Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, 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 Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir Fock. 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.
Quantum many-body states 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 r i {\displaystyle \mathbf {r} _{i}} and different configurations of the set of r i {\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. r i ↔ r j {\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:
Ψ B ( ⋯ , r i , ⋯ , r j , ⋯ ) = + Ψ B ( ⋯ , r j , ⋯ , r i , ⋯ ) {\displaystyle \Psi _{\rm {B}}(\cdots ,\mathbf {r} _{i},\cdots ,\mathbf {r} _{j},\cdots )=+\Psi _{\rm {B}}(\cdots ,\mathbf {r} _{j},\cdots ,\mathbf {r} _{i},\cdots )} if the particles are bosons,
Ψ F ( ⋯ , r i , ⋯ , r j , ⋯ ) = − Ψ F ( ⋯ , r j , ⋯ , r i , ⋯ ) {\displaystyle \Psi _{\rm {F}}(\cdots ,\mathbf {r} _{i},\cdots ,\mathbf {r} _{j},\cdots )=-\Psi _{\rm {F}}(\cdots ,\mathbf {r} _{j},\cdots ,\mathbf {r} _{i},\cdots )} if the particles are fermions. This exchange symmetry property imposes a constraint on the many-body wave function. Each time a particle is added or removed from the many-body system, the wave function must be properly symmetrized or anti-symmetrized to satisfy the symmetry constraint. In the first quantization formalism, this constraint is guaranteed by representing the wave function as linear combination of permanents (for bosons) or determinants (for fermions) of single-particle states. In the second quantization formalism, the issue of symmetrization is automatically taken care of by the creation and annihilation operators, such that its notation can be much simpler.
First-quantized many-body wave function Consider a complete set of single-particle wave functions ψ α ( r ) {\displaystyle \psi _{\alpha }(\mathbf {r} )} labeled by α {\displaystyle \alpha } (which may be a combined index of a number of quantum numbers). The following wave function
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