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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,^[1] and were developed, most notably, by Vladimir Fock and Pascual Jordan later.^[2][3]

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 ri{displaystyle mathbf {r} _{i}} and different configurations of the set of ri{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. ri↔rj{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:

ΨB(⋯,ri,⋯,rj,⋯)=+ΨB(⋯,rj,⋯,ri,⋯){displaystyle Psi _{B}(cdots ,mathbf {r} _{i},cdots ,mathbf {r} _{j},cdots )=+Psi _{B}(cdots ,mathbf {r} _{j},cdots ,mathbf {r} _{i},cdots )} if the particles are bosons,

ΨF(⋯,ri,⋯,rj,⋯)=−ΨF(⋯,rj,⋯,ri,⋯){displaystyle Psi _{F}(cdots ,mathbf {r} _{i},cdots ,mathbf {r} _{j},cdots )=-Psi _{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 ψα(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

Ψ[ri]=∏i=1Nψαi(ri)≡ψα1⊗ψα2⊗⋯⊗ψαN{displaystyle Psi [mathbf {r} _{i}]=prod _{i=1}^{N}psi _{alpha _{i}}(mathbf {r} _{i})equiv psi _{alpha _{1}}otimes psi _{alpha _{2}}otimes cdots otimes psi _{alpha _{N}}}

represents an N-particle state with the ith particle occupying the single-particle state |αi⟩{displaystyle |{alpha _{i}}rangle }. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ith single-particle wave function describes the state of the ith particle. The wave function Ψ{displaystyle Psi } has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles. However, it can be brought to the symmetrized (anti-symmetrized) form by operators S{displaystyle {mathcal {S}}} for symmetrizer, and A{displaystyle {mathcal {A}}} for antisymmetrizer.

For bosons, the many-body wave function must be symmetrized,

ΨB[ri]=NSΨ[ri]=N∑π∈SN∏i=1Nψαπ(i)(ri)=N∑π∈SNψαπ(1)⊗ψαπ(2)⊗⋯⊗ψαπ(N);{displaystyle Psi _{B}[mathbf {r} _{i}]={mathcal {N}}{mathcal {S}}Psi [mathbf {r} _{i}]={mathcal {N}}sum _{pi in S_{N}}prod _{i=1}^{N}psi _{alpha _{pi (i)}}(mathbf {r} _{i})={mathcal {N}}sum _{pi in S_{N}}psi _{alpha _{pi (1)}}otimes psi _{alpha _{pi (2)}}otimes cdots otimes psi _{alpha _{pi (N)}};}

while for fermions, the many-body wave function must be anti-symmetrized,

ΨF[ri]=NAΨ[ri]=N∑π∈SN(−1)π∏i=1Nψαπ(i)(ri)=N∑π∈SN(−1)πψαπ(1)⊗ψαπ(2)⊗⋯⊗ψαπ(N).{displaystyle Psi _{F}[mathbf {r} _{i}]={mathcal {N}}{mathcal {A}}Psi [mathbf {r} _{i}]={mathcal {N}}sum _{pi in S_{N}}(-1)^{pi }prod _{i=1}^{N}psi _{alpha _{pi (i)}}(mathbf {r} _{i})={mathcal {N}}sum _{pi in S_{N}}(-1)^{pi }psi _{alpha _{pi (1)}}otimes psi _{alpha _{pi (2)}}otimes cdots otimes psi _{alpha _{pi (N)}}.}

Here π{displaystyle pi } is an element in the N-body permutation group (or symmetric group) SN{displaystyle S_{N}}, which performs a permutation among the state labels αi{displaystyle alpha _{i}}, and (−1)π{displaystyle (-1)^{pi }} denotes the corresponding permutation sign. N{displaystyle {mathcal {N}}} 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 single-particle wave functions in a matrix U{displaystyle U}, such that the row-i column-j matrix element is Uij=ψαj(ri)≡⟨ri|αj⟩{displaystyle U_{ij}=psi _{alpha _{j}}(mathbf {r} _{i})equiv langle mathbf {r} _{i}|alpha _{j}rangle }, then the boson many-body wave function can be simply written as a permanent ΨB=Nperm⁡U{displaystyle Psi _{B}={mathcal {N}}operatorname {perm} U}, and the fermion many-body wave function as a determinant ΨF=Ndet⁡U{displaystyle Psi _{F}={mathcal {N}}operatorname {det} U} (also known as the Slater determinant).

