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| {{Quantum field theory}}
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| '''Second quantization''' is a powerful procedure used in [[quantum field theory]] for describing the [[n-body problem|many-particle]] systems by [[canonical quantization|quantizing]] the fields using a basis that describes the '''number of particles''' occupying each state in a complete set of single-particle states. This differs from the [[first quantization]], which uses the single-particle states as basis.
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| ==Introduction==
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| The starting point of this formalism is the notion of [[indistinguishability]] of particles that bring us to use [[determinant]]s of single-particle states as a basis of the [[Hilbert space]] of N-particles states{{Clarify|date=April 2013}}. Quantum theory can be formulated in terms of occupation numbers (number of particles occupying one determined energy state) of these [[wave function|single-particle states]]. The formalism was introduced in 1927 by [[Paul Dirac|Dirac]].<ref name="Dirac">{{cite doi|10.1098/rspa.1927.0039}}</ref>
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| ===The occupation number representation=== | |
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| Consider an ordered and complete single-particle basis
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| <math> \left\{| \nu_1 \rang, | \nu_2 \rang, | \nu_3 \rang, ...\right\} </math>, where <math>| \nu_i \rang</math> is the set of all states <math>\nu</math> available for the <math>i</math>-th particle. In an N-particle system, only the occupied single-particle states play a role. So it is simpler to formulate a representation where one just counts how many particles there are in each orbital <math>| \nu \rang</math>. This simplification is achieved with the occupation number representation.
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| The basis states for an N-particle system in this representation are obtained simply by listing the occupation numbers of each basis state,
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| <math>|n_{\nu_1}, n_{\nu_2}, n_{\nu_3},\dots \rang</math>, where <math> \sum_j n_{\nu_j} = N</math> The notation means that there are <math> n_{\nu_j}</math> particles in the state <math> \nu_j</math>. It is therefore natural to define the occupation number operator <math> \hat{n}_{\nu_j}</math> which obeys
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| :<math> \hat{n}_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang</math>
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| For [[fermions]] <math> n_{\nu_j}</math> can be 0 or 1, while for [[bosons]] it can be any non negative number
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| :<math>n_{\nu_j}= \begin{cases}
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| \ 0, 1. &\text{fermions}\\
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| 0,1,2,... &\text{bosons}
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| \end{cases}
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| </math>
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| The space spanned by the occupation number basis is denoted the [[Fock space]].
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| ==Creation and annihilation operators==
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| The [[creation and annihilation operators]] are the way to connect the [[first quantization|first]] and second quantizations. It is fundamental for the many-body theory that every operator can be expressed in terms of annihilation and creation operators. Originally constructed in the context of the [[quantum harmonic oscillator]], these operators are the most general form to describe quantum fields.<ref name="Mahan">{{cite book|last=Mahan|first=GD|authorlink=|title=Many Particle Physics|publisher=Springer|location=New York|isbn=0306463385|year=1981}}</ref> Depending on the nature of the fields we can use two different approaches:
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| ===Bosons===
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| Given the occupation number, we introduce the annihilation <math>b_{\nu_j}</math> and creation <math>{b^{\dagger}}_{\nu_j}</math> operators that lowers(raises) the occupation number in the state <math>| \nu_j \rang</math> by 1,
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| :<math>b_{\nu_j}|\dots,n_{\nu_{j-1}}, n_{\nu_j}, n_{\nu_{j+1}},\dots \rang=\sqrt{n_{\nu_j}}|\dots,n_{\nu_{j-1}}, n_{\nu_j}-1, n_{\nu_{j+1}},\dots \rang</math>
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| :<math>{b^{\dagger}}_{\nu_j}|\dots,n_{\nu_{j-1}}, n_{\nu_j}, n_{\nu_{j+1}},\dots \rang=\sqrt{n_{\nu_j}+1}|\dots,n_{\nu_{j-1}}, n_{\nu_j}+1, n_{\nu_{j+1}},\dots \rang</math>
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| Since bosons are symmetric in the single-particle state index <math>\nu_j</math> we demand that <math>b_{\nu_j}</math> and <math>{b^{\dagger}}_{\nu_j}</math> [[commutation|commute]],
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| So, we can obtain the mean properties of these operators:
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| :<math>\begin{matrix}
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| [{b^{\dagger}}_{\nu_j},{b^{\dagger}}_{\nu_k}] = 0 & [b_{\nu_j},b_{\nu_k}]=0 & [b_{\nu_j},{b^{\dagger}}_{\nu_k}]=\delta_{\nu_j\nu_k}\\
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| {b^{\dagger}}_{\nu_j}|n_{\nu_j}\rang=\sqrt{n_{\nu_j}+1}|n_{\nu_j}+1 \rang & b_{\nu_j}|n_{\nu_j}\rang=\sqrt{n_{\nu_j}}|n_{\nu_j} -1\rang & b_{\nu_j}|0\rang=0\\
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| {b^{\dagger}}_{\nu_j}b_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang &\left({b^{\dagger}}_{\nu_j}\right)^{n_{\nu_j}}|0 \rang=\sqrt{(n_{\nu_j})!}|n_{\nu_j} \rang & n_{\nu_j}=0,1,2,\dots\\
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| \end{matrix}</math>
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| and therefore identify the [[identical particles|first]] and second quantized states,
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| :<math> \hat{S}_+|\psi_{n_{\nu_1}}(\bold{r}_1)\rang|\psi_{n_{\nu_2}}(\bold{r}_2)\rang\dots |\psi_{n_{\nu_N}}(\bold{r}_N)\rang= {b^{\dagger}}_{n_{\nu_1}}{b^{\dagger}}_{n_{\nu_2}}\dots{b^{\dagger}}_{n_{\nu_N}}|0\rang</math>
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| with <math> \hat{S}</math> the [[symmetrization]] operator. Here, both contain N-particle state-kets completely symmetric in the single-particle state index <math>\psi_{\nu_j}</math>. Because the creation and annihilation operators of the [[quantum harmonic oscillator]] obey these properties, one can classify the field associated to it as bosonic.
