In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operators in the product. The process of putting a product into normal order is called normal ordering (also called Wick ordering). The terms antinormal order and antinormal ordering are analogously defined, where the annihilation operators are placed to the left of the creation operators. Normal ordering of a product of quantum fields or creation and annihilation operators can also be defined in many other ways. Which definition is most appropriate depends on the expectation values needed for a given calculation. Most of this article uses the most common definition of normal ordering as given above, which is appropriate when taking expectation values using the vacuum state of the creation and annihilation operators. The process of normal ordering is particularly important for a quantum mechanical Hamiltonian. When quantizing a classical Hamiltonian there is some freedom when choosing the operator order, and these choices lead to differences in the ground state energy. That's why the process can also be used to eliminate the infinite vacuum energy of a quantum field.
Notation If O ^ {\displaystyle {\hat {O}}} denotes an arbitrary product of creation and/or annihilation operators (or equivalently, quantum fields), then the normal ordered form of O ^ {\displaystyle {\hat {O}}} is denoted by : O ^ : {\displaystyle {\mathopen {:}}{\hat {O}}{\mathclose {:}}} . An alternative notation is N ( O ^ ) {\displaystyle {\mathcal {N}}({\hat {O}})} . Note that normal ordering is a concept that only makes sense for products of operators. Attempting to apply normal ordering to a sum of operators is not useful as normal ordering is not a linear operation.
Bosons Bosons are particles which satisfy Bose–Einstein statistics. We will now examine the normal ordering of bosonic creation and annihilation operator products.
Single bosons If we start with only one type of boson there are two operators of interest:
b ^ † {\displaystyle {\hat {b}}^{\dagger }} : the boson's creation operator.
b ^ {\displaystyle {\hat {b}}} : the boson's annihilation operator. These satisfy the commutator relationship
[ b ^ † , b ^ † ] − = 0 {\displaystyle \left[{\hat {b}}^{\dagger },{\hat {b}}^{\dagger }\right]_{-}=0}
[ b ^ , b ^ ] − = 0 {\displaystyle \left[{\hat {b}},{\hat {b}}\right]_{-}=0}
[ b ^ , b ^ † ] − = 1 {\displaystyle \left[{\hat {b}},{\hat {b}}^{\dagger }\right]_{-}=1}
where [ A , B ] − ≡ A B − B A {\displaystyle \left[A,B\right]_{-}\equiv AB-BA} denotes the commutator. We may rewrite the last one as: b ^ b ^ † = b ^ † b ^ + 1. {\displaystyle {\hat {b}}\,{\hat {b}}^{\dagger }={\hat {b}}^{\dagger }\,{\hat {b}}+1.}
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