Second-quantized Fock states

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. In the first quantization language, the many-body state is described by answering a series of questions like "Which particle is in 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 ψ1⊗ψ2{displaystyle psi _{1}otimes psi _{2}} and ψ2⊗ψ1{displaystyle psi _{2}otimes psi _{1}} are actually redundant names of the same quantum many-body state. So the symmetrization (or anti-symmetrization) 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 in 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 many-body state. In this approach, the many-body state is represented in the occupation number basis, and the basis state is labeled by the set of occupation numbers, denoted

|[nα]⟩≡|n1,n2,⋯,nα,⋯⟩,{displaystyle |[n_{alpha }]rangle equiv |n_{1},n_{2},cdots ,n_{alpha },cdots rangle ,}

meaning that there are nα{displaystyle n_{alpha }} particles in the single-particle state |α⟩{displaystyle |alpha rangle } (or as ψα{displaystyle psi _{alpha }}). The occupation numbers sum to the total number of particles, i.e. ∑αnα=N{displaystyle sum _{alpha }n_{alpha }=N}. For fermions, the occupation number nα{displaystyle n_{alpha }} can only be 0 or 1, due to the Pauli exclusion principle; while for bosons it can be any non-negative integer

nα={0,1fermions,0,1,2,3,...bosons.{displaystyle n_{alpha }={begin{cases}0,1&{text{fermions,}}\0,1,2,3,...&{text{bosons.}}end{cases}}}

The occupation number states |[nα]⟩{displaystyle |[n_{alpha }]rangle } are also known as Fock states. All the Fock states form a complete basis of the many-body Hilbert space, or Fock space. Any generic quantum many-body 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 n-particle bosonic or fermionic tensor spaces described in the previous section, including a one-dimensional zero-particle space ℂ.

The Fock state with all occupation numbers equal to zero is called the vacuum state, denoted |0⟩≡|⋯,0α,⋯⟩{displaystyle |0rangle equiv |cdots ,0_{alpha },cdots rangle }. The Fock state with only one non-zero occupation number is a single-mode Fock state, denoted |nα⟩≡|⋯,0,nα,0,⋯⟩{displaystyle |n_{alpha }rangle equiv |cdots ,0,n_{alpha },0,cdots rangle }. In terms of the first quantized wave function, the vacuum state is the unit tensor product and can be denoted |0⟩=1{displaystyle |0rangle =1}. The single-particle state is reduced to its wave function |1α⟩=ψα{displaystyle |1_{alpha }rangle =psi _{alpha }}. Other single-mode many-body (boson) states are just the tensor product of the wave function of that mode, such as |2α⟩=ψα⊗ψα{displaystyle |2_{alpha }rangle =psi _{alpha }otimes psi _{alpha }} and
|nα⟩=ψα⊗n{displaystyle |n_{alpha }rangle =psi _{alpha }^{otimes n}}. For multi-mode Fock states (meaning more than one single-particle state |α⟩{displaystyle |alpha rangle } is involved), the corresponding first-quantized wave function will require proper symmetrization according to the particle statistics, e.g. |11,12⟩=(ψ1ψ2+ψ2ψ1)/2{displaystyle |1_{1},1_{2}rangle =(psi _{1}psi _{2}+psi _{2}psi _{1})/{sqrt {2}}} for a boson state, and |11,12⟩=(ψ1ψ2−ψ2ψ1)/2{displaystyle |1_{1},1_{2}rangle =(psi _{1}psi _{2}-psi _{2}psi _{1})/{sqrt {2}}} for a fermion state (the symbol ⊗{displaystyle otimes } between ψ1{displaystyle psi _{1}} and ψ2{displaystyle psi _{2}} is omitted for simplicity). In general, the normalization is found to be ∏αnα!N!{displaystyle {sqrt {tfrac {prod _{alpha }n_{alpha }!}{N!}}}}, where N is the total number of particles. For fermion, this expression reduces to 1N!{displaystyle {tfrac {1}{sqrt {N!}}}} as nα{displaystyle n_{alpha }} can only be either zero or one. So the first-quantized wave function corresponding to the Fock state reads

