Editing Second quantization (section)

==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.<ref name="Mahan2000">{{cite book
 | author = Mahan, Gerald D.
 | title = Many-Particle Physics
 | publisher = Springer
 | location = New York
 | edition = 3rd
 | series = Physics of Solids and Liquids
 | doi = 10.1007/978-1-4757-5714-9
 | year = 2000
 | isbn = 978-1-4757-5714-9
}}</ref> 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 <math>\psi_\alpha</math> be a single-particle state, let 1 be the tensor identity (it is the generator of the zero-particle space '''C''' and satisfies <math>\psi_\alpha\equiv1\otimes\psi_\alpha\equiv\psi_\alpha\otimes1</math> in the [[tensor algebra]] over the fundamental Hilbert space), and let <math>\Psi =\psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots</math> be a generic tensor product state. The insertion <math>\otimes_\pm</math>  and the deletion <math>\oslash_\pm</math> operators are linear operators defined by the following recursive equations
:<math>\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);</math>
:<math>\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).</math>
Here <math>\delta_{\alpha\beta}</math> is the [[Kronecker delta]] symbol, which gives 1 if <math>\alpha=\beta</math>, and 0 otherwise. The subscript <math>\pm</math> 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 <math>b_{\alpha}^\dagger</math> (resp. <math>b_{\alpha}</math>). The creation operator <math>b_{\alpha}^\dagger</math> adds a boson to the single-particle state <math>|\alpha\rangle</math>, and the annihilation operator <math>b_{\alpha}</math> removes a boson from the single-particle state <math>|\alpha\rangle</math>. The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (<math>b_\alpha\neq b_\alpha^\dagger</math>).

==== Definition ====

The boson creation (annihilation) operator is a linear operator, whose action on a ''N''-particle first-quantized wave function <math>\Psi</math> is defined as
:<math>b_\alpha^\dagger \Psi = \frac{1}{\sqrt{N+1}}\psi_\alpha\otimes_+\Psi,</math>
:<math>b_\alpha\Psi = \frac{1}{\sqrt{N}}\psi_\alpha\oslash_+\Psi,</math>
where <math>\psi_\alpha\otimes_+</math> inserts the single-particle state <math>\psi_\alpha</math> in <math>N+1</math> possible insertion positions symmetrically, and <math>\psi_\alpha\oslash_+</math> deletes the single-particle state <math>\psi_\alpha</math> from <math>N</math> possible deletion positions symmetrically.

===== Examples =====
Hereinafter the tensor symbol <math>\otimes</math> between single-particle states is omitted for simplicity. Take the state <math>|1_1,1_2\rangle=(\psi_1\psi_2+\psi_2\psi_1)/\sqrt{2}</math>, create one more boson on the state <math>\psi_1</math>,
:<math>\begin{array}{rl}b_1^\dagger|1_1,1_2\rangle=&\frac{1}{\sqrt{2}}(b_1^\dagger\psi_1\psi_2+b_1^\dagger\psi_2\psi_1)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}\psi_1\otimes_+\psi_1\psi_2+\frac{1}{\sqrt{3}}\psi_1\otimes_+\psi_2\psi_1\right)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}(\psi_1\psi_1\psi_2+\psi_1\psi_1\psi_2+\psi_1\psi_2\psi_1)+\frac{1}{\sqrt{3}}(\psi_1\psi_2\psi_1+\psi_2\psi_1\psi_1+\psi_2\psi_1\psi_1)\right)\\=&\frac{\sqrt{2}}{\sqrt{3}}(\psi_1\psi_1\psi_2+\psi_1\psi_2\psi_1+\psi_2\psi_1\psi_1)\\
=&\sqrt{2}|2_1,1_2\rangle.\end{array}</math>
Then annihilate one boson from the state <math>\psi_1</math>,
:<math>\begin{array}{rl}b_1|2_1,1_2\rangle=&\frac{1}{\sqrt{3}}(b_1\psi_1\psi_1\psi_2+b_1\psi_1\psi_2\psi_1+b_1\psi_2\psi_1\psi_1)\\=&\frac{1}{\sqrt{3}}\left(\frac{1}{\sqrt{3}}\psi_1\oslash_+\psi_1\psi_1\psi_2+\frac{1}{\sqrt{3}}\psi_1\oslash_+\psi_1\psi_2\psi_1+\frac{1}{\sqrt{3}}\psi_1\oslash_+\psi_2\psi_1\psi_1\right)\\=&\frac{1}{\sqrt{3}}\left(\frac{1}{\sqrt{3}}(\psi_1\psi_2+\psi_1\psi_2+0)+\frac{1}{\sqrt{3}}(\psi_2\psi_1+0+\psi_1\psi_2)+\frac{1}{\sqrt{3}}(0+\psi_2\psi_1+\psi_2\psi_1)\right)\\=&\psi_1\psi_2+\psi_2\psi_1\\=&\sqrt{2}|1_1,1_2\rangle.\end{array}</math>

