Recollections of Some Notable Texts

In the fall of 1951, I entered graduate school at Harvard to study mathematics, which had been my undergraduate major. But, truth be told, my real interest was physics. Even the mathematics I studied, such as the theory of Hilbert spaces, had an orientation in physics. After I completed my master’s degree, the department chairman informed me that I had to choose between mathematics and physics. I chose physics.

I had already taken some courses in the physics department. These included an introduction to quantum mechanics by Julian Schwinger, one of the giants in the field of quantum electrodynamics. Schwinger’s course had something of a Dr. Jekyll and Mr. Hyde character to its development. During the fall semester, Schwinger provided a marvelous introduction to the experiments that had led to quantum mechanics. I vividly recall him describing Albert Einstein’s argument that the uncertainty principle connecting energy and time could be violated.¹ It was Niels Bohr who observed that Einstein, in framing his criticism, had neglected his own theory of general relativity. Once gravitational effects were properly taken into account, Einstein’s critique lost its point.

That was the Dr. Jekyll part of the course, now for Mr. Hyde.

Schwinger was developing his own version of quantum theory. His approach used propagators—Green’s functions—to describe the time evolution of matrix elements. As it turned out, Schwinger’s lectures could not have been a worse introduction to the theory. Schrödinger’s equation was somehow derived and then ignored. During one lecture, Schwinger became stuck and had to defer to the following lecture.² When I think of all the things that Schwinger could have taught us, not the least of which was the Dirac equation and some of its applications, I very much regret the loss.

Freeman Dyson later filled in some of the gaps. Dyson had done his graduate work with Hans Bethe at Cornell during the late 1940s. Bethe had been impressed by Dyson’s extraordinary abilities. Having returned to Cornell in 1951 as a member of the faculty, Dyson taught courses in advanced quantum mechanics. The notes from Dyson’s lectures began circulating in samizdat form; I might have paid two dollars for my copy. These lectures remained unpublished until the appearance of an elegant edition by World Scientific in 2007. A number of errors and infelicities in the original notes were corrected by the transcriber, David Derbes.

Advanced Quantum Mechanics

Let me begin by describing what this book is not. There are very few problems to be solved, and there is no discussion of the interpretation of quantum mechanics. In this respect, it resembles Paul Dirac’s The Principles of Quantum Mechanics.³ Dirac once remarked that he thought his was a good book, but that it lacked a first chapter describing the interpretation of quantum mechanics. Steven Weinberg’s Lectures on Quantum Mechanics provides both.⁴ Readers may find it disconcerting to learn that Weinberg has not found a satisfactory interpretation of quantum mechanics. I agree with him.

And if one insists on using quantum theory to describe the past, I think Dyson would too. The past has always been a vexed issue in quantum theory. Examples can be easily constructed that apparently violate the uncertainty principles. Suppose you have a system that ejects a particle at a precise location, A. The particle then travels unimpeded to a detector at B, which measures its momentum. If the momentum at B is taken to be the momentum at A, a violation of the uncertainty principle would seem to be the result, since the particle has a precise location and momentum at A. Dyson has nothing to say about any of this in Advanced Quantum Mechanics.

The book begins by making the case for field theories, which are needed to accommodate many-particle systems. Such systems arise whenever quantum mechanics is merged with special relativity. Consider a spin zero particle that obeys the Klein–Gordon equation,

$\frac{1}{c^2}\frac{{\partial }^2\psi }{\partial t^2}=\frac{{\partial }^2\psi }{\partial x^2}-{\mu }^2\psi$.

This is a second order equation. Both the wave function and its time derivative can be positive or negative. It follows that the probability density derived from the conserved current,

$\rho =\frac{i\hslash }{2m{c}^2}\left({\psi }^{*}\frac{\partial }{\partial t}\psi -\psi \frac{\partial }{\partial t}{\psi }^{*}\right)$,

is not necessarily positive. It must, for this reason, be reinterpreted as a charge density in a quantum theory of fields.

Dyson provides a brief, but clear, introduction to the Dirac equation, and its solution in the classic case of the hydrogen atom. The Dirac equation, Dyson notes, had negative-energy solutions. This led Wolfgang Pauli and Victor Weisskopf to claim that the equation lacked any physical sense.⁵ Dyson offers a simple argument against negative-energy particles. Such particles keep gaining energy and can never be stopped by ordinary matter at rest. The solution to this dilemma is well known. A negative-energy electron is really a positively charged electron, the positron, in disguise.⁶

Dyson’s book is mostly concerned with quantum electrodynamics. This is hardly a coincidence given that the original lectures took place in 1951. While there was other important physics going on at the time, it was quantum electrodynamics that made the deepest impression. This was especially true at Harvard, where Schwinger, one of its developers, was resident.⁷ At the time, I was working with Abraham Klein on the theory of mesons. None of our calculations made much sense. The constant coupling mesons to nucleons had a value of about ten, and this seemed to rule out the standard technique of perturbation expansions. On the other hand, the fine structure constant, α = ~1/137, which characterized electrodynamics, was small enough to permit an expansion series. At one point, Dyson thought that this expansion might converge; if so, the various functions of α would be analytic, and, by analytic continuation, extended to –α. This would have been a disaster, electrons attracting electrons, and positrons, positrons.⁸ The series cannot converge. Dyson never mentions this.