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| ===Fermions===
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| [[Fermions]] have similar annihilation <math>c_{\nu_j}</math> and creation <math>{c^{\dagger}}_{\nu_j}</math> operators:
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| :<math>\begin{matrix} | |
| c_{\nu_j}|\dots,1,\dots \rang= |\dots,0,\dots \rang & c_{\nu_j}|\dots,0,\dots \rang=0 \\
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| {c^{\dagger}}_{\nu_j}|\dots,0,\dots \rang=|\dots,1,\dots\rang& {c^{\dagger}}_{\nu_j}|\dots,1,\dots \rang=0
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| \end{matrix}</math>
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| (Note that the only permitted number of occupation are 0 or 1.) To maintain the fundamental fermionic antisymmetry upon exchange of orbitals, the operators must [[Anticommutativity|anticommute]], rather than commute:
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| :<math>\begin{matrix}
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| \{ {c^{\dagger}}_{\nu_j},{c^{\dagger}}_{\nu_k}\} = 0 & \{c_{\nu_j},c_{\nu_k}\}=0 & \{c_{\nu_j}, {c^{\dagger}}_{\nu_k}\}=\delta_{\nu_j\nu_k}\\
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| \left({c^{\dagger}}_{\nu_j}\right)^2=\left(c_{\nu_j}\right)^2=0 & {c^{\dagger}}_{\nu_j}c_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang & n_{\nu_j}=0,1\\
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| \end{matrix}</math>
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| Where we have used the [[Commutator#Anticommutator|anticommutator]] <math>\{,\}</math>
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| Therefore one can identify the [[identical particles|first]] quantized states in terms of the second quantized:
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| :<math> \hat{S}_-|\psi_{n_{\nu_1}}(\bold{r}_1)\rang|\psi_{n_{\nu_2}}(\bold{r}_2)\rang\dots |\psi_{n_{\nu_N}}(\bold{r}_N)\rang= {c^{\dagger}}_{n_{\nu_1}}{c^{\dagger}}_{n_{\nu_2}}\dots{c^{\dagger}}_{n_{\nu_N}}|0\rang</math>
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| with <math> \hat{S}_-</math> the [[Antisymmetrizer]] operator. Here, both contain N-particle state-kets completely anti-symmetric in the single-particle state index <math>\psi_{\nu_j}</math> in accordance with [[Pauli exclusion principle]].
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| ==Quantum field operators==
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| Defining <math>{a^{(\dagger)}}_{\nu}</math> as a general annihilation(creation) operator that could be either fermionic <math>({c^{(\dagger)}}_{\nu})</math> or bosonic <math>({b^{(\dagger)}}_{\nu})</math>, the [[Position and momentum space|real space representation]] of the operators defines the [[quantum]] field [[operators]] <math> \Psi(\bold{r})</math> and <math>\Psi^{\dagger}(\bold{r})</math> by
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| :<math> \Psi(\bold{r})=\sum_{\nu} \psi_{\nu} \left( \bold{r} \right) a_{\nu}</math>
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| :<math> \Psi^{\dagger}(\bold{r})=\sum_{\nu} {\psi^*}_{\nu} \left( \bold{r} \right) {a^{\dagger}}_{\nu}</math>
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| Second quantization operators, while the coefficients <math>\psi_{\nu} \left( \bold{r} \right)</math> and <math> {\psi^*}_{\nu} \left( \bold{r} \right)</math> are the ordinary [[first quantization]] [[wavefunctions]]. Loosely speaking, <math>\Psi^{\dagger}(\bold{r})</math> is the sum of all possible ways to add a particle to the system at position '''r''' through any of the basis states <math>\psi_{\nu}\left(\bold{r}\right)</math>. Since <math> \Psi(\bold{r})</math> and <math>\Psi^{\dagger}(\bold{r})</math> 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,
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| :<math>\begin{align}
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| \left[\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\right]=\delta (\bold{r}_1-\bold{r}_2) &\text{ boson fields,}\\
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| \{\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\}=\delta (\bold{r}_1-\bold{r}_2)&\text{ fermion fields.}
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| \end{align}</math>
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| In homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in [[Fourier transform|Fourier basis]] yields:
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| :<math> \Psi(\bold{r})={1\over \sqrt {V}} \sum_{\bold{k}} e^{i\bold{k\cdot r}}a_{\bold{k}}</math>
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| :<math> \Psi^{\dagger}(\bold{r})={ 1\over \sqrt{V}} \sum_{\bold{k}} e^{-i\bold{k\cdot r}}{a^{\dagger}}_{\bold{k}}</math>
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| == See also ==
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| * [[Canonical quantization]]
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| * [http://www.cond-mat.de/events/correl13/manuscripts/koch.pdf Many-Electron States] in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013, ISBN 978-3-89336-884-6
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| [[Category:Quantum field theory]]
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| [[Category:Mathematical quantization]]
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| [[es:segunda cuantización]]
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| [[ja:第二量子化]]
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| [[uk:Вторинне квантування ферміонів]]
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