Editing Second quantization (section)

==== 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.