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.
Contents
- Notation
- Bosons
- Single bosons
- Examples
- Multiple bosons
- Fermions
- Single fermions
- Multiple fermions
- Uses in quantum field theory
- Free fields
- Wicks theorem
- Alternative definitions
- References
Normal ordering of a product 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, that 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.
Notation
If
An alternative notation is
Note that the normal ordering is a concept that only makes sense for products of operators. Attempting to apply normal ordering to a sums 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:
These satisfy the commutator relationship
where
Examples
1. We'll consider the simplest case first. This is the normal ordering of
The expression
2. A more interesting example is the normal ordering of
Here the normal ordering operation has reordered the terms by placing
These two results can be combined with the commutation relation obeyed by
or
This equation is used in defining the contractions used in Wick's theorem.
3. An example with multiple operators is:
4. A more complicated example shows how we can normal order functions of operators by expanding them out in a series and normal ordering each term:
5. A simple example shows that normal ordering cannot be extended by linearity from the monomials to all operators in a self-consistent way:
The implication is that normal ordering is not well defined as a function on operators. Indeed, normal ordering only serves as a definition of the LHS purely as a symbolic expression.
Multiple bosons
If we now consider
Here
These satisfy the commutation relations:
where
These may be rewritten as:
Examples
1. For two different bosons (
2. For three different bosons (
Notice that since (by the commutation relations)
Fermions
Fermions are particles which satisfy Fermi–Dirac statistics. We will now examine the normal ordering of fermionic creation and annihilation operator products.
Single fermions
For a single fermion there are two operators of interest:
These satisfy the anticommutator relationships
where
To define the normal ordering of a product of fermionic creation and annihilation operators we must take into account the number of interchanges between neighbouring operators. We get a minus sign for each such interchange.
Examples
1. 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.
2. The normal order of any more complicated cases gives zero because there will be at least one creation or annihilation operator appearing twice. For example:
Multiple fermions
For
Here
These satisfy the commutation relations:
where
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. It is as if we pretend the creation and annihilation operators anticommute and then we reorder the expression to ensure the creation operators are on the left and the annihilation operators are on the right - all the time taking account of the anticommutation relations.
Examples
1. For two different fermions (
Here the expression is already normal ordered so nothing changes.
Here we introduce a minus sign because we have interchanged the order of two operators.
Note that the order in which we write the operators here, unlike in the bosonic case, does matter.
2. For three different fermions (
Notice that since (by the anticommutation relations)
Similarly we have
Uses in quantum field theory
The vacuum expectation value of a normal ordered product of creation and annihilation operators is zero. This is because, denoting the vacuum state by
(here
Any normal ordered operator therefore has a vacuum expectation value of zero. Although an operator
we always have
This is 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:
Free fields
With two free fields φ and χ,
where
Wick's theorem
Wick's theorem states the existence of a relationship between the time ordered product of
where the summation is over all the distinct ways in which one may pair up fields. The result for
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.
Alternative definitions
The most general definition of normal ordering involves splitting all quantum fields into two parts (for example see Evans and Steer 1996)
It is also important for practical calculations that all the commutators (anti-commutator for fermionic fields) of all
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
So here the number operator