I felt compelled to learn something about quantum electrodynamics.⁹ The obvious place to begin was with Schwinger’s papers. They look impenetrable, both then and now.¹⁰ Another student advised me to read Richard Feynman’s papers. Feynman’s mathematics was less intimidating than Schwinger’s mathematics, which conveyed Green’s function to a transcendental level. But I still could not understand what he was doing. There were a few odd diagrams that were somehow connected to the equations. I was about to give up when I came across Dyson’s paper, “The Radiation Theories of Tomanaga, Schwinger, and Feynman,” which was published in 1949.¹¹ The diagrams began to make sense, and I could see how they were connected to the theory.

One might well describe Advanced Quantum Mechanics as an introduction to quantum electrodynamics, because that is precisely what it is.

QED

Quantum electrodynamics is nearly as old as quantum theory itself. During the 1930s, a problem emerged in the application of quantum mechanics. The theory gave infinite answers. When performing a calculation in quantum electrodynamics (QED), physicists insert a parameter into their equations representing the bare electron mass, m₀. The actual mass, m, of an electron is, by way of contrast, influenced by the sea of electrodynamic fluctuations in which it moves. As it turned out, the actual mass, m, differed from its bare mass, m₀, by an infinite amount. This is an absurdity. The same observation can be made about the electron’s charge. This problem was well known in the 1930s, but there was little incentive to resolve it. Before the war, Hans Kramers had provided an important clue about how infinite corrections might be handled by mass renormalization. Consider a calculation using the bare mass together with a correction designating kinetic energy. If the actual kinetic energy was originally expressed in terms of the bare mass plus this correction, it would be necessary to subtract this infinite term to extract a finite physical answer. This simple algebraic insight is now called renormalization. A theory which can be rendered finite by a finite number of such renormalizations is termed renormalizable. The theory of gravitation is notoriously not renormalizable.

From Dyson’s work, it was known that the only other quantity in QED that required renormalization was the electric charge. Before presenting many of the standard QED calculations, such as the Lamb shift and the magnetic moment of the electron, Dyson makes the following observation:

In this course we follow the pedestrian route of logical development, starting from general principles of quantizations applied to covariant field equations, and deriving from these principles first the existence of particles and later the results of the Feynman theory. Feynman by the use of imagination and intuition was able to build a correct theory and get the right answers to problems much quicker than we can. It is safer and better for us to use the Feynman space-time pictures not as the basis for our calculations but only as a help in visualizing the formulae which we derive rigorously from the field theory. In this way we have the advantages of the Feynman theory, its concreteness and its simplification of calculations, without its logical disadvantages.¹²

This is precisely what I was missing when I tried to read Feynman’s papers all those years ago. Where did his formulae and space-time pictures come from?

Dyson’s book remains a wonderful introduction to QED, as it was in the early 1950s. But QED has, of course, moved on. Consider the electron charge. One might attempt to measure it by electron scattering experiments. These suggest that the charge being measured depends on the energies of the electrons. This results from vacuum polarization. While the vacuum is the state of lowest energy, it is far from empty.¹³ The vacuum swarms with electrons and positrons that appear briefly in accordance with Werner Heisenberg’s uncertainty principle. Observations occur at different length scales, and the effective charge reflects this. Any modern text on QED covers this, but there is something even more fundamental.

At the time of Dyson’s lectures, QED was an island unto itself. As he recently wrote to me:

The notes were written at a time when quantum electrodynamics was a hastily constructed method for calculating radiation processes with an obviously inconsistent mathematical basis. We all expected that the theory would fail and be replaced by something better as soon as the experiments would become more precise. I expected the theory would have a useful life of about five years. The big surprise is that the theory has lasted sixty five years and now agrees with experiments within a few parts per trillion.¹⁴

It was later found that QED had to be combined with the theory of weak interactions to yield a consistent renormalizable theory. This was first established by Weinberg. Since weak interactions are indeed weak, they do not much influence the effects that Dyson computed in his lectures. The converse is not true. The theory of beta decay includes radiative corrections.

