Second quantization
Quantum field theory 

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Second quantization is a formalism used to describe and analyze quantum manybody 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,^{[1]} and were developed, most notably, by Fock and Jordan later.^{[2]}^{[3]}
In this approach, the quantum manybody states are represented in the Fock state basis, which are constructed by filling up each singleparticle 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 manybody theory.
Quantum manybody 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 and different configurations of the set of 's correspond to different manybody states, in quantum mechanics, the particles are identical, such that exchanging two particles, i.e. , does not lead to a different manybody quantum state. This implies that the quantum manybody wave function must be invariant (up to a phase factor) under the exchange of two particles. According to the statistics of the particles, the manybody wave function can either be symmetric or antisymmetric under the particle exchange:
This exchange symmetry property imposes a constraint on the manybody wave function. Each time a particle is added or removed from the manybody system, the wave function must be properly symmetrized or antisymmetrized 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 singleparticle 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.
Firstquantized manybody wave function
Consider a complete set of singleparticle wave functions labeled by (which may be a combined index of a number of quantum numbers). The following wave function
represents an Nparticle state with the ith particle occupying the singleparticle state . In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ith singleparticle wave function describes the state of the ith particle. The wave function has not been symmetrized or antisymmetrized, thus in general not qualified as a manybody wave function for identical particles. However, it can be brought to the symmetrized (antisymmetrized) form by the symmetrization (antisymmetrization) operators, denoted ().
For bosons, the manybody wave function must be symmetrized,
while for fermions, the manybody wave function must be antisymmetrized,
Here is an element in the Nbody permutation group (or symmetric group) , which performs a permutation among the state labels , and denotes the corresponding permutation sign. is the normalization operator that normalizes the wave function. (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.)
If one arranges the singleparticle wave functions in a matrix , such that the rowi columnj matrix element is , then the boson manybody wave function can be simply written as a permanent , and the fermion manybody wave function as a determinant (also known as the Slater determinant).
Secondquantized Fock states
First quantized wave functions involve complicated symmetrization procedures to describe physically realizable manybody states because the language of first quantization is redundant for indistinguishable particles. In the first quantization language, the manybody state is described by answering a series of questions like "which particle is on which state". However these are not physical questions, because the particles are identical, and it is impossible to tell which particle is which in the first place. The seemingly different states and are actually redundant names of the same quantum manybody state. So the symmetrization (or antisymmetrization) must be introduced to eliminate this redundancy in the first quantization description.
In the second quantization language, instead of asking "each particle on which state", one asks "how many particles are there on each state". Because this description does not refer to the labeling of particles, it contains no redundant information, and hence leads to a precise and simpler description of the quantum manybody state. In this approach, the manybody state is represented in the occupation number basis, and the basis state is labeled by the set of occupation numbers, denoted
meaning that there are particles in the singleparticle state (or as ). The occupation numbers sum up to the total number of particles, i.e. . For fermions, the occupation number 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 are also known as the Fock states. All the Fock states form a complete set of basis of the manybody Hilbert space, or the Fock space. Any generic quantum manybody state can be expressed as a linear combination of Fock states.
Note that besides providing a more efficient language, Fock space allows for a variable number of particles. As a Hilbert space, it is isomorphic to the sum of the nparticle bosonic or fermionic tensor spaces described in the previous section, including a onedimensional zeroparticle space ℂ.
The Fock state with all occupation numbers equal to zero is called the vacuum state, denoted . The Fock state with only one nonzero occupation number is a singlemode Fock state, denoted . In terms of the first quantized wave function, the vacuum state is the unit of tensor product, and can be denoted as . The singleparticle state is reduced to its wave function . Other singlemode manybody (boson) state are just the tensor product of the wave function of that mode, such as and . For multimode Fock states (meaning more than one singleparticle state is involved), the corresponding firstquantized wave function will require proper symmetrization according to the particle statistics, e.g. for a boson state, and for a fermion state (the symbol between and is omitted for simplicity). In general, the normalization is found to be , where N is the total number of particles. For fermion, this expression reduces to as can only be either zero or one. So the firstquantized wave function corresponding to the Fock state reads
for bosons and
for fermions. Note that for fermions, only, so the tensor product above is effectively just a product over all occupied singleparticle states.
