A little while ago I was asked to give a lecture at the very elegant Weisman Art Museum in Minneapolis on an assigned title: “Elegance in Physics.” As I get older, the things I'm asked to do get stranger, so I wasn't surprised. Alarmingly, the older I get, the stronger is my inclination to do the peculiar ones. So I accepted the invitation, and soon found myself brooding, not, as I had imagined, about the glory of the eternal verities, but about the highly contentious nature of elegance in physics. Here is the first such difference of opinion I came upon. In a lecture at Fermilab with a title similar to the one I was given, Subrahmanyan Chandrasekhar talked about “harmoniously organizing a domain of science with order, pattern, and coherence.” He cited five examples of such pinnacles of exposition, one of them being Paul Dirac's celebrated book, Principles of Quantum Mechanics. “The translucence of the eternal splendor through material phenomena,” Chandrasekhar remarked, “[is] made iridescent in these books.” Keeping that iridescent translucence firmly in mind, consider the following remarks of the eminent mathematician Jean Dieudonné: When one arrives at the mathematical theories on which quantum mechanics is based, one realizes that the attitude of certain physicists in the handling of these theories truly borders on delirium … One has to ask oneself what remains in the mind of a student who has absorbed this unbelievable accumulation of nonsense, real hogwash! It would appear that today's physicists are only at ease in the vague, the obscure, and the contradictory [1]. What is Dieudonné talking about? He is addressing the approach to quantum mechanics laid out in Dirac's book. Elegance in physics is as much in the eye of the beholder as it is in any other field of human endeavor. Dirac's formulation appeals to physicists because, by being a little vague and ambiguous about its precise mathematical structure, it enables them to grasp and manipulate the physical content of the theory with a clarity and power that would be greatly diminished if one were distracted by certain complicating but fundamentally uninteresting mathematical technicalities. But for mathematicians, those minor technical matters lie at the heart of the subject. Quantum mechanics becomes ill-formulated and grotesque if it does not properly rest on impeccable mathematical foundations.
A little while ago I was asked to give a lecture at the very elegant Weisman Art Museum in Minneapolis on an assigned title: “Elegance in Physics.” As I get older, the things I'm asked to do get stranger, so I wasn't surprised. Alarmingly, the older I get, the stronger is my inclination to do the peculiar ones. So I accepted the invitation, and soon found myself brooding, not, as I had imagined, about the glory of the eternal verities, but about the highly contentious nature of elegance in physics. Here is the first such difference of opinion I came upon. In a lecture at Fermilab with a title similar to the one I was given, Subrahmanyan Chandrasekhar talked about “harmoniously organizing a domain of science with order, pattern, and coherence.” He cited five examples of such pinnacles of exposition, one of them being Paul Dirac's celebrated book, Principles of Quantum Mechanics. “The translucence of the eternal splendor through material phenomena,” Chandrasekhar remarked, “[is] made iridescent in these books.” Keeping that iridescent translucence firmly in mind, consider the following remarks of the eminent mathematician Jean Dieudonné: When one arrives at the mathematical theories on which quantum mechanics is based, one realizes that the attitude of certain physicists in the handling of these theories truly borders on delirium … One has to ask oneself what remains in the mind of a student who has absorbed this unbelievable accumulation of nonsense, real hogwash! It would appear that today's physicists are only at ease in the vague, the obscure, and the contradictory [1]. What is Dieudonné talking about? He is addressing the approach to quantum mechanics laid out in Dirac's book. Elegance in physics is as much in the eye of the beholder as it is in any other field of human endeavor. Dirac's formulation appeals to physicists because, by being a little vague and ambiguous about its precise mathematical structure, it enables them to grasp and manipulate the physical content of the theory with a clarity and power that would be greatly diminished if one were distracted by certain complicating but fundamentally uninteresting mathematical technicalities. But for mathematicians, those minor technical matters lie at the heart of the subject. Quantum mechanics becomes ill-formulated and grotesque if it does not properly rest on impeccable mathematical foundations.
