Second Quantization

a method used in quantum mechanics and quantum field theory to study systems consisting of many or an infinite number of particles (or quasiparticles). In this method the state of a quantum system is described by means of occupation numbers—quantities that characterize the average number of particles in a system in each possible state.

The method of second quantization is particularly important in quantum field theory when the number of particles in a given physical system is not constant but may change during various processes that transpire in the system. Therefore, quantum radiation theory and quantum theory of elementary particles and systems of various quasiparticles are the most important areas of application of the method of second quantization. Systems containing light quanta (photons), the number of which changes during the processes of emission, absorption, and scattering, are examined in radiation theory. In the theory of elementary particles the necessity of using the method of second quantization is connected with the possibility of mutual transformations of particles—for example, the processes of conversion of electrons and positrons into photons, and the reverse process. The method of second quantization is most effective in quantum electrodynamics—the quantum theory of electromagnetic processes, as well as in solid-state theory, which is based on the concept of quasiparticles. The use of second quantization is less effective for describing mutual transformations of particles caused by nonelec-tromagnetic interactions.

In the mathematical apparatus of second quantization, the wave function of a system is taken to be a function of the occupation numbers. Here the primary role is played by “creation” and “annihilation” operators of particles. The annihilation operator is an operator under whose influence the wave function of some state of a given physical system is converted into the wave function of another state having one less particle. Similarly, the creation operator increases the number of particles in the state by 1.

The fundamental aspect of the method of second quantization does not depend on whether the particles of the system conform to Bose-Einstein statistics (for example, photons) or Fermi-Dirac statistics (electrons and positrons). The specific mathematical apparatus of the method, including the main properties of the creation and annihilation operators, differs significantly in these cases because the number of particles that may exist in a given state is in no way limited in Bose-Einstein statistics (so that the occupation numbers may assume arbitrary values), whereas in Fermi-Dirac statistics no more than one particle may be in each state (and the occupation numbers may have only the values 0 and 1).

The method of second quantization was first developed by the English physicist P. Dirac (1927) in his theory of radiation; and further work was done by the Soviet physicist V. A. Fok (1932). The term “second quantization” appeared because the method developed after “ordinary,” or “first,” quantization, the purpose of which was to determine the wave properties of particles. The necessity of systematic consideration of the corpuscular properties of fields as well (since particle-wave duality is inherent in all types of matter) led to the appearance of methods of second quantization.