Editing Second quantization (section)

==== 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.