==== Action on Fock states ====
Starting from the single-mode vacuum state <math>|0_\alpha\rangle=1</math>, applying the creation operator <math>b_\alpha^\dagger</math> repeatedly, one finds
:<math>b_\alpha^\dagger|0_\alpha\rangle=\psi_\alpha\otimes_+ 1=\psi_\alpha=|1_\alpha\rangle,</math>
:<math>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+1\rangle.</math>
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
:<math>|n_\alpha\rangle=\frac{1}{\sqrt{n_\alpha!}}(b_{\alpha}^\dagger)^{n_\alpha}|0_\alpha\rangle.</math>
On the other hand, the annihilation operator <math>b_\alpha</math> lowers the boson occupation number by 1
:<math>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-1\rangle.</math>
It will also quench the vacuum state <math>b_\alpha|0_\alpha\rangle=0</math> as there has been no boson left in the vacuum state to be annihilated. Using the above formulae, it can be shown that
:<math>b_\alpha^\dagger b_\alpha|n_\alpha\rangle=n_\alpha|n_\alpha\rangle,</math>
meaning that <math>\hat{n}_\alpha = b_\alpha^\dagger b_\alpha</math> defines the boson number operator.

The above result can be generalized to any Fock state of bosons.
:<math>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.</math>
:<math>b_\alpha|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= \sqrt{n_\alpha}|\cdots,n_\beta,n_\alpha-1,n_\gamma,\cdots\rangle.</math>
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,
:<math>[b_\alpha^\dagger,b_\beta^\dagger]=[b_\alpha,b_\beta]=0,\quad [b_\alpha,b_\beta^\dagger]=\delta_{\alpha\beta}.</math>
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 satisfy the same set of commutation relations, implying that the bosons can be interpreted as the energy quanta (phonons) of an oscillator. The position and momentum operators of a Harmonic oscillator (or a collection of Harmonic oscillating modes) are given by Hermitian combinations of phonon creation and annihilation operators,
:<math>x_{\alpha}=(b_{\alpha}+b_{\alpha}^\dagger)/\sqrt{2},\quad p_{\alpha}=(b_{\alpha}-b_{\alpha}^\dagger)/(\sqrt{2}\mathrm{i}), </math>
which reproduce the canonical commutation relation between position and momentum operators (with <math>\hbar=1</math>)
:<math>[x_{\alpha},p_{\beta}]=\mathrm{i}\delta_{\alpha\beta},\quad [x_{\alpha},x_{\beta}]=[p_{\alpha},p_{\beta}]=0.</math>
This idea is generalized in the [[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 <math>c_{\alpha}^\dagger</math> (<math>c_{\alpha}</math>). The creation operator <math>c_{\alpha}^\dagger</math> adds a fermion to the single-particle state <math>|\alpha\rangle</math>, and the annihilation operator <math>c_{\alpha}</math> removes a fermion from the single-particle state <math>|\alpha\rangle</math>.

==== Definition ====

The fermion creation (annihilation) operator is a linear operator, whose action on a ''N''-particle first-quantized wave function <math>\Psi</math> is defined as
:<math>c_\alpha^\dagger \Psi = \frac{1}{\sqrt{N+1}}\psi_\alpha\otimes_-\Psi,</math>
:<math>c_\alpha\Psi = \frac{1}{\sqrt{N}}\psi_\alpha\oslash_-\Psi,</math>
where <math>\psi_\alpha\otimes_-</math> inserts the single-particle state <math>\psi_\alpha</math> in <math>N+1</math> possible insertion positions anti-symmetrically, and <math>\psi_\alpha\oslash_-</math> deletes the single-particle state <math>\psi_\alpha</math> from <math>N</math> possible deletion positions anti-symmetrically.

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. A few illustrative operations are given in the example below.  The complete algebra for creation and annihilation operators on a two-fermion state can be found in ''Quantum Photonics''.<ref name="Pearsall2020">{{cite book
 | author = Pearsall, Thomas P.
 | title = Quantum Photonics
 | publisher = Springer
 | location = Cham, Switzerland
 | edition = 2nd
 | series = Graduate Texts in Physics
 | isbn = 978-3-030-47325-9
 | doi = 10.1007/978-3-030-47325-9
 | year = 2020
 | pages = 301–302
| bibcode = 2020quph.book.....P
 }}</ref>