Green Master

These reflections have been prompted by news of a forthcoming edition of Schwinger’s Quantum Mechanics, to be published next year in commemoration of the centenary of his birth.¹⁵ The book is composed of course notes from classes Schwinger taught on quantum mechanics. It is comprised of two parts: “Fall Quarter: Quantum Kinematics,” and “Winter Quarter: Quantum Dynamics.”¹⁶ One of the most striking aspects of the original edition of the book is the plethora of student problems. Solutions to these problems will be included in the centenary edition. Still, the book is not a trot. It is an account of Schwinger’s version of quantum mechanics. The prologue is reminiscent of Schwinger’s Dr. Jekyll. Here, he places particular emphasis on the difference between quantum mechanical and classical measurement:

[P]hysics is an experimental science; it is concerned only with those statements which in some sense can be verified by an experiment. The purpose of the theory is to provide a unification, a codification, or however you want to say it, of those results which can be tested by means of some experiment. Therefore, what is fundamental to any theory of a specific department of nature is the theory of measurement within that domain.¹⁷

It is disappointing that the book does not contain any discussion of the paradoxical fact that standard quantum mechanics does not and cannot describe the measurement of quantum-mechanical properties. Suppose we have a system described by a wave function $\psi \left(x,t\right)$ which obeys the Schrödinger equation,

$i\hslash \frac{\partial }{\partial t}\psi \left(x,t\right)=\mathrm{H}\psi \left(x,t\right)$,

where H is the Hamiltonian. Solutions are perfectly time reversible. If we know $\psi$ at some time we can both predict and retrodict its value at any past or future time. Associated with the system are the observables of momentum, angular momentum, and the like. These are represented by Hermitian operators A, B, …, which take eigenvalues a, b, …. These are the predicted results of any possible quantum mechanical measurement. The wave function, $\psi$, can be expanded in a basis of any of these eigenfunctions,

$\psi =\mathrm{\Sigma }{\psi }_a{c}_a$,

where the c_a’s are complex coefficients. We have known since the work of Max Born that the absolute values of these coefficients represent the relative probability of finding the system in a particular state. Such is the collapse of the wave function. The term does not appear in Schwinger’s book. Before measurement, the wave function is in a superposition of quantum states, and afterwards, in just one. The collapse is irreversible and cannot be described by Schrödinger’s equation itself.

Bohr argued that measurement was classical, while the state it was measuring was quantum mechanical. He was never able to provide a precise boundary between the two. This issue is ignored by most physicists working today since they have no trouble in making the distinction on a practical level. Schwinger also ignores the question. Weinberg, on the other hand, does consider this question in his Lectures on Quantum Mechanics, but concludes that none of the proposals on offer is satisfactory.

An interpretation that Weinberg does not consider in his book is the quantum mechanics of David Bohm.¹⁸ When Bohm published the first of his papers in 1952, I discussed it briefly with Schwinger. I suggested that it looked too complicated. Schwinger replied it looked too simple. Einstein was of a similar mind when he wrote to Max Born in 1952:

Have you noticed that Bohm believes (as [Louis] de Broglie did, by the way, 25 years ago) that he is able to interpret the quantum theory in deterministic terms? That way seems too cheap to me. But you, of course, can judge this better than I.¹⁹

In his work, Bohm introduced a deterministic pilot wave guiding the evolution of a quantum system. All the results of non-relativistic quantum mechanics are reproduced. There is nothing special about an apparatus. John Bell was the first to point out that if you had, say, two interacting particles, the behavior of one would be influenced by the instantaneous behavior of the other. This is a clear violation of relativity. Yet within its limited domain Bohm did provide another way of approaching non-relativistic quantum mechanics.

Eugene Wigner once suggested that attaching a small piece to the Hamiltonian would provoke the collapse of the wave function. Variations have also been suggested. Weinberg does not consider this possibility. Instead, he focuses on what is termed the many-world, or many-history, interpretation of quantum mechanics, first introduced by Hugh Everett III in his 1957 Princeton dissertation. In measurement, the wave function does not collapse, but one branch is singled out, and the rest propagate freely, only to split into more and more branches as time goes on. I do not see anything logically wrong with this, but it is certainly inelegant.²⁰ Weinberg concludes that none of these approaches is satisfactory and that a new theory may be required, to which quantum mechanics is an excellent approximation.

So, we have three books.

Dyson’s Advanced Quantum Mechanics is a superb introduction to quantum electrodynamics.

Schwinger’s Quantum Mechanics is a virtuoso display of mathematics applied to familiar problems.

Weinberg’s Lectures on Quantum Mechanics is the one to choose if you want to learn how quantum mechanics is used by the common man with a PhD in physics.