Editing Second quantization (section)

=== 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.