Introduction

The many-body systems which we consider in this book are met in several different physical contexts, and so the ideas presented here can be applied to problems which arise in various branches of physics. In nuclear physics, atomic physics, solid-state physics, and low-temperature physics, the types of problem which we discuss arise frequently. It is not so obvious that they also occur in elementary particle physics, but the fact that ideas developed for the study of elementary particles have been applied to many-body problems is an indication that they do indeed occur.

We discuss only those systems for which the symmetry or antisymmetry of the wave functions with respect to interchange of the particles has a dominating influence on the properties of the system. There are, of course, other many-body systems whose properties can only be explained in terms of quantum mechanics; a gas of diatomic molecules is such a system. For such a system, quantum theory is needed only to explain the properties of the individual molecules, and the many-body problem can then be solved without further reference to quantum mechanics. The properties of the systems with which we are concerned here can only be understood in terms of the quantum theory of many-body systems. It is not possible to regard the understanding of the quantum effects and of the many-body properties as two distinct problems. With such systems, it is of the utmost importance whether the constituent particles are bosons, so that the wave function is symmetric, or fermions, so that the wave function is antisymmetric.

The best examples of this sort of system are the two isotopic forms of liquid helium. Both appear to be liquids right down to zero temperature, and this is a fact which cannot be explained by classical mechanics. There is a striking difference between the properties of the two isotopes, much greater than would be expected from the difference in mass between them. Liquid He4 is superfluid below 2.2 °K, flowing in narrow tubes with no apparent viscosity, while liquid He3 behaves like a normal fluid down to less than 0.003&#176K. This difference is almost certainly due to the fact that the atoms of He4 are bosons, while the atoms of He3 are fermions.

Apart from liquid He3, there are several other examples of many-fermion systems whose properties are strongly affected by the antisymmetry of the wave functions. The electrons in an atom are fermions, but we only give a brief discussion of the theory of atomic structure. The properties of atoms and molecules are not very similar to the properties of other many-fermion systems, since the nuclei, so much heavier than the electrons, affect their structure so profoundly. Atomic nuclei are also composed of fermions — protons and neutrons — and some of the ideas used in nuclear theory are borrowed from atomic theory. It is also possible to regard certain nuclei as composed of alpha-particles, which are bosons, but this is awkward for a number of reasons, and the alpha-particle model has only a limited range of usefulness. The valence electrons in a metal are also fermions, and it might be possible to produce plasmas whose properties are influenced strongly by the Pauli exclusion principle. The so-called "electron gas" serves as a model for either of these systems. A neutron star, a sphere of neutrons bound together by gravitational forces, is another many-fermion system; pulsars are widely believed to be neutron stars.

There is no example other than He4 of a system of many bosons of finite mass which displays quantum properties, but there are many systems that contain zero-mass bosons at finite temperatures. Phonons can exist in almost all systems, whatever sort of molecules they are composed of. Spin-waves exist in ferromagnetic or antiferromagnetic materials. In many-fermion systems at very low temperatures, there are phonons of "zero sound."

At first sight it would seem presumptuous to make any attempt to construct a detailed theory of such systems. In the first place, the quantum mechanics of even a three-body system is poorly...