|[nα]⟩B=(∏αnα!N!)1/2S⨂αψα⊗nα{displaystyle |[n_{alpha }]rangle _{B}=left({frac {prod _{alpha }n_{alpha }!}{N!}}right)^{1/2}{mathcal {S}}bigotimes limits _{alpha }psi _{alpha }^{otimes n_{alpha }}}

for bosons and

|[nα]⟩F=1N!A⨂αψα⊗nα{displaystyle |[n_{alpha }]rangle _{F}={frac {1}{sqrt {N!}}}{mathcal {A}}bigotimes limits _{alpha }psi _{alpha }^{otimes n_{alpha }}}

for fermions. Note that for fermions, nα=0,1{displaystyle n_{alpha }=0,1} only, so the tensor product above is effectively just a product over all occupied single-particle states.

Creation and annihilation operators

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. Applying the creation (annihilation) operator to a first-quantized many-body wave function will insert (delete) a single-particle state from the wave function in a symmetrized way depending on the particle statistics. On the other hand, all the second-quantized 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 many-body theory, in the sense that every many-body operator (including the Hamiltonian of the many-body 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 single-particle state from the first quantized wave function in an either symmetric or anti-symmetric manner. Let ψα{displaystyle psi _{alpha }} be a single-particle state, let 1 be the tensor identity (it is the generator of the zero-particle space ℂ and satisfies ψα≡1⊗ψα≡ψα⊗1{displaystyle psi _{alpha }equiv 1otimes psi _{alpha }equiv psi _{alpha }otimes 1} in the tensor algebra over the fundamental Hilbert space), and let Ψ=ψα1⊗ψα2⊗⋯{displaystyle Psi =psi _{alpha _{1}}otimes psi _{alpha _{2}}otimes cdots } be a generic tensor product state. The insertion ⊗±{displaystyle otimes _{pm }} and the deletion ⊘±{displaystyle oslash _{pm }} operators are linear operators defined by the following recursive equations

ψα⊗±1=ψα,ψα⊗±(ψβ⊗Ψ)=ψα⊗ψβ⊗Ψ±ψβ⊗(ψα⊗±Ψ);{displaystyle psi _{alpha }otimes _{pm }1=psi _{alpha },quad psi _{alpha }otimes _{pm }(psi _{beta }otimes Psi )=psi _{alpha }otimes psi _{beta }otimes Psi pm psi _{beta }otimes (psi _{alpha }otimes _{pm }Psi );}

ψα⊘±1=0,ψα⊘±(ψβ⊗Ψ)=δαβΨ±ψβ⊗(ψα⊘±Ψ).{displaystyle psi _{alpha }oslash _{pm }1=0,quad psi _{alpha }oslash _{pm }(psi _{beta }otimes Psi )=delta _{alpha beta }Psi pm psi _{beta }otimes (psi _{alpha }oslash _{pm }Psi ).}

Here δαβ{displaystyle delta _{alpha beta }} is the Kronecker delta symbol, which gives 1 if α=β{displaystyle alpha =beta }, and 0 otherwise. The subscript ±{displaystyle pm } of the insertion or deletion operators indicates whether symmetrization (for bosons) or anti-symmetrization (for fermions) is implemented.

Boson creation and annihilation operators

The boson creation (resp. annihilation) operator is usually denoted as bα†{displaystyle b_{alpha }^{dagger }} (resp. bα{displaystyle b_{alpha }}). The creation operator bα†{displaystyle b_{alpha }^{dagger }} adds a boson to the single-particle state |α⟩{displaystyle |alpha rangle }, and the annihilation operator bα{displaystyle b_{alpha }} removes a boson from the single-particle state |α⟩{displaystyle |alpha rangle }. The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (bα≠bα†{displaystyle b_{alpha }neq b_{alpha }^{dagger }}).

Definition

The boson creation (annihilation) operator is a linear operator, whose action on a N-particle first-quantized wave function Ψ{displaystyle Psi } is defined as

bα†Ψ=1N+1ψα⊗+Ψ,{displaystyle b_{alpha }^{dagger }Psi ={frac {1}{sqrt {N+1}}}psi _{alpha }otimes _{+}Psi ,}

bαΨ=1Nψα⊘+Ψ,{displaystyle b_{alpha }Psi ={frac {1}{sqrt {N}}}psi _{alpha }oslash _{+}Psi ,}

where ψα⊗+{displaystyle psi _{alpha }otimes _{+}} inserts the single-particle state ψα{displaystyle psi _{alpha }} in N+1{displaystyle N+1} possible insertion positions symmetrically, and ψα⊘+{displaystyle psi _{alpha }oslash _{+}} deletes the single-particle state ψα{displaystyle psi _{alpha }} from N{displaystyle N} possible deletion positions symmetrically.

