Wick's theorem is a method of reducing high-
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s to a
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
problem. It is named after Italian physicist
Gian-Carlo Wick. It is used extensively in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
to reduce arbitrary products of
creation and annihilation operators
Creation operators 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 (usually d ...
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
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s in the field under study. A more general idea in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
is
Isserlis' theorem
In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis.
This t ...
.
In perturbative quantum field theory, Wick's theorem is used to quickly rewrite each
time ordered
In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter:
:\mathcal P \left\
\equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_).
H ...
summand in the
Dyson series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
as a sum of
normal ordered terms. In the limit of asymptotically free ingoing and outgoing states, these terms correspond to
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s.
Definition of contraction
For two operators
and
we define their contraction to be
:
where
denotes the
normal order of an operator
. Alternatively, contractions can be denoted by a line joining
and
, like
.
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
.
They satisfy the commutation relations for bosonic operators
, or the anti-commutation relations for fermionic operators
where
denotes the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
.
We then have
:
:
:
:
where
.
These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.
Examples
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.
Suppose
and
are
bosonic operators satisfying the
commutation relations:
:
:
:
where
,
denotes the
commutator
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
The commutator of two elements, ...
, and
is 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 shortcut.
Statement of the theorem
A product of creation and annihilation operators
can be expressed as
:
In other words, 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" in Wick's paper).
Example:
If we have two fermions (
) with creation and annihilation operators
and
(
) then
:
Note that the term with contractions of the two creation operators and of the two annihilation operators is not included because their contractions vanish.
Proof of Wick's theorem
We use induction to prove the theorem for bosonic creation and annihilation operators. The
base case is trivial, because there is only one possible contraction:
:
In general, the only non-zero contractions are between an annihilation operator on the left and a creation operator on the right. Suppose that Wick's theorem is true for
operators
, and consider the effect of adding an ''N''th operator
to the left of
to form
. By Wick's theorem applied to
operators, we have:
:
is either a creation operator or an annihilation operator. If
is a creation operator, all above products, such as
, are already normal ordered and require no further manipulation. Because
is to the left of all annihilation operators in
, any contraction involving it will be zero. Thus, we can add all contractions involving
to the sums without changing their value. Therefore, if
is a creation operator, Wick's theorem holds for
.
Now, suppose that
is an annihilation operator. To move
from the left-hand side to the right-hand side of all the
products, we repeatedly swap
with the operator immediately right of it (call it
), each time applying
to account for noncommutativity. Once we do this, all terms will be normal ordered. All terms added to the sums by pushing
through the products correspond to additional contractions involving
. Therefore, if
is an annihilation operator, Wick's theorem holds for
.
We have proved the base case and the induction step, so the theorem is true. By introducing the appropriate minus signs, the proof can be extended to fermionic creation and annihilation operators. The theorem applied to fields is proved in essentially the same way.
Wick's theorem applied to fields
The correlation function that appears in quantum field theory can be expressed by a contraction on the field operators:
:
where the operator
are the amount that do not annihilate the vacuum state
. Which means that
. This means that
is a contraction over
. Note that the contraction of a time-ordered string of two field operators is a c-number.
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
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
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
Isserlis' theorem
In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis.
This t ...
in statistics for the
moments of a
Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
.
Note that this discussion is in terms of the usual definition of normal ordering which is appropriate for the
vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
s (VEV's) of fields. (Wick's theorem provides as a way of expressing VEV's of ''n'' fields in terms of VEV's of two 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
In theoretical physics, thermal quantum field theory (thermal field theory for short) 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
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 operato ...
.
See also
*
Isserlis' theorem
In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis.
This t ...
References
Further reading
* (ยง4.3)
* {{cite book , first=Silvan S. , last=Schweber , author-link=Silvan S. Schweber , title=An Introduction to Relativistic Quantum Field Theory , url=https://archive.org/details/introductiontore0000schw , url-access=registration , publisher=Harper and Row , location=New York , year=1962 (Chapter 13, Sec c)
Quantum field theory
Scattering theory
Physics theorems