Second Quantization

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second quantization

[′sek·ənd ‚kwän·tə′zā·shən]
(quantum mechanics)
A procedure in which the dependent variables of a classical field or a quantum-mechanical wave function are regarded as operators on which commutation rules are imposed; this produces a formalism in which particles may be created and destroyed.

Quantization, Second

 

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.

References in periodicals archive ?
Our first step towards a covariant canonical quantization begins with defining a quantized space-time and its quanta.
M, (2010), Fractional Canonical Quantization of the Free Electromagnetic Lagrangian Density, Jordan journal of physics, volume3, number2, 2010.pp.
Then Heisenberg, Born, Jordan, and Dirac realized that noncommutativity is the raison d'etre of quantum mechanics and they have introduced the so-called canonical quantization. It means that the basic relations of classical mechanics,
The WDW equation is obtained by means of canonical quantization of Hamiltonian H according to the standard canonical rule, this leads to a difficulty known as the problem of time [6].
Here I focus on canonical quantization, where the initial input is a classical theory in Hamiltonian form, and the output is a quantum theory which - one hopes - has the original classical theory as its h [right arrow] 0 limit.
The canonical quantization of such a system transforms its Hamiltonian to an Hermitian operator, which now contains the parameter h, in such a way that in the limit h [right arrow] 0, the quantum Hamiltonian [H.sub.h] returns to its classical counterpart.
By the Dirac canonical quantization procedure, we have
In the absence of a complete theory of quantum gravity, it is relevant to try their quantum cosmological version within the approach of canonical quantization, in sense that the wave function for a quantum cosmological model under consideration is found.
This scheme could lead toward a full canonical quantization.
There has been some work done recently along this line of investigation in the framework of the canonical quantization in time-dependent gauges that attempted to interpret the [sigma]-models as tree level bosonic string theories (The de Sitter space-time is not a proper string background as it is not a solution of the string [beta]-function equations.).