Action on Fock states

Starting from the single-mode vacuum state |0α⟩=1{displaystyle |0_{alpha }rangle =1}, applying the creation operator bα†{displaystyle b_{alpha }^{dagger }} repeatedly, one finds

bα†|0α⟩=ψα⊗+1=ψα=|1α⟩,{displaystyle b_{alpha }^{dagger }|0_{alpha }rangle =psi _{alpha }otimes _{+}1=psi _{alpha }=|1_{alpha }rangle ,}

bα†|nα⟩=1nα+1ψα⊗+ψα⊗nα=nα+1ψα⊗(nα+1)=nα+1|nα+1⟩.{displaystyle b_{alpha }^{dagger }|n_{alpha }rangle ={frac {1}{sqrt {n_{alpha }+1}}}psi _{alpha }otimes _{+}psi _{alpha }^{otimes n_{alpha }}={sqrt {n_{alpha }+1}}psi _{alpha }^{otimes (n_{alpha }+1)}={sqrt {n_{alpha }+1}}|n_{alpha }+1rangle .}

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

|nα⟩=1nα!(bα†)nα|0α⟩.{displaystyle |n_{alpha }rangle ={frac {1}{sqrt {n_{alpha }!}}}(b_{alpha }^{dagger })^{n_{alpha }}|0_{alpha }rangle .}

On the other hand, the annihilation operator bα{displaystyle b_{alpha }} lowers the boson occupation number by 1

bα|nα⟩=1nαψα⊘+ψα⊗nα=nαψα⊗(nα−1)=nα|nα−1⟩.{displaystyle b_{alpha }|n_{alpha }rangle ={frac {1}{sqrt {n_{alpha }}}}psi _{alpha }oslash _{+}psi _{alpha }^{otimes n_{alpha }}={sqrt {n_{alpha }}}psi _{alpha }^{otimes (n_{alpha }-1)}={sqrt {n_{alpha }}}|n_{alpha }-1rangle .}

It will also quench the vacuum state bα|0α⟩=0{displaystyle b_{alpha }|0_{alpha }rangle =0} as there has been no boson left in the vacuum state to be annihilated. Using the above formulae, it can be shown that

bα†bα|nα⟩=nα|nα⟩,{displaystyle b_{alpha }^{dagger }b_{alpha }|n_{alpha }rangle =n_{alpha }|n_{alpha }rangle ,}

meaning that n^α=bα†bα{displaystyle {hat {n}}_{alpha }=b_{alpha }^{dagger }b_{alpha }} defines the boson number operator.

The above result can be generalized to any Fock state of bosons.

bα†|⋯,nβ,nα,nγ,⋯⟩=nα+1|⋯,nβ,nα+1,nγ,⋯⟩.{displaystyle b_{alpha }^{dagger }|cdots ,n_{beta },n_{alpha },n_{gamma },cdots rangle ={sqrt {n_{alpha }+1}}|cdots ,n_{beta },n_{alpha }+1,n_{gamma },cdots rangle .}

bα|⋯,nβ,nα,nγ,⋯⟩=nα|⋯,nβ,nα−1,nγ,⋯⟩.{displaystyle b_{alpha }|cdots ,n_{beta },n_{alpha },n_{gamma },cdots rangle ={sqrt {n_{alpha }}}|cdots ,n_{beta },n_{alpha }-1,n_{gamma },cdots rangle .}

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.

