Editing Canonical quantization (section)

====Other fields====

All other fields can be quantized by a generalization of this procedure. Vector or tensor fields simply have more components, and independent creation and destruction operators must be introduced for each independent component. If a field has any [[internal symmetry]], then creation and destruction operators must be introduced for each component of the field related to this symmetry as well. If there is a [[gauge symmetry]], then the number of independent components of the field must be carefully analyzed to avoid over-counting equivalent configurations, and [[gauge-fixing]] may be applied if needed.

It turns out that commutation relations are useful only for quantizing ''bosons'', for which the occupancy number of any state is unlimited. To quantize ''fermions'', which satisfy the [[Pauli exclusion principle]], anti-commutators are needed.  These are defined by {{math|{''A'', ''B''} {{=}} ''AB'' + ''BA''}}.

When quantizing fermions, the fields are expanded in creation and annihilation operators, {{math|''θ<sub>k</sub>''<sup>†</sup>}}, {{math|''θ<sub>k</sub>''}}, which satisfy
<math display="block">\{\theta_k,\theta_l^\dagger\} = \delta_{kl}, \ \ \{\theta_k, \theta_l\} = 0, \ \ \{\theta_k^\dagger, \theta_l^\dagger\} = 0. </math>

The states are constructed on a vacuum <math>|0\rangle</math> annihilated by the {{math|''θ<sub>k</sub>''}}, and the [[Fock space]] is built by applying all products of creation operators {{math|''θ<sub>k</sub>''<sup>†</sup>}} to {{ket|0}}.  Pauli's exclusion principle is satisfied, because <math>(\theta_k^\dagger)^2|0\rangle = 0</math>, by virtue of the anti-commutation relations.