===== Examples =====
Hereinafter the tensor symbol <math>\otimes</math> between single-particle states is omitted for simplicity. Take the state <math>|1_1,1_2\rangle=(\psi_1\psi_2-\psi_2\psi_1)/\sqrt{2}</math>, attempt to create one more fermion on the occupied <math>\psi_1</math> state will quench the whole many-body wave function,
:<math>\begin{array}{rl}c_1^\dagger|1_1,1_2\rangle=&\frac{1}{\sqrt{2}}(c_1^\dagger\psi_1\psi_2-c_1^\dagger\psi_2\psi_1)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}\psi_1\otimes_-\psi_1\psi_2-\frac{1}{\sqrt{3}}\psi_1\otimes_-\psi_2\psi_1\right)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}(\psi_1\psi_1\psi_2-\psi_1\psi_1\psi_2+\psi_1\psi_2\psi_1)-\frac{1}{\sqrt{3}}(\psi_1\psi_2\psi_1-\psi_2\psi_1\psi_1+\psi_2\psi_1\psi_1)\right)\\=&0.\end{array}</math>
Annihilate a fermion on the <math>\psi_2</math> state,
take the state <math>|1_1,1_2\rangle=(\psi_1\psi_2-\psi_2\psi_1)/\sqrt{2}</math>,
:<math>\begin{array}{rl}c_2|1_1,1_2\rangle=&\frac{1}{\sqrt{2}}(c_2\psi_1\psi_2-c_2\psi_2\psi_1)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}\psi_2\oslash_-\psi_1\psi_2-\frac{1}{\sqrt{2}}\psi_2\oslash_-\psi_2\psi_1\right)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}(0-\psi_1)-\frac{1}{\sqrt{2}}(\psi_1-0)\right)\\=&-\psi_1\\=&-|1_1,0_2\rangle.\end{array}</math>
The minus sign (known as the fermion sign) appears due to the anti-symmetric property of the fermion wave function.

==== Action on Fock states ====
Starting from the single-mode vacuum state <math>|0_\alpha\rangle=1</math>, applying the fermion creation operator <math>c_\alpha^\dagger</math>, 
:<math>c_\alpha^\dagger|0_\alpha\rangle=\psi_\alpha\otimes_- 1=\psi_\alpha=|1_\alpha\rangle,</math>
:<math>c_\alpha^\dagger|1_\alpha\rangle=\frac{1}{\sqrt{2}}\psi_\alpha\otimes_- \psi_\alpha=0.</math>
If the single-particle state <math>|\alpha\rangle</math> 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 <math>c_\alpha</math>, 
:<math>c_\alpha|1_\alpha\rangle=\psi_\alpha\oslash_-\psi_\alpha=1=|0_\alpha\rangle,</math>
:<math>c_\alpha|0_\alpha\rangle =0.</math>
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
:<math>|n_\alpha\rangle=(c_{\alpha}^\dagger)^{n_\alpha}|0_\alpha\rangle.</math>
It is easy to check (by enumeration) that
:<math>c_\alpha^\dagger c_\alpha|n_\alpha\rangle=n_\alpha|n_\alpha\rangle,</math>
meaning that <math>\hat{n}_\alpha = c_\alpha^\dagger c_\alpha</math> defines the fermion number operator.

The above result can be generalized to any Fock state of fermions.
:<math>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.</math><ref>Book "Nuclear Models" of Greiner and Maruhn p53 equation 3.47 : http://xn--webducation-dbb.com/wp-content/uploads/2019/02/Walter-Greiner-Joachim-A.-Maruhn-D.A.-Bromley-Nuclear-Models-Springer-Verlag-1996.pdf </ref>
:<math>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.</math>
Recall that the occupation number <math>n_\alpha</math> 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 <math>(-1)^{\sum_{\beta<\alpha}n_\beta} </math>, also known as the [[Jordan–Wigner transformation|Jordan-Wigner string]], requires there to exist a predefined ordering of the single-particle states (the [[spin structure]]){{clarify|reason=There is no mention of an ordering of single-particle states in the link, or it is difficult to find. Could it be some spin quantum number instead of spin structure?|date=April 2015}} 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.<ref>{{Cite journal | doi = 10.1103/PhysRevB.67.245316| title = Fermions, strings, and gauge fields in lattice spin models| journal = Physical Review B| volume = 67| issue = 24| year = 2003| last1 = Levin | first1 = M. | last2 = Wen | first2 = X. G. | page = 245316| arxiv = cond-mat/0302460| bibcode = 2003PhRvB..67x5316L| s2cid = 29180411}}</ref>

==== Operator identities ====
The following operator identities follow from the action of the fermion creation and annihilation operators on the Fock state,
:<math>\{c_\alpha^\dagger,c_\beta^\dagger\}=\{c_\alpha,c_\beta\}=0,\quad \{c_\alpha,c_\beta^\dagger\}=\delta_{\alpha\beta}.</math>
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.

The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (<math>c_\alpha\neq c_\alpha^\dagger</math>). The Hermitian combination of the fermion creation and annihilation operators
:<math>\chi_{\alpha,\text{Re}}=(c_\alpha+c_\alpha^\dagger)/\sqrt{2}, \quad \chi_{\alpha,\text{Im}}=(c_\alpha-c_\alpha^\dagger)/(\sqrt{2}\mathrm{i}),</math>
are called [[Majorana fermion]] operators. They can be viewed as the fermionic analog of position and momentum operators of a "fermionic" Harmonic oscillator. They satisfy the anticommutation relation
:<math>\{\chi_{i},\chi_{j}\}=\delta_{ij},</math>
where <math>i,j</math> labels any Majorana fermion operators on equal footing (regardless their origin from Re or Im combination of complex fermion operators <math>c_{\alpha}</math>). The anticommutation relation indicates that Majorana fermion operators generates a [[Clifford algebra]], which can be systematically represented as Pauli operators in the many-body Hilbert space.