Operator identities

The following operator identities follow from the action of the boson creation and annihilation operators on the Fock state,

[bα†,bβ†]=[bα,bβ]=0,[bα,bβ†]=δαβ.{displaystyle [b_{alpha }^{dagger },b_{beta }^{dagger }]=[b_{alpha },b_{beta }]=0,quad [b_{alpha },b_{beta }^{dagger }]=delta _{alpha beta }.}

These commutation relations can be considered as the algebraic definition of the boson creation and annihilation operators. The fact that the boson many-body 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 cα†{displaystyle c_{alpha }^{dagger }} (cα{displaystyle c_{alpha }}). The creation operator cα†{displaystyle c_{alpha }^{dagger }} adds a fermion to the single-particle state |α⟩{displaystyle |alpha rangle }, and the annihilation operator cα{displaystyle c_{alpha }} removes a fermion from the single-particle state |α⟩{displaystyle |alpha rangle }. The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (cα≠cα†{displaystyle c_{alpha }neq c_{alpha }^{dagger }}). The Hermitian combination of the fermion creation and annihilation operators

χα,Re=(cα+cα†)/2,χα,Im=(cα−cα†)/(2i),{displaystyle chi _{alpha ,{text{Re}}}=(c_{alpha }+c_{alpha }^{dagger })/2,quad chi _{alpha ,{text{Im}}}=(c_{alpha }-c_{alpha }^{dagger })/(2mathrm {i} ),}

are called Majorana fermion operators.

Definition

The fermion creation (annihilation) operator is a linear operator, whose action on a N-particle first-quantized wave function Ψ{displaystyle Psi } is defined as

cα†Ψ=1N+1ψα⊗−Ψ,{displaystyle c_{alpha }^{dagger }Psi ={frac {1}{sqrt {N+1}}}psi _{alpha }otimes _{-}Psi ,}

cαΨ=1Nψα⊘−Ψ,{displaystyle c_{alpha }Psi ={frac {1}{sqrt {N}}}psi _{alpha }oslash _{-}Psi ,}

where ψα⊗−{displaystyle psi _{alpha }otimes _{-}} inserts the single-particle state ψα{displaystyle psi _{alpha }} in N+1{displaystyle N+1} possible insertion positions anti-symmetrically, and ψα⊘−{displaystyle psi _{alpha }oslash _{-}} deletes the single-particle state ψα{displaystyle psi _{alpha }} from N{displaystyle N} possible deletion positions anti-symmetrically.

Action on Fock states

Starting from the single-mode vacuum state |0α⟩=1{displaystyle |0_{alpha }rangle =1}, applying the fermion creation operator cα†{displaystyle c_{alpha }^{dagger }},

cα†|0α⟩=ψα⊗−1=ψα=|1α⟩,{displaystyle c_{alpha }^{dagger }|0_{alpha }rangle =psi _{alpha }otimes _{-}1=psi _{alpha }=|1_{alpha }rangle ,}

cα†|1α⟩=12ψα⊗−ψα=0.{displaystyle c_{alpha }^{dagger }|1_{alpha }rangle ={frac {1}{sqrt {2}}}psi _{alpha }otimes _{-}psi _{alpha }=0.}

If the single-particle state |α⟩{displaystyle |alpha rangle } 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 cα{displaystyle c_{alpha }},

cα|1α⟩=ψα⊘−ψα=1=|0α⟩,{displaystyle c_{alpha }|1_{alpha }rangle =psi _{alpha }oslash _{-}psi _{alpha }=1=|0_{alpha }rangle ,}

cα|0α⟩=0.{displaystyle c_{alpha }|0_{alpha }rangle =0.}

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

|nα⟩=(cα†)nα|0α⟩.{displaystyle |n_{alpha }rangle =(c_{alpha }^{dagger })^{n_{alpha }}|0_{alpha }rangle .}

It is easy to check (by enumeration) that

cα†cα|nα⟩=nα|nα⟩,{displaystyle c_{alpha }^{dagger }c_{alpha }|n_{alpha }rangle =n_{alpha }|n_{alpha }rangle ,}

meaning that n^α=cα†cα{displaystyle {hat {n}}_{alpha }=c_{alpha }^{dagger }c_{alpha }} defines the fermion number operator.

The above result can be generalized to any Fock state of fermions.

cα†|⋯,nβ,nα,nγ,⋯⟩=(−1)∑β<αnβ1−nα|⋯,nβ,1−nα,nγ,⋯⟩.{displaystyle c_{alpha }^{dagger }|cdots ,n_{beta },n_{alpha },n_{gamma },cdots rangle =(-1)^{sum _{beta <alpha }n_{beta }}{sqrt {1-n_{alpha }}}|cdots ,n_{beta },1-n_{alpha },n_{gamma },cdots rangle .}

cα|⋯,nβ,nα,nγ,⋯⟩=(−1)∑β<αnβnα|⋯,nβ,1−nα,nγ,⋯⟩.{displaystyle c_{alpha }|cdots ,n_{beta },n_{alpha },n_{gamma },cdots rangle =(-1)^{sum _{beta <alpha }n_{beta }}{sqrt {n_{alpha }}}|cdots ,n_{beta },1-n_{alpha },n_{gamma },cdots rangle .}