Creation and annihilation operators
The creation and annihilation operators are introduced to add or remove a particle from the manybody system. These operators lie at the core of the second quantization formalism, bridging the gap between the first and the secondquantized states. Applying the creation (annihilation) operator to a firstquantized manybody wave function will insert (delete) a singleparticle state from the wave function in a symmetrized way depending on the particle statistics. On the other hand, all the secondquantized Fock states can be constructed by applying the creation operators to the vacuum state repeatedly.
The creation and annihilation operators (for bosons) are originally constructed in the context of the quantum harmonic oscillator as the raising and lowering operators, which are then generalized to the field operators in the quantum field theory.^{[4]} They are fundamental to the quantum manybody theory, in the sense that every manybody operator (including the Hamiltonian of the manybody system and all the physical observables) can be expressed in terms of them.
Insertion and deletion operation
The creation and annihilation of a particle is implemented by the insertion and deletion of the singleparticle state from the first quantized wave function in an either symmetric or antisymmetric manner. Let be a singleparticle state, let 1 be the tensor identity (it is the generator of the zeroparticle space ℂ and satisfies in the tensor algebra over the fundamental Hilbert space), and let be a generic tensor product state. The insertion and the deletion operators are linear operators defined by the following recursive equations
Here is the Kronecker delta symbol, which gives 1 if , and 0 otherwise. The subscript of the insertion or deletion operators indicates whether symmetrization (for bosons) or antisymmetrization (for fermions) is implemented.
Boson creation and annihilation operators
The boson creation (resp. annihilation) operator is usually denoted as (resp. ). The creation operator adds a boson to the singleparticle state , and the annihilation operator removes a boson from the singleparticle state . The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators ().
Definition
The boson creation (annihilation) operator is a linear operator, whose action on a Nparticle firstquantized wave function is defined as
where inserts the singleparticle state in possible insertion positions symmetrically, and deletes the singleparticle state from possible deletion positions symmetrically.
Hereinafter the tensor symbol between singleparticle states is omitted for simplicity. Take the state , create one more boson on the state ,
Then annihilate one boson from the state ,
Action on Fock states
Starting from the singlemode vacuum state , applying the creation operator repeatedly, one finds
The creation operator raises the boson occupation number by 1. Therefore, all the occupation number states can be constructed by the boson creation operator from the vacuum state
On the other hand, the annihilation operator lowers the boson occupation number by 1
It will also quench the vacuum state as there has been no boson left in the vacuum state to be annihilated. Using the above formulae, it can be shown that
meaning that defines the boson number operator.
The above result can be generalized to any Fock state of bosons.
These two equations can be considered as the defining properties of boson creation and annihilation operators in the secondquantization formalism. The complicated symmetrization of the underlying firstquantized wave function is automatically taken care of by the creation and annihilation operators (when acting on the firstquantized wave function), so that the complexity is not revealed on the secondquantized level, and the secondquantization formulae are simple and neat.
Operator identities
The following operator identities follow from the action of the boson creation and annihilation operators on the Fock state,
These commutation relations can be considered as the algebraic definition of the boson creation and annihilation operators. The fact that the boson manybody wave function is symmetric under particle exchange is also manifested by the commutation of the boson operators.
The raising and lowering operators of the quantum harmonic oscillator also satisfies the same set of commutation relations, implying that the bosons can be interpreted as the energy quanta (phonons) of an oscillator. This is indeed the idea of quantum field theory, which considers each mode of the matter field as an oscillator subject to quantum fluctuations, and the bosons are treated as the excitations (or energy quanta) of the field.
Fermion creation and annihilation operators
The fermion creation (annihilation) operator is usually denoted as (). The creation operator adds a fermion to the singleparticle state , and the annihilation operator removes a fermion from the singleparticle state . The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (). The Hermitian combination of the fermion creation and annihilation operators
are called Majorana fermion operators.
