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.

WHAT TO LOOK FOR

Although the classical theory developed in Chapter 2 can be used to describe a great many phenomena in the real world, there is little doubt that the proper description of the world is quantum. To a large measure we perceive the world classically and we must find ways to uncover the underlying quantum theory. The process of “quantization” normally consists of turning a classical theory into a corresponding quantum theory. In turn, the “classical limit” is the process by which a quantum theory is brought to its associated classical theory. The parameter ħ = 1.0545 × 10−27 (≃ 10−27) erg-seconds (cgs units), referred to as Planck's constant, sets the scale of quantum phenomena. In many applications it will prove useful to use “natural” units in which ħ = 1 just so that formulas are less cumbersome. For most typical formulations of quantum mechanics the classical limit means the limit that ħ → 0, and in this text we shall have many occasions to use this definition as well. In addition, we shall also learn to speak about the classical and quantum theories simultaneously coexisting – as they do in the real world – without the need to take the limit ħ → 0.

There are three generally accepted “royal” routes to quantization, due principally to Heisenberg, Schrödinger, and Feynman, respectively, and we shall exploit them all. In each case we begin with a single degree of freedom (N = 1).

WHAT TO LOOK FOR

In its simplest characterization, the quantum theory of scalar fields is nothing but the quantum mechanics of N canonical degrees of freedom in the limit that N → ∞. Of course, things are not quite that simple since sequences do not always have the virtue of converging, and even if they do converge they do not always converge to acceptable limits. However, in this chapter we take a simple, pragmatic point of view and assume that any needed limits converge and, in addition, that the resultant answers are acceptable. Partway through this chapter we introduce units in which ħ = 1.

Classical scalar fields

For purposes of the present section we let g(x) denote a real scalar field defined for x ∈ ℝn, where n ≥ 1 denotes the dimension of space-time. If n = 1 then that single dimension is the time dimension and so there is no space dimension – hence no space – and thus one is really back in the case of classical mechanics. We shall have essentially no occasion to study the case n = 1 in this chapter. For n ≥ 2 the number of space dimensions is s ≡ n − 1 ≥ 1. (We note that the symbol n has several uses in this chapter, often as a dummy index of summation, but the context is generally clear in each case.)