the specific details of, for example, summing up the effect of having small circular loops—so small we could never measure them—swirling around the more normal straight trajectories between two points. The effect of these additions may be to change, by a small calculable amount, what would be the action associated with a straight trajectory without the loops. After having done the summation (or in the case of an infinite number of such additional paths, the integral), we can then forget about such extra loop trajectories and focus on only trajectories that are more straight, as long as we use the new, altered action for this trajectory in our calculations. This process is called integrating out parts of the system.
This may seem, at first sight, like a technical detail not worth mentioning. However, as we will see later, it ultimately would allow almost all of the most important theoretical advances in fundamental physics in the twentieth century to occur, and it would allow us to totally quantitatively revolutionize what are otherwise such vague notions as scientific truth.
But for the moment, in 1942 as he completed this thesis, titled “The Principle of Least Action in Quantum Mechanics,” Feynman had other things on his mind. Preparing to graduate that June, he had received word that he was to move to Los Alamos to focus on the actual building of the atomic bomb. He was also busy planning for his long hoped-for marriage following his graduation. He thus had to put his immediate physics concerns to rest and focus on sorting out the rest of his life. Perhaps all of the diversions explain why, even as he acknowledged Professor Wheeler for his advice and encouragement, he never took the opportunity to add what surely would have been a far more poetic acknowledgment to the connection made between the subject of the thesis (and ultimately the work that would win him the Nobel Prize) and that fateful afternoon in his high school physics class when Mr. Bader awakened his mind to the subtle beauties of theoretical physics.
Three years and what undoubtedly seemed like a lifetime later, the war had ended and he finally got around to writing up his thesis for publication. He still did not make this connection. Instead he was able to clearly enunciate what undoubtedly had been the very same hopes he carried with him as he left Princeton, and which had continued to buoy him through the various immediate insanities of the world of human affairs over which he had so little control, until the day he was finally free to explore full-time the more intoxicating insanities of the quantum universe, which he felt much more confident he could conquer:
The formulation is mathematically equivalent to the more usual formulations. There are therefore no fundamentally new results. However there is pleasure in recognizing old things from a new viewpoint. Also there are problems for which the new viewpoint offers a distinct advantage. . . . In addition, there is always the hope that the new point of view will inspire an idea for the modification of present theories, a modification necessary to encompass present experiments.
CHAPTER 6
Loss of Innocence
He’s another Dirac. Only this time human.
—EUGENE WIGNER, SPEAKING
ABOUT RICHARD FEYNMAN
Richard Feynman graduated with a PhD from Princeton in 1942 as a relatively naive and hopeful young man, known to his fellow students and professors as a brilliant and brash intellect, but largely unknown outside the university. He emerged three years later, from Los Alamos, as a well-tested physicist highly regarded by most of the major players in physics around the world, and a somewhat jaded and world-weary adult. Along the way, he experienced incredible personal loss, as well as the loss of intellectual and moral innocence that is the inevitable by-product of war.
THE INK HAD barely dried on Fe
ynman’s diploma when he began to execute his decision, outlined in that dispassionate letter to his mother, to marry Arline. The opposition by his parents and Arline’s, who were more concerned about his health and Arline’s than about their mutual love, was futile. Both he and Arline felt the other was a bastion against any onslaught from the rest of the world. Together anything was possible, and they refused to be pessimistic about the future. As Arline wrote to Richard shortly after he had moved into a new flat in Princeton and made final arrangements for the ceremony, “We’re not little people—we’re giants. . . . I know we both have a future ahead of us—with a world of happiness—now and forever.”
Every aspect of their brief lives together is, in retrospect, heart wrenching. On what would be their wedding day, Richard borrowed a station wagon from a friend, which he outfitted with mattresses so Arline could lie down. Then he drove from Princeton to her parents’ home and picked her up in her wedding dress, and together they drove to Staten Island for a wedding ceremony with no family or acquaintances, and from there to what would become Arline’s temporary new home, a charity hospital in New Jersey.
Shortly thereafter, without any fanfare or honeymoon, Feynman returned to work at Princeton, except there was nothing to do. The project with Wilson had been closed down and the team was waiting for new orders. Since the main activity at the time was taking place in Chicago, where Enrico Fermi and Wheeler were working on building a nuclear reactor, Feynman was sent to Chicago to learn what was going on there.
His trip in 1943 began what would be a succession of opportunities to ultimately meet and impress his peers and his bosses. While the war disrupted all lives, in at least two senses it provided Feynman with incredible opportunities he would not otherwise have had.
First, since the best and brightest minds were being gathered to spend two years in close quarters, Feynman was given a chance to shine in front of individuals he would have otherwise had to travel around the world to meet. He had already, through his attendance since 1942 at periodic group meetings in New York and at the MIT Radiation Laboratory in Massachusetts, impressed the brilliant, if troubled, physicist Robert Oppenheimer, who would shortly be chosen to lead the entire atomic bomb project. In Chicago, while carrying out his job of gathering information, he blew away members of the theory group there when he was able to perform a calculation that had eluded them for over a month.
Quantum Man: Richard Feynman's Life in Science Page 7