Bibliography

Walter Thirring. Introduction to Quantum Electrodynamics (Einführung in die Quantenelektrodynamik. Walter Thirring. Wien, Franz Deuticke, 1955).

Thirring’s book Introduction to Quantum Electrodynamics, unlike all other existing monographs on quantum electrodynamics, is very small in size—only VIII + 122 small pages. Moreover, again unlike all other similar books, the formulas in the text are comparatively not so complicated. Nevertheless, the material considered in the book is very extensive.

In the introduction to the book (14 pp.) some formulas of classical (non-quantum) relativistic electrodynamics are given, and, by means of the uncertainty relation, certain phenomena of atomic physics are considered, giving an idea of the order of magnitude of the quantities with which quantum electrodynamics operates.

In the first part of the book, “Quantization of Free Fields” (39 pp.), the general principles of quantum field theory are successfully formulated; the Lagrange function and the tensors of energy-momentum density and angular momentum are introduced; and the basic relations of second quantization are discussed, as well as the connection between spin and statistics according to Pauli. In the course of the exposition, various commutation functions are then introduced; after this, certain concrete fields are considered—scalar, vector, and spinor fields. In the two remaining sections of this chapter, vacuum fluctuations are analyzed by means of normal products, and the limits of measurability of the electromagnetic field according to Bohr–Rosenfeld are discussed.

In the second part, “Interacting Fields” (37 pp.), the general equations of quantum electrodynamics and their solution by the perturbation method are given. The scattering matrix is introduced and the various types of Feynman diagrams are explained. Next, some of the most important applications of the formulas obtained are considered—the emission of light, the interaction of electrons (and positrons), the Compton effect and the Klein–Nishina formula, nonlinear corrections to Maxwell’s equations, vacuum polarization, and others.

In the third part, “Limits of the Theory” (12 pp.), divergences, the electron’s self-energy, the Lamb shift, the vacuum magnetic moment of the electron, the theory of mass and charge renormalization, the convergence of the expansion of the \(S\)-matrix, and so forth are discussed.

In addition, the book contains a list of notations, two appendices—on Dirac matrices and on singular functions with their Fourier representations in the form of contour integrals—as well as 25 problems with solutions. In the problems a number of calculations omitted in the text are given, connected with the clarification of the properties of the energy tensor, commutation relations, and the definition of the vacuum. Other problems concern the calculation of energy losses by an electron due to radiation in linear and cyclic accelerators, the calculation of effective cross sections for a number of processes, including the scattering of light by light. Of course, in difficulty these problems go beyond the scope of ordinary exercises and constitute a kind of supplement, unfortunately presented too briefly.

It should be emphasized that in this book the general physical principles, the basic mathematical methods, and the most important results of quantum electrodynamics are brought to the fore, rather than its various applications to concrete problems. At the same time, a more detailed examination of the book shows that all this material is set forth not merely qualitatively and in general outline, but quite concretely, with all the mathematical rigor customary at the level of quantum field theory.

One is struck by the author’s ability to present an enormous and complex body of material at such a high level in a small book. He succeeded in this above all thanks to a very compressed style. In the whole book, one may say, there are no superfluous words. Guiding thoughts of the reader, for example, there are very few; although very precisely, definitions of the basic concepts are given and principles are formulated, and all the proofs of the conclusions are outlined. At the same time, inevitably, a certain part of the questions—for example, concerning the vacuum (anomalous) magnetic moment of the electron and the renormalization of mass and charge—is only explained, but not exhaustively solved.

The second and undoubtedly more substantial reason enabling the author to achieve the stated aim with a small volume of the book is the successful choice of the initial principles of the theory. Apparently, quantum electrodynamics is a discipline that has become fairly well established; at least over the last five years no substantial changes have occurred here. Nevertheless, in the choice of its basic propositions there is a certain freedom. The author formulated these propositions in such a way that they could serve as a solid foundation for a general and logically consistent theory.

As an example we shall cite one of Thirring’s initial propositions: “To every canonical transformation of any classical system \(S\) there corresponds a unitary transformation of the quantum system \(S\) in such a way that the generator of the infinitesimal unitary transformation is constructed similarly to the generator of the corresponding infinitesimal canonical transformation.” This principle, which is a certain physical interpretation of Schwinger’s principle, is cited by the author after he has accepted that to every state of the system \(S\) there corresponds a certain vector \(D\) in the representation space \(E\), and to every dynamical variable \(o\) there corresponds a certain Hermitian operator \(O\) in \(E\), and that the possible results of measuring the quantity \(o\) and their probabilities are connected in the usual way with the vector of the state \(D\), as well as with the eigenvalues and eigenvectors of the operator \(O\). In view of the definitions given by Thirring, the content of this principle is as follows.