Recall that the occupation number nα{displaystyle n_{alpha }} 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 (−1)∑β<αnβ{displaystyle (-1)^{sum _{beta <alpha }n_{beta }}}, 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.^[5]

Operator identities

The following operator identities follow from the action of the fermion creation and annihilation operators on the Fock state,

{cα†,cβ†}={cα,cβ}=0,{cα,cβ†}=δαβ.{displaystyle {c_{alpha }^{dagger },c_{beta }^{dagger }}={c_{alpha },c_{beta }}=0,quad {c_{alpha },c_{beta }^{dagger }}=delta _{alpha beta }.}

These anti-commutation relations can be considered as the algebraic definition of the fermion creation and annihilation operators. The fact that the fermion many-body wave function is anti-symmetric under particle exchange is also manifested by the anti-commutation of the fermion operators.

Quantum field operators

Defining aν†{displaystyle a_{nu }^{dagger }} as a general annihilation(creation) operator for a single-particle state ν{displaystyle nu } that could be either fermionic (cν†){displaystyle (c_{nu }^{dagger })} or bosonic (bν†){displaystyle (b_{nu }^{dagger })}, the real space representation of the operators defines the quantum field operators Ψ(r){displaystyle Psi (mathbf {r} )} and Ψ†(r){displaystyle Psi ^{dagger }(mathbf {r} )} by

Ψ(r)=∑νψν(r)aν{displaystyle Psi (mathbf {r} )=sum _{nu }psi _{nu }left(mathbf {r} right)a_{nu }}

Ψ†(r)=∑νψν∗(r)aν†{displaystyle Psi ^{dagger }(mathbf {r} )=sum _{nu }psi _{nu }^{*}left(mathbf {r} right)a_{nu }^{dagger }}

These are second quantization operators, with coefficients ψν(r){displaystyle psi _{nu }left(mathbf {r} right)} and ψν∗(r){displaystyle psi _{nu }^{*}left(mathbf {r} right)} that are ordinary first-quantization wavefunctions. Thus, for example, any expectation values will be ordinary first-quantization wavefunctions. Loosely speaking, Ψ†(r){displaystyle Psi ^{dagger }(mathbf {r} )} is the sum of all possible ways to add a particle to the system at position r through any of the basis states ψν(r){displaystyle psi _{nu }left(mathbf {r} right)}.

Since Ψ(r){displaystyle Psi (mathbf {r} )} and Ψ†(r){displaystyle Psi ^{dagger }(mathbf {r} )} are second quantization operators defined in every point in space they are called quantum field operators. They obey the following fundamental commutator and anti-commutator relations,

[Ψ(r1),Ψ†(r2)]=δ(r1−r2){displaystyle left[Psi (mathbf {r} _{1}),Psi ^{dagger }(mathbf {r} _{2})right]=delta (mathbf {r} _{1}-mathbf {r} _{2})} boson fields,

{Ψ(r1),Ψ†(r2)}=δ(r1−r2){displaystyle {Psi (mathbf {r} _{1}),Psi ^{dagger }(mathbf {r} _{2})}=delta (mathbf {r} _{1}-mathbf {r} _{2})} fermion fields.

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:

Ψ(r)=1V∑keik⋅rak{displaystyle Psi (mathbf {r} )={1 over {sqrt {V}}}sum _{mathbf {k} }e^{imathbf {kcdot r} }a_{mathbf {k} }}

Ψ†(r)=1V∑ke−ik⋅ra†k{displaystyle Psi ^{dagger }(mathbf {r} )={1 over {sqrt {V}}}sum _{mathbf {k} }e^{-imathbf {kcdot r} }{a^{dagger }}_{mathbf {k} }}

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 quantum-mechanical one.

See also

• Schrödinger functional

References

Further reading

• 
  Second quantization Carlo Maria Becchi, Scholarpedia, 5(6):7902. doi:10.4249/scholarpedia.7902
  

External links

Second quantization on Wikiversity

• 
  Many-Electron States in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013,
  ISBN 978-3-89336-884-6