Definition
The fermion creation (annihilation) operator is a linear operator, whose action on a Nparticle firstquantized wave function is defined as
where inserts the singleparticle state in possible insertion positions antisymmetrically, and deletes the singleparticle state from possible deletion positions antisymmetrically.
Hereinafter the tensor symbol between singleparticle states is omitted for simplicity. Take the state , attempt to create one more fermion on the occupied state will quench the whole manybody wave function,
Annihilate a fermion on the state, take the state ,
The minus sign (known as the fermion sign) appears due to the antisymmetric property of the fermion wave function.
Action on Fock states
Starting from the singlemode vacuum state , applying the fermion creation operator ,
If the singleparticle state is empty, the creation operator will fill the state with a fermion. However, if the state is already occupied by a fermion, further application of the creation operator will quench the state, demonstrating the Pauli exclusion principle that two identical fermions can not occupy the same state simultaneously. Nevertheless, the fermion can be removed from the occupied state by the fermion annihilation operator ,
The vacuum state is quenched by the action of the annihilation operator.
Similar to the boson case, the fermion Fock state can be constructed from the vacuum state using the fermion creation operator
It is easy to check (by enumeration) that
meaning that defines the fermion number operator.
The above result can be generalized to any Fock state of fermions.
Recall that the occupation number can only take 0 or 1 for fermions. These two equations can be considered as the defining properties of fermion creation and annihilation operators in the second quantization formalism. Note that the fermion sign structure , also known as the JordanWigner string, requires there to exist a predefined ordering of the singleparticle states (the spin structure) and involves a counting of the fermion occupation numbers of all the preceding states; therefore the fermion creation and annihilation operators are considered nonlocal in some sense. This observation leads to the idea that fermions are emergent particles in the longrange entangled local qubit system.^{[5]}
Operator identities
The following operator identities follow from the action of the fermion creation and annihilation operators on the Fock state,
These anticommutation relations can be considered as the algebraic definition of the fermion creation and annihilation operators. The fact that the fermion manybody wave function is antisymmetric under particle exchange is also manifested by the anticommutation of the fermion operators.
Quantum field operators
Defining as a general annihilation(creation) operator for a singleparticle state that could be either fermionic or bosonic , the real space representation of the operators defines the quantum field operators and by
These are second quantization operators, with coefficients and that are ordinary firstquantization wavefunctions. Thus, for example, any expectation values will be ordinary firstquantization wavefunctions. Loosely speaking, is the sum of all possible ways to add a particle to the system at position r through any of the basis states .
Since and are second quantization operators defined in every point in space they are called quantum field operators. They obey the following fundamental commutator and anticommutator relations,
 boson fields,
 fermion fields.
In homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in Fourier basis yields:
Comment on nomenclature
The term "second quantization" is a misnomer that has persisted for historical reasons. One is not quantizing "again", as the term "second" might suggest; the field that is being quantized is not a Schrödinger wave function that was produced as the result of quantizing a particle, but is a classical field (such as the electromagnetic field or Dirac spinor field) that was not previously quantized. One is merely shifting from a semiclassical treatment of the system to a fully quantummechanical one.
See also
References
 ↑ Dirac, P. A. M. (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 114 (767): 243. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039.
 ↑ V. Fock, Z. Phys. 75 (1932), 622647
 ↑ M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. p. 328.
 ↑ Mahan, GD (1981). Many Particle Physics. New York: Springer. ISBN 0306463385.
 ↑ Levin, M.; Wen, X. G. (2003). "Fermions, strings, and gauge fields in lattice spin models". Physical Review B. 67 (24). doi:10.1103/PhysRevB.67.245316.
Further reading
 Second quantization Carlo Maria Becchi, Scholarpedia, 5(6):7902. doi:10.4249/scholarpedia.7902
External links
Second quantization on Wikiversity 
 ManyElectron States in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013, ISBN 9783893368846