in the expansion of the Hamiltonian. This procedure is known as Wick ordering or normal ordering. 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 . When quantizing fermions, the fields are expanded in creation and annihilation operators, †, , which satisfy The states are constructed on a vacuum |0> annihilated by the , and the Fock space is built by applying all products of creation operators † to |0>. Pauli's exclusion principle is satisfied, because , by virtue of the anti-commutation relations. Condensates The construction of the scalar field states above assumed that the potential was minimized at = 0, so that the vacuum minimizing the Hamiltonian satisfies 〈 〉= 0, indicating that the vacuum expectation value (VEV) of the field is zero. In cases involving spontaneous symmetry breaking, it is possible to have a non-zero VEV, because the potential is minimized for a value = . This occurs for example, if with > 0 and 2 > 0, for which the minimum energy is found at . The value of in one of these vacua may be considered as condensate of the field . Canonical quantization then can be carried out for the shifted field , and particle states with respect to the shifted vacuum are defined by quantizing the shifted field. This construction is utilized in the Higgs mechanism in the standard model of particle physics. Mathematical quantization Deformation quantization The classical theory is described using a spacelike foliation of spacetime with the state at each slice being described by an element of a symplectic manifold with the time evolution given by the symplectomorphism generated by a Hamiltonian function over the symplectic manifold. The quantum