Fundamentals of Condensed Matter Physics

In Chapter 6, the many-electron problem was discussed and electron–electron interactions were considered using the standard approaches of many-body quantum theory. The quasi-particle and collective excitations of the interacting system are expected to emerge as elementary excitations of the interacting system, and their properties should dominate the spectra associated with various response functions. Since the elementary excitations are emergent properties, which can be viewed in terms of excitations or particles that can be created or destroyed, it is convenient to use techniques associated with the fields of quantum electrodynamics (QED) and statistical physics. The essential point is that in QED, because relativistic effects are included, particles can be created and destroyed, and the formalism accounts for the changes associated with these events.

For condensed matter systems, we are not usually probing the systems in energy ranges where relativistic effects are important. However, the techniques of QED are useful if they are applied to cases where elementary excitations are created or destroyed. In this chapter, we focus on recipes for calculations of this kind and refer the reader to books in this area that provide the proofs and more complete discussions.

General formalism

The first step in setting up the formalism for a many-electron- or many-phonon-type study is to rely on second quantization techniques. The approach, which is described in many texts on quantum mechanics, consists in replacing operators in the usual quantum theory, such as the kinetic energy, that act on the coordinate variables in a wavefunction by other operators that operate on a wavefunction describing the number of particles in a given state. These new operators change the number of particles in specific states, and the result is the same as would be obtained when using ordinary quantum theory techniques. So second quantization is a convenient method, which although in principle is not required for the study of solids, allows efficient calculations and provides new insights. Analogies like the one comparing electron–hole excitations by photons with Dirac's electron–positron theory, and viewing a collection of phonons as a Bose system, are very useful. Because particles can be created and destroyed at will, the grand canonical ensemble approach of statistical physics is employed. Another very convenient aspect of second quantization is that the task of symmetrizing wavefunctions for many-body systems is achieved through commutation relations.