Page xii, “the belief in an objective reality … ”: M. Kakutani, “How technology has changed modern culture”, New York Times, 28 November 1989.

Page 2, “[the theory of relativity] is a modification … ”: Herbert Goldstein, Classical Mechanics (Addison-Wesley, Reading, Massachusetts, 1950) page 185.

Page 2, “Nobody feels perfectly comfortable with it.”: Murray Gell-Mann, “Is the whole universe composed of superstrings?”, in Modern Physics in America, edited by W. Fickinger and K. Kiwakski (American Institute of Physics, New York, 1987) page 186.

Page 2, “I can safely say … ”: Richard Feynman, The Character of Physical Law (MIT Press, Cambridge, Massachusetts, 1965) page 129.

Page 3, The threshold wavelength for stripping one electron from a helium atom: Experiment: S.D. Bergeson et al., “Measurement of the He Ground State Lamb Shift”, Physical Review Letters, 80 (1998) 3475–3478. Calculation: G.W.F. Drake, “High precision calculations for helium”, in Atomic, Molecular, and Optical Physics Handbook, edited by G.W.F. Drake (American Institute of Physics, Woodbury, New York, 1996) pages 154–171.

Page 4, “One can popularize the quantum theory … ”: N.D. Mermin, Space and Time in Special Relativity (Waveland Press, Prospect Heights, Illinois, 1968) page vii.

R.P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University Press, Princeton, New Jersey, 1985). My favorite book about physics, and the best place to turn if you want to learn more about quantum mechanics after finishing this book. After an inspiring introduction (pages 1–12), Feynman skillfully sets up the framework of quantum mechanics (pages 13–83) and then goes on to give the specific rules — within that framework — for assigning amplitudes for a class of phenomena called “electrodynamics” (pages 83–130). The remainder of the book (pages 130–152) surveys those parts of nature that fall outside of the domain of electrodynamics, and briefly shows how they, too, fit into the quantal framework.

R.P. Feynman, The Character of Physical Law (MIT Press, Cambridge, Massachusetts, 1965). Chapter 6, “Probability and uncertainty — the quantum mechanical view of nature”, is the best one-hour summary of quantum mechanics that I know. It is the transcript of a lecture that was also recorded on film, and viewing the film is even better than reading the transcript. The video recording is distributed by Education Development Center, Inc.; 55 Chapel Street; Newton, Massachusetts 02158–1060.

P.C.W. Davies and J.R. Brown, The Ghost in the Atom (Cambridge University Press, Cambridge, UK, 1986). Interviews with quantum physicists at the popular level.

Up to now this book has focused on the behavior of nature. I could say more: more about measurement, more about the classical limit, more about different rules for assigning amplitudes, and so forth, but the main points have been made. So instead of talking more about nature I'm going to talk about people — about how people discovered quantum mechanics.

Warnings

I am not a historian of science. The history of science is a very difficult field. A historian of science must be just as proficient at science as a scientist is, but must also have a good understanding of personalities, and a good knowledge of the social and political background that affects developments in science and that is in turn affected by those developments. He or she has to know not only the outcome of the historical process, namely the science that is generally accepted today, but also the many false turns and blind alleys that scientists tripped across in the process of discovering what we believe today. He or she must understand not only the cleanest and most direct experimental evidence supporting our current theories (like the evidence presented in this book), but must understand also how those theories came to be accepted through a tightly interconnected web of many experiments, no one of which was completely convincing but which taken together presented an overwhelming argument.

The first section of this chapter shows that the mathematical representation of amplitude cannot be as simple as a real number, but must be at least as complicated as a two-dimensional arrow. If you're willing to accept this as fact, then you may skip that rather technical and involved section. But in no case should you skip over the second section of this chapter, which makes a simple but subtle and important general point.

Amplitude is represented by an arrow

I'm going to introduce one more type of analyzer: the “front–back analyzer” (also called the “y analyzer”). This will be the last new analyzer, I promise. The left half of this analyzer is just like the left half of a traditional Stern–Gerlach analyzer, with its traditional non-uniform magnetic field. But while the right half of the traditional Stern–Gerlach analyzer contains only plumbing to make sure the atoms come out parallel to the sides of the box, the right half of the front–back analyzer contains also a magnetic field that changes direction slowly from place to place. Along the path towards the upper exit, the magnetic field starts by pointing straight up. A little farther on it tilts a bit to the right. The tilt angle of the field increases gradually until, just before the exit, the field points directly to the right. The path towards the lower exit is similar, except that in this case the field starts out pointing down and gradually tilts until it points directly to the left.

