Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems.
The key ideas of this method were introduced in 1927 by Paul Dirac,[1] and were later developed, most notably, by Pascual Jordan[2] and Vladimir Fock.
[5] 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.
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: 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.
has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles.
(It is the operator that applies a suitable numerical normalization factor to the symmetrized tensors of degree n; see the next section for its value.)
[6] First quantized wave functions involve complicated symmetrization procedures to describe physically realizable many-body states because the language of first quantization is redundant for indistinguishable particles.
can only be 0 or 1, due to the Pauli exclusion principle; while for bosons it can be any non-negative integer The occupation number states
Note that besides providing a more efficient language, Fock space allows for a variable number of particles.
In terms of the first quantized wave function, the vacuum state is the unit tensor product and can be denoted
Other single-mode many-body (boson) states are just the tensor product of the wave function of that mode, such as
is involved), the corresponding first-quantized wave function will require proper symmetrization according to the particle statistics, e.g.
So the first-quantized wave function corresponding to the Fock state reads for bosons and for fermions.
The creation and annihilation operators are introduced to add or remove a particle from the many-body system.
These operators lie at the core of the second quantization formalism, bridging the gap between the first- and the second-quantized states.
The creation and annihilation of a particle is implemented by the insertion and deletion of the single-particle state from the first quantized wave function in an either symmetric or anti-symmetric manner.
These two equations can be considered as the defining properties of boson creation and annihilation operators in the second-quantization formalism.
The complicated symmetrization of the underlying first-quantized wave function is automatically taken care of by the creation and annihilation operators (when acting on the first-quantized wave function), so that the complexity is not revealed on the second-quantized level, and the second-quantization formulae are simple and neat.
It is particularly instructive to view the results of creation and annihilation operators on states of two (or more) fermions, because they demonstrate the effects of exchange.
The complete algebra for creation and annihilation operators on a two-fermion state can be found in Quantum Photonics.
These two equations can be considered as the defining properties of fermion creation and annihilation operators in the second quantization formalism.
, also known as the Jordan-Wigner string, requires there to exist a predefined ordering of the single-particle states (the spin structure)[clarification needed] and involves a counting of the fermion occupation numbers of all the preceding states; therefore the fermion creation and annihilation operators are considered non-local in some sense.
This observation leads to the idea that fermions are emergent particles in the long-range entangled local qubit system.
They obey the following fundamental commutator and anti-commutator relations, For homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in Fourier basis yields: The term "second quantization", introduced by Jordan,[11] is a misnomer that has persisted for historical reasons.
One is merely quantizing each oscillator in this assembly, shifting from a semiclassical treatment of the system to a fully quantum-mechanical one.