Editing Second quantization (section)

==== 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>