Overview

We have uncovered the first central principle of quantum mechanics, which is that the outcome of an experiment cannot, in general, be predicted exactly; only the probabilities of the various outcomes can be found. In particular, for the magnetic arrow of a silver atom, we know:

How do you like it? Do you feel liberated from the shackles of classical determinism? Or do you feel like Matthew Arnold, who wrote in Dover Beach that

Regardless of your personal reaction, it is our job as scientists to describe nature, not to dictate to it!

In particular, we know that the model of a magnetic needle as an arrow, so carefully developed in chapter 2 and so correct within the domain of classical mechanics, must be wrong.

I interpreted the repeated measurement experiments of the previous chapter by saying that quantum mechanics can find probabilities only, not certainties (that is, that quantum mechanics is “probabilistic”, not “deterministic”). You may object, maintaining that the world is deterministic, but that my particular deterministic scheme (the “magnetic arrow”) is incorrect. The next chapter presents an ingenious argument, invented by Einstein, which shows that no local deterministic scheme could give the results observed by experiment. In order to understand that argument you need some background in probability.

But in fact, a knowledge of probability is generally useful in day-to-day life as well as in physics. You walk across a street — what is the probability of your being hit by a car? You are advised to undergo elective surgery — what is the probability that the surgery will extend your life, and what is the probability that the surgery will go wrong and injure you? You breath some asbestos or smoke a cigarette — what is the probability of contracting cancer? Misconceptions about probability abound and can lead to disastrous public policy decisions. A knowledge of quantum mechanics is good for your soul, but it is of practical importance only to the designers of lasers, transistors, and superconductors. A knowledge of probability is of practical importance to everyone.

Another book on quantum field theory?

There is absolutely no question that there are many excellent texts on the general subject of Quantum Field Theory. The field is vast and there are good reasons for such a large number of books. Generally, each book adopts a point of view favored by the author, and this is how it should be (indeed, this book will be no exception). Despite the fact that there are so many different important topics to address, it is no exaggeration to state, generally speaking, that most of these books cover common ground. There are chapters on canonical quantization techniques, free boson and fermion fields, Abelian gauge fields, interactions with external sources, coupled fields, perturbation theory with an emphasis on Feynman graphs, dealings with divergences: regularization and renormalization, path integral and functional integration formulations, non-Abelian gauge fields, and more modern topics such as the renormalization group and dimensional regularization. These and related tools and techniques are well honed and serve magnificently for studying a wide variety of interactions that describe the real world. But perhaps less appreciated is the fact that these methods do not provide a satisfactory quantization scheme for all classical field theories that one would like to deal with. At present there is, in the author's opinion, no satisfactory quantum theory of self-interacting relativistic scalar fields in four and more space-time dimensions, and while there is no lack of attempts there is no consensus on a satisfactory theory of the quantum gravitational field.

WHAT TO LOOK FOR

In the previous chapters, we have developed a rather standard view toward classical and quantum theory illustrating the three canonical quantization procedures, namely those of Schrӧdinger, Heisenberg, and Feynman. In this chapter we want to return to Hilbert space and analyze several new issues as well as some old issues in greater depth than was previously the case. These issues will include operators and their domains, bilinear forms, infinite-product representations, irreducible representations of the canonical operators, as well as the important notion of “tags” (unitary invariants of operator representations). Some general remarks on measures, probability distributions, characteristic functions, and infinitely divisible distributions are also included here. Additionally, a few comments about Brownian motion are included. We close with some general remarks regarding canonical quantum scalar fields. The properties developed in this chapter will find application in our study of model quantum field theories in subsequent chapters.

Hilbert space and operators, revisited

In Chapter 3 we already introduced and discussed at some length properties of Hilbert spaces suitable for quantum mechanical applications. In all earlier applications we have used the elegant notation of Dirac, and this notation is generally appropriate for most applications.

WHAT TO LOOK FOR

In this chapter we introduce the subject of Lagrangian and Hamiltonian formulations, following descriptions that focus on action functionals and their stationary variation to arrive at the relevant equations of motion. Initially we deal with one degree of freedom, and then with finitely many degrees of freedom. These studies are standard and introductory to the case of an infinite number of degrees of freedom, namely the case of fields. In the classical theory, the infinite number of variables case can be approached as a limit of the finite number of variables cases as the number of those variables tends to infinity. This seems completely reasonable, and, more or less, may be taken as an unwritten rule of how to treat classical systems with an infinite number of variables. Unfortunately, as we shall learn much later, this unwritten rule will not always carry over to the quantum theory, at least in any obvious and straightforward fashion.

Lagrangian classical mechanics

Without undue exaggeration, the goal of classical dynamics may be said to be the introduction of dynamical equations of motion for point particles, or for entities that may be idealized as point-like particles, and the analysis of properties of the solution to such equations, i.e., the time-dependent path x(t) describing the classical motion.