Attitudes toward quantum mechanics differ interestingly from one generation of physicists to the next. The first generation are the founding fathers, who struggled through the welter of confusing and self-contradictory constructions to emerge with the modern theory of the atomic world and supply it with the “Copenhagen interpretation.” On the whole they seem to have taken the view that while the theory is extraordinarily strange (Bohr is said to have remarked that if it didn't make you dizzy then you didn't really understand it), the strangeness arises out of some deeply ingrained but invalid modes of thought. Once these are recognized and abandoned the theory makes sense in a perfectly straightforward way. The word “irrational,” which appears frequently in Bohr's early writings about the quantum theory, is almost entirely absent from his later essays. The second generation, those who were students of the founding fathers in the early postrevolutionary period, seem firmly—at times even ferociously—committed to the position that there is really nothing peculiar about the quantum world at all. Far from making bons mots about dizziness, or the opposite of deep truths being deep truths, they appear to go out of their way to make quantum mechanics sound as boringly ordinary as possible. The third generation—mine—were born a decade or so after the revolution and learned about the quantum as kids from popular books like George Gamow's. We seem to be much more relaxed about it than the other two. Few of us brood about what it all means, any more than we worry about how to define mass or time when we use classical mechanics. In contemplative moments some of us think the theory is wonderfully strange and others think it isn't; but we don't hold these views with great passion. Most of us, in fact, feel irritated, bored, or downright uncomfortable when asked to articulate what we really think about quantum mechanics. I'm one of the uncomfortable ones. If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, it would be “Shut up and calculate!” But I won't shut up. I would rather celebrate the strangeness of quantum theory than deny it, because I believe it still has interesting things to teach us about how certain powerful but flawed verbal and mental tools we once took for granted continue to infect our thinking in subtly hidden ways.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
reprinted in the British trade journal Physics World in 1990, three separate and 5 lengthy replies from establishment physicists were printed in subsequent issues. For outsiders, especially scientists who rely on physicist's theories in their own fields, this situation is disquieting. Moreover, many recall their introduction to quantum mechanics as a startling, if not shocking, experience. A molecular biologist related how he had started in theoretical physics but, after hearing the ideology of quantum mechanics, marched straight to the Reg istrar's office and switched fields. A colleague recalled how her undergraduate chemistry professor religiously entertained queries from the class - until one day he began with the words: "No questions will be permitted on today's lecture." The topic, of course, was quantum mechanics. My father, an organic chemist at a Midwestern university, also had to give that dreaded annual lecture. Around age 16, I picked up a little book he used to prepare and was perplexed by the author's tone, which seemed apologetic to the point of pleading. It was my first brush with the quantum theory. 6 Eventually, I went to graduate school in physics. By then I had acquired an historical bent, which developed out of an episode in my freshman year in college. To relieve the tedium of the introductory physics course, I set out to understand Einstein's theory of relativity (the so-called Special Theory of 1905, not the later and more difficult General Theory of 1915). This went badly at first.
Written for advanced undergraduates, physicists, and historians and philosophers of physics, this book tells the story of the development of our understanding of quantum phenomena through the extraordinary years of the first three decades of the twentieth century. Rather than following the standard axiomatic approach, this book adopts a historical perspective, explaining clearly and authoritatively how pioneers such as Heisenberg, Schrodinger, Pauli and Dirac developed the fundamentals of quantum mechanics and merged them into a coherent theory, and why the mathematical infrastructure of quantum mechanics has to be as complex as it is. The author creates a compelling narrative, providing a remarkable example of how physics and mathematics work in practice. The book encourages an enhanced appreciation of the interaction between mathematics, theory and experiment, helping the reader gain a deeper understanding of the development and content of quantum mechanics than any other text at this level.
How is it that physicists can invent equations, so rich in structure, so detailed in consequences, and so closely in agreement with experience? What does it mean when we create equations that agree with experiment to ten decimal places? Do these equations have a reality of their own? This essay is based on a lecture originally presented at the Trieste conference in honor of Paul Dirac in 1972 . It is more than fifty years since Paul Dirac gave us one of the most beautiful inventions of the twentieth century: H = c a · p + β mc 2 I once asked Professor Dirac how he came upon his masterpiece. If my memory serves (I hope he will correct me if it does not), he said that in the late 1920s most physicists, trying to calculate with relativistic quantum mechanics, were content to use the Klein–Gordon equation modified with ad-hoc corrections. In his attempt, Dirac did not try to put in the electron spin but rather to treat in a consistent manner the simplest possible problem: that of a spinless, relativistic electron. The critical moment came when he realized that 4 × 4 matrices were required. “How long did it take you from that point?” “About two weeks”. I couldn't refrain: “why so long?” He was a bit taken aback; then, his attention focused perhaps on those two glorious past weeks, he nodded as though to agree in silence why so long?