Normal ordering of a product of quantum fields or creation and annihilation operators can also be defined in many other ways.
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.
denotes an arbitrary product of creation and/or annihilation operators (or equivalently, quantum fields), then the normal ordered form of
Note that normal ordering is a concept that only makes sense for products of operators.
We will now examine the normal ordering of bosonic creation and annihilation operator products.
If we start with only one type of boson there are two operators of interest: These satisfy the commutator relationship where
A simple example shows that normal ordering cannot be extended by linearity from the monomials to all operators in a self-consistent way.
Assume that we can apply the commutation relations to obtain: Then, by linearity, a contradiction.
and are therefore normal ordered by construction, such that the Newton series expansion of an operator function
We will now examine the normal ordering of fermionic creation and annihilation operator products.
For a single fermion there are two operators of interest: These satisfy the anticommutator relationships where
We again start with the simplest cases: This expression is already in normal order so nothing is changed.
In the reverse case, we introduce a minus sign because we have to change the order of two operators: These can be combined, along with the anticommutation relations, to show or This equation, which is in the same form as the bosonic case above, is used in defining the contractions used in Wick's theorem.
The normal order of any more complicated cases gives zero because there will be at least one creation or annihilation operator appearing twice.
These may be rewritten as: When calculating the normal order of products of fermion operators we must take into account the number of interchanges of neighbouring operators required to rearrange the expression.
Note that the order in which we write the operators here, unlike in the bosonic case, does matter.
Similarly we have The vacuum expectation value of a normal ordered product of creation and annihilation operators is zero.
Although this may satisfy we have Normal ordered operators are particularly useful when defining a quantum mechanical Hamiltonian.
If the Hamiltonian of a theory is in normal order then the ground state energy will be zero:
Wick's theorem states the relationship between the time ordered product of
odd looks the same except for the last line which reads This theorem provides a simple method for computing vacuum expectation values of time ordered products of operators and was the motivation behind the introduction of normal ordering.
The most general definition of normal ordering involves splitting all quantum fields into two parts (for example see Evans and Steer 1996)
These two properties means that we can apply Wick's theorem in the usual way, turning expectation values of time-ordered products of fields into products of c-number pairs, the contractions.
The simplest example is found in the context of thermal quantum field theory (Evans and Steer 1996).
In this case the expectation values of interest are statistical ensembles, traces over all states weighted by
is normal ordered in the usual sense used in the rest of the article yet its thermal expectation values are non-zero.
Applying Wick's theorem and doing calculation with the usual normal ordering in this thermal context is possible but computationally impractical.