Second quantization

Second quantization is a formalism used to describe and analyze quantum many-body systems. It is also known as canonical quantization in quantum field theory, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Dirac, and were developed, most notably, by Fock and Jordan later.

In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

The starting point of the second quantization formalism is the notion of indistinguishability of particles in quantum mechanics. Unlike in classical mechanics, where each particle is labeled by a distinct position vector ${\displaystyle {\mathbf {r}}_{i}}$ and different configurations of the set of ${\displaystyle {\mathbf {r}}_{i}}$s correspond to different many-body states, in quantum mechanics, the particles are identical, such that exchanging two particles, i.e. ${\displaystyle {\mathbf {r}}_{i}\leftrightarrow {\mathbf {r}}_{j}}$, does not lead to a different many-body quantum state. This implies that the quantum many-body wave function must be invariant (up to a phase factor) under the exchange of two particles. According to the statistics of the particles, the many-body wave function can either be symmetric or antisymmetric under the particle exchange:

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