Wick's theorem
Encyclopedia
Wick's theorem is a method of reducing high-order derivative
s to a combinatorics
problem (Philips, 2001). It is named after Gian-Carlo Wick
. It is used extensively in quantum field theory
to reduce arbitrary products of creation and annihilation operators
to sums of products of pairs of these operators. This allows for the use of Green's function methods
, and consequently the use of Feynman diagram
s in the field under study.
and 
we define their contraction to be
where
denotes the normal order
of operator
There is alternative notation for this as a line joining
and 
.
We shall look in detail at four special cases where
and 
are equal to creation and annihilation operators. For 
particles we'll denote the creation operators by 
and the annihilation operators by 
(
).
We then have
where
and 
denotes the Kronecker delta.
These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.
and 
(
) are bosonic operators satisfying the commutation relations:
where
, 
denotes the commutator
and
denotes the Kronecker delta.
We can use these relations, and the above definition of contraction, to express products of
and 
in other ways.
Example 1
Note that we have not changed
but merely re-expressed it in another form as 
Example 2
Example 3
In the last line we have used different numbers of
symbols to denote different contractions. By repeatedly applying the commutation relations it takes a lot of work, as you can see, to express 
in the form of a sum of normally ordered products. It is an even lengthier calculation for more complicated products.
Luckily Wick's theorem provides a short cut.
we can express it as
In words, this theorem states that a string of creation and annihilation operators can be rewritten as the normal ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.
Applying the theorem to the above examples provides a much quicker method to arrive at the final expressions.
A warning: In terms on the right hand side containing multiple contractions care must be taken when the operators are fermionic. In this case an appropriate minus sign must be introduced according to the following rule: rearrange the operators (introducing minus signs whenever the order of two fermionic operators is swapped) to ensure the contracted terms are adjacent in the string. The contraction can then be applied (See Rule C
Example:
If we have two fermions (
) with creation and annihilation operators 
and 
(
) then
Which means that
In the end, we arrive at Wick's theorem:
The T-product of a time-ordered free fields string can be expressed in the following manner:
Applying this theorem to S-matrix
elements, we discover that normal-ordered terms acting on vacuum state
give a null contribution to the sum. We conclude that m is even and only completely contracted terms remain.
where p is the number of interaction fields (or, equivalently, the number of interacting particles) and n is the development order (or the number of vertices of interaction). For example, if
This is analogous to the corresponding theorem in statistics for the moments
of a Gaussian distribution.
Note that this discussion is in terms of the usual definition of normal ordering which is appropriate for the vacuum expectation values of fields. There are any other possible definitions of normal ordering, and Wick's theorem is valid irrespective. However Wick's theorem only simplifies computations if the definition of normal ordering used is changed to match the type of expectation value wanted. That is we always want the expectation value of the normal ordered product to be zero. For instance in
thermal field theory
a different type of expectation value, a thermal trace over the density matrix, requires a different definition of normal ordering is needed (Evans and Steer, 1996).
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s to a combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
problem (Philips, 2001). It is named after Gian-Carlo Wick
Gian-Carlo Wick
Gian Carlo Wick was an Italian theoretical physicist who made important contributions to quantum field theory...
. It is used extensively in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
to reduce arbitrary products of creation and annihilation operators
Creation and annihilation operators
Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one...
to sums of products of pairs of these operators. This allows for the use of Green's function methods
Green's function (many-body theory)
In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators....
, and consequently the use of Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s in the field under study.
Definition of contraction
For two operators and we define their contraction to bewhere
denotes the normal orderNormal order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered 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...
of operator
There is alternative notation for this as a line joining
and .We shall look in detail at four special cases where
and are equal to creation and annihilation operators. For particles we'll denote the creation operators by and the annihilation operators by ().We then have
where
and denotes the Kronecker delta.These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.
Wick's theorem
We can use contractions and normal ordering to express any product of creation and annihilation operators as a sum of normal ordered terms. This is the basis of Wick's theorem. Before stating the theorem fully we shall look at some examples.Examples
Suppose and () are bosonic operators satisfying the commutation relations:where
, denotes the commutatorCommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
and
denotes the Kronecker delta.We can use these relations, and the above definition of contraction, to express products of
and in other ways.Example 1
Note that we have not changed
but merely re-expressed it in another form as Example 2
Example 3
In the last line we have used different numbers of
symbols to denote different contractions. By repeatedly applying the commutation relations it takes a lot of work, as you can see, to express in the form of a sum of normally ordered products. It is an even lengthier calculation for more complicated products.Luckily Wick's theorem provides a short cut.
Statement of the theorem
For a product of creation and annihilation operators we can express it asIn words, this theorem states that a string of creation and annihilation operators can be rewritten as the normal ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.
Applying the theorem to the above examples provides a much quicker method to arrive at the final expressions.
A warning: In terms on the right hand side containing multiple contractions care must be taken when the operators are fermionic. In this case an appropriate minus sign must be introduced according to the following rule: rearrange the operators (introducing minus signs whenever the order of two fermionic operators is swapped) to ensure the contracted terms are adjacent in the string. The contraction can then be applied (See Rule C
Example:
If we have two fermions (
) with creation and annihilation operators and () thenWick's theorem applied to fields
Which means that
In the end, we arrive at Wick's theorem:
The T-product of a time-ordered free fields string can be expressed in the following manner:
Applying this theorem to S-matrix
S matrix
In physics, the scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process...
elements, we discover that normal-ordered terms acting on vacuum state
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
give a null contribution to the sum. We conclude that m is even and only completely contracted terms remain.
where p is the number of interaction fields (or, equivalently, the number of interacting particles) and n is the development order (or the number of vertices of interaction). For example, if
This is analogous to the corresponding theorem in statistics for the moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
of a Gaussian distribution.
Note that this discussion is in terms of the usual definition of normal ordering which is appropriate for the vacuum expectation values of fields. There are any other possible definitions of normal ordering, and Wick's theorem is valid irrespective. However Wick's theorem only simplifies computations if the definition of normal ordering used is changed to match the type of expectation value wanted. That is we always want the expectation value of the normal ordered product to be zero. For instance in
thermal field theory
Thermal quantum field theory
In theoretical physics, thermal quantum field theory or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature....
a different type of expectation value, a thermal trace over the density matrix, requires a different definition of normal ordering is needed (Evans and Steer, 1996).