DOI link for SECOND QUANTIZATION
SECOND QUANTIZATION book
As mentioned in previous chapters, processes like electron-positron pair creation or annihilation cannot be understood in the framework of single particle wave equations. Such processes are best understood in terms of a quantum field theory for the electron, similar to the one we have for the photon. The electrons may then be treated as quanta of the electron field (or Dirac field) in the same sense that photons are regarded as the quanta of the electromagnetic field. When we treat the Dirac equation (or Klein-Gordon equation) on the same footing as the Maxwell equations and subject the wave field or the wave function to quantization again in order to obtain the quantum field operators from it then the process is called second quantization. As a consequence, single-particle probability and current densities formally go over to particle density and current density operators. However, the analogy is purely formal as the former are real functions and the latter operators.1 Though there is a basic difference between quantizing an electromagnetic field, which is a classical field and quantizing a Dirac field which is not a classical field, second quantization appears to be the only way to understand a large number of phenomenon pertaining to the interaction of radiation and matter. The need for second quantization, in fact, followed from the difficulties faced in the attempts to construct a relativistic single particle wave equation. We may recall, for example, that the relativistic generalization of the Schro¨dinger equation, the Klein Gordon (KG) equation, yielded an equation of continuity in which the probability density was not a positive definite quantity. For this reason the KG equation was initially discarded; it came to be accepted only after Pauli and Weisskopf suggested that it could well serve as a field equation whose quanta are zero-spin particles of mass m. Once the KG equation was accepted as a field equation, the concepts of probability density and equation of continuity (which represents conservation of total probability) did not have the same significance they did in single-particle theory as particles could be created or destroyed in the framework of field theory. In the Dirac single-particle wave equation we find, of course, that the probability density ψ†ψ is positive definite. However when we use this equation for finding the energy E of a free particle we find the answer to be2
E = ± √ c2p2 + m2 c4 ,
so that the existence of states with negative energy (E < −mc2) has to be accommodated in the quantum theory. For this purpose, Dirac devised an ingenious hypothesis that the continuum of negative energy states is normally completely occupied by the electrons and this produces no observable effects. If energy > 2mc2 is made available to a negative
energy electron to go to a positive energy state the void or hole created in the negative energy continuum manifests itself as a positron and we have an electron positron pair. The subsequent experimental discovery of the positron by Anderson (1932) confirmed Dirac’s hypothesis. Dirac’s single-particle equation had several other successes as well [Chapter 12]. But the fact remains that the concept of occupied negative energy states was introduced to save single-particle relativistic theory and in this process one had to postulate an infinite number of particles in the negative energy states. Hence it was thought that the Dirac equation, instead of being regarded as a single-particle wave equation should be regarded as a field equation subject to a second quantization.