Collective Classical and Quantum Fields

The following sections are included:

• Dedication
• Preface
• Contents
• List of Figures
• List of Tables

An important goal of many-body physics is the study of collective phenomena in systems of many bosons or fermions. The interactions are typically caused by two-body forces. In their field-theoretic description, such forces emerge in a perturbation theory from the exchange of virtual particles, such as photons or phonons. More complicated forces can also be generated by the exchange of virtual particles carried by higher tensor fields. So far, all forces in nature between particles can be reduced to such exchange processes. Depending on the detailed properties of forces and thermodynamic parameters such as density, pressure, and temperature, bosons or fermions may exhibit different collective behaviors. In an electron gas, for example, one may observe density fluctuations or pair condensation. The first type is found if the exchanged particles couple strongly to other particles or holes. Examples are plasma oscillations in a degenerate electron gas. The second type of behavior is found if the forces favor the formation of bound states between pairs of particles. This is usually observed below a certain critical temperature T_c. Examples are excitons in a semiconductor or Cooper pairs in a superconductor.

In this chapter we develop a collective quantum field theory for a gas of many electrons which interact only via long-range Coulomb forces. The Coulomb forces give rise to collective modes called plasmons.

Superconductors are made from materials which do not pose any resistance to the flow of electricity. The phenomenon was first observed in 1911 by the Dutch physicist Heike Kamerlingh Onnes at Leiden University. When he cooled mercury down to the temperature of liquid helium, which appears at about 4 degree Kelvin (1 degree Kelvin −273.15°C), its resistance suddenly disappeared. For this discovery, Onnes won the Nobel Prize in physics in 1913.

The explanation of the phenomenon of superconductivity by Bardeen, Cooper, and Schrieffer in 1957 [1], and a little later by Bogoliubov and his school [2], prompted a search for similar phenomena in other Fermi systems, such as fermionic nuclei [3] and, in particular, liquid ³He [4]. While nuclear forces did, in principle, allow a direct application of the BCS formalism [5], it was soon realized [6] that in ³He the strong repulsive core of the interatomic potential would not permit the formation of s-wave Cooper pairs as in superconductors. Thus, if anything similar to Cooper pairing should occur, it had to be in a nonzero angular momentum.

In 1888, Friedrich Reinitzer investigated the thermodynamic properties of crystals of cholesteryl benzoate [1]. He observed that they melt at 145.5°C to form a cloudy liquid. This was stable up to 178.5°C, before it melted again to form a clear liquid. The cloudy liquid is a new phase of matter intermediate between a crystal and a liquid which is now referred to as a liquid crystal. A liquid crystal is a system of rod-like or disk-like molecules which behave under translations in the same way as the molecules in an ordinary liquid, while their molecular orientations can undergo phase transitions into states of long-range order, a typical property of crystals.

The techniques developed in the previous chapters can be understood better by observing how they work in some simple exactly solvable models whose physics is well known on the basis of conventional techniques. This will be illustrated in this chapter in several typical cases.

The following section is included:

• Index