Quantum Electrodynamics

Author: V.B. Berestetskii et al.
Publisher:
Publish Date: 1999-06-01
Features: THE first edition of this volume of the Course of Theoretical Physics was published in two parts (1971 and 1974) under the title "Relativistic Quantum Theory". It contained not only the basic material on quantum electrodynamics but also chapters on weak interactions and certain topics in the theory of strong interactions. The inclusion of those chapters now seems to us inopportune. The theory of strong and weak interactions is undergoing a vigorous development founded on new physical ideas, and the situation in this field is changing very rapidly, so that the time for a consistent exposition of the theory has not yet arrived. In the present edition, therefore, we have retained only quantum electrodynamics, and accordingly changed the title of the volume. This book is an English edition.
Excerpt:
INTRODUCTION?1. The uncertainty principle in the relativistic case
THE quantum theory described in Volume 3 (Quantum Mechanics) is essentially non-relativistic throughout, and is not applicable to phenomena involving motion at velocities comparable with that of light. At first sight, one might expect that the change to a relativistic theory is possible by a fairly direct generalization of the formalism of non-relativistic quantum mechanics. But further considerations show that logically complete relativistic theory cannot be constructed without invoking new physical principles. Let us recall some of the physical concepts forming the basis of non-relativistic quantum mechanics (QM,?). We saw that one fundamental concept is that of measurement, by which is meant the process of interaction between a quantum system and a classical object or apparatus, causing the quantum system to acquire definite values of some particular dynamical variables (coordinates, velocities, etc.). We saw also that quantum mechanics greatly restricts the possibility that an electron simultaneously possesses values of different dynamical variables. For example, the uncertainties Δq and Δp, simultaneously existing values of the coordinate and the momentum, are related by the expression ΔqΔp ~ ?; the greater the accuracy with which one of these quantities is measured, the less the accuracy with which the other can be measured at the same time. It is important to note, however, that any of the dynamical variables of the electron can individually be measured with arbitrarily high accuracy, and in an arbitrarily short period of time. This fact is of fundamental importance throughout non-relativistic quantum mechanics. It is the only justification for using the concept of the wave function, which is a basic part of the formalism. The physical significance of the wave function (q) is that the square of its modulus gives the probability of finding a particular value of the electron coordinate as the result of a measurement made at a given instant. The concept of such a probability clearly requires that the coordinate can in principle be measured with any specified accuracy and rapidity, since otherwise this concept would be purposeless and devoid of physical significance. The existence of a limiting velocity (the velocity of light, denoted by c) leads to new fundamental limitations on the possible measurements of various physical quantities (L.D. Landau and R.E. Peierls, 1930). In QM,?4, the following relationship has been derived: relating the uncertainty Δp in the measurement of the electron momentum and the duration of the measurement process itself; v and v' are the velocities of the electron before and after the measurement. From this relationship it follows that a momentum measurement of high accuracy made during a short time (i.e. with Δp and Δt both small) can occur only if there is a large change in the velocity as a result of the measurement process itself. In the non-relativistic theory, this showed that the measurement of momentum cannot be repeated at short intervals of time, but it did not at all diminish the possibility, in principle, of making a single measurement of the momentum with arbitrarily high accuracy, since the difference v' - v could take any value, no matter how large. The existence of a limiting velocity, however, radically alters the situation. The difference v' - v, like the velocities themselves, cannot now exceed c (or rather 2c). Replacing v' - v in (1.1) by c, we obtain which determines the highest accuracy theoretically attainable when the momentum is measured by a process occupying a given time. In the relativistic theory, therefore, it is in principle impossible to make an arbitrarily accurate and rapid measurement of the momentum. An exact measurement is possible only in the limit as the duration of the measurement tends to infinity. There is reason to suppose that the concept of the measurability of the electron coordinate itself must also undergo modification. In the mathematical formalism of the theory, this situation is shown by the fact that an accurate measurement of the coordinate is incompatible with the assertion that the energy of a free particle is positive. It will be seen later that the complete set of eigenfunctions of the relativistic wave equation of a free particle includes, as well as solutions having the "correct" time dependence, also solutions having a "negative frequency". These functions will generally appear in the expansion of the wave packet corresponding to an electron localized in a small region of space. It will be shown that the wave functions having a "negative frequency" correspond to the existence of antiparticles (positrons). The appearance of these functions in the expansion of the wave packet expresses the (in general) inevitable production of electron-positron pairs in the process of measuring the coordinates of an electron. The formation of new particles in a way that cannot be detected by the process itself renders meaningless the measurement of the electron coordinates.