Wick's theorem

Summary

Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem.[1] It is named after Italian physicist Gian-Carlo Wick.[2] 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 diagrams in the field under study. A more general idea in probability theory is Isserlis' theorem.

In perturbative quantum field theory, Wick's theorem is used to quickly rewrite each time ordered summand in the Dyson series as a sum of normal ordered terms. In the limit of asymptotically free ingoing and outgoing states, these terms correspond to Feynman diagrams.

Definition of contractionEdit

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.

We then have

 
 
 
 

where  .

These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.

ExamplesEdit

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, 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 1Edit

 

Note that we have not changed   but merely re-expressed it in another form as  

Example 2Edit

 

Example 3Edit

 
 
 
 
 

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 theoremEdit

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 theoremEdit

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 Nth 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.[3]

Wick's theorem applied to fieldsEdit

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 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 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 (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.[4]) 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.[5]

See alsoEdit

ReferencesEdit

  1. ^ Tony Philips (November 2001). "Finite-dimensional Feynman Diagrams". What's New In Math. American Mathematical Society. Retrieved 2007-10-23.
  2. ^ Wick, G. C. (1950). "The Evaluation of the Collision Matrix". Phys. Rev. 80 (2): 268–272. doi:10.1103/PhysRev.80.268.
  3. ^ Coleman, Sydney (2019). Quantum Field Theory: Lectures of Sidney Coleman. World Scientific Publishing. p. 158.
  4. ^ See for example also: Mrinal Dasgupta: An introduction to Quantum Field Theory, Lectures presented at the RAL School for High Energy Physics, Somerville College, Oxford, September 2008, section 5.1 Wick's Theorem (downloaded 3 December 2012)
  5. ^ Evans, T. S.; Steer, D. A. (1996). "Wick's theorem at finite temperature". Nucl. Phys. B. 474: 481–496. arXiv:hep-ph/9601268. doi:10.1016/0550-3213(96)00286-6.

Further readingEdit

  • Peskin, M. E.; Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Perseus Books. (§4.3)
  • Schweber, Silvan S. (1962). An Introduction to Relativistic Quantum Field Theory. New York: Harper and Row. (Chapter 13, Sec c)