Let \(\psi^\alpha(x_i)\) be \(N\) field operators in \(E\), each of which depends on the coordinates \(x_i\) in Minkowski space \(R\). Let \(L(\psi^\alpha,\ \psi_i^\alpha)\) be a Lagrange function depending on the operators \(\psi^\alpha\) and on their derivatives \(\psi_i^\alpha\) with respect to \(x_i\). Let \(\sigma\) be a space-like surface in \(R\), and

\[ W_{12}=\int_{\sigma_1}^{\sigma_2} L(dx_i) \]

the generalized action.

Each operator \(O\) is regarded as an integral over \(\sigma\) of some function of the operators \(\psi^\alpha\): \(O=O(\psi^\alpha,\sigma)\). With this notation, the above-mentioned Thirring principle means, first, that if \(O(\psi_1^\alpha,\sigma_1)\) and \(O(\psi_2^\alpha,\sigma_2)\) are one and the same operator \(O\) for different choices of the surface \(\sigma\) and of the representation of the operators \(\psi^\alpha\), then there exists a relation of the form

\[ O(\psi_2^\alpha,\sigma_2)=U_{12}^{-1}O(\psi_1^\alpha,\sigma_1)U_{12}, \]

where \(U_{12}\) is some unitary matrix, and, secondly, that if \(\sigma_1,\sigma_2\) and \(\psi^\alpha(x_i)\) are subjected to an infinitely small variation, then the corresponding increments \(U_{12}\) and \(W_{12}\) will be connected by the relation

\[ \delta U_{12}=\delta W_{12}U_{12}. \]

From this principle there are derived, first, the Euler equations of motion

\[ -\frac{\partial L}{\partial \psi_i^\alpha}-\frac{\partial}{\partial x_i}\frac{\partial L}{\partial \psi_i^\alpha}=0, \]

secondly, formulas expressing the charge \(Q\), the energy-momentum vector \(P_i\), the angular-momentum tensor \(J_{ih}\), and also the densities of these quantities \(j_i, T_{ik}, M_{ikl}\) through the function \(L\); thirdly, commutation relations for the functions \(\psi^\alpha\); and, fourthly, commutation relations of the type \([\psi_k^\alpha,P_k]=i\psi_k^\alpha\) and analogous relations for the commutators \([\psi^\alpha,Q]\) and \([\psi^\alpha,J_{ih}]\).

Further, the book contains a number of original approaches, for example to the definition of the vacuum that preserves the definite metric in \(E\), the introduction of normal products already at the specification of the Lagrange function, the derivation of the basic properties of \(\psi^r\) as creation and annihilation operators for different particles independently of the choice of \(L\), and so on.

Nevertheless, for a better understanding of the book the author ought to have given more explanations of the basic concepts and principles. For example, it would have been good to give some representations of the operators \(\psi^\alpha\) for a definite choice of coordinate vectors in \(E\); to clarify the question of the existence of operators \(O\) and \(U\) satisfying the above-mentioned principle for any choice of the surfaces \(\sigma_1\) and \(\sigma_2\); to define how one should understand derivatives of the function \(L\) when it is a function, in particular a non-analytic one, of noncommuting operators; to specify how one should understand the assertion that \(\psi^\alpha(x_i)\) are creation and annihilation operators of particles at the point \(x_i\), without having an operator that gives the number of particles at a given point or in a given region, etc.

Although the brevity of the book is one of its merits, and should not be impaired, one would nevertheless like to see here a few additional comments also on the derivation of the Pauli theorem, which connects spin with statistics in the spirit of Schwinger, as well as on the influence of the finiteness of the charge on the limits of measurability.

The author was able, in a number of places, to note the principal difficulties of the theory, but did not indicate possible ways of overcoming them. However, there are at present no established views on this question. Let us note, incidentally, that the successful development of quantum mesodynamics over the last year and a half or two years, connected with the establishment of dispersion relations, the equation of \(J_0\), and the refinement of the coupling constant, makes pessimistic statements by Thirring about the state of meson theory obsolete.

The author calls this book an “Introduction” and says in the preface that he assumes the reader is acquainted only with mathematical analysis, linear algebra, special relativity, and elementary quantum mechanics. It seems to us, however, that this book, by an authoritative and active author, with its original derivations and its many interesting remarks, may be more useful to specialists already somewhat familiar with quantum electrodynamics.

This book is especially useful now, when the theory has become so complex and almost all courses so bulky. It will undoubtedly enable the reader—whether a student or graduate student in theoretical physics, or a mature specialist—to clarify more deeply the basic concepts and laws of quantum electrodynamics and, in particular, of quantum field theory, and thereby to contribute to the further creative development of this most important branch of modern theoretical physics.

We consider a Russian translation of Thirring’s book highly desirable. In view of the small size of the book, it seems advisable to publish it together with another, likewise comparatively small but somewhat more elementary and interesting book on quantum electrodynamics by Prof. Dyson, a well-known specialist in the latest quantum electrodynamics.

D. Ivanenko and Chr. Christov