Wick's theorem

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Wick's theorem is a method of reducing high-order derivatives 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 diagrams in the field under study.

[edit] Definition of contraction

For two operators $\hat{A}$ and $\hat{B}$ we define their contraction to be

$\hat{A}^\bullet\, \hat{B}^\bullet \equiv \hat{A}\,\hat{B}\, - : \hat{A}\,\hat{B} :$

where $: \hat{O} :$ denotes the normal order of operator $\hat{O}$

There is alternative notation for this as a line joining $\hat{A}$ and $\hat{B}$ .

We shall look in detail at four special cases where $\hat{A}$ and $\hat{B}$ are equal to creation and annihilation operators. For $N$ particles we'll denote the creation operators by $\hat{a}_i^\dagger$ and the annihilation operators by $\hat{a}_i$ ( $i=1,\ldots,N$ ).

We then have

$\hat{a}_i^\bullet \,\hat{a}_j^\bullet = \hat{a}_i \,\hat{a}_j \,- :\,\hat{a}_i\, \hat{a}_j\,:\, = 0$

$\hat{a}_i^{\dagger\bullet}\, \hat{a}_j^{\dagger\bullet} = \hat{a}_i^\dagger\, \hat{a}_j^\dagger \,-\,:\hat{a}_i^\dagger\,\hat{a}_j^\dagger\,:\, = 0$

$\hat{a}_i^{\dagger\bullet}\, \hat{a}_j^\bullet = \hat{a}_i^\dagger\, \hat{a}_j \,- :\,\hat{a}_i^\dagger \,\hat{a}_j\, :\,= 0$

$\hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet}= \hat{a}_i\, \hat{a}_j^\dagger \,- :\,\hat{a}_i\,\hat{a}_j^\dagger \,:\, = \delta_{ij}$

where $i,j = 1,\ldots,N$ and $δ i j$ denotes the Kronecker delta.

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

[edit] 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.

[edit] Examples

Suppose $\hat{a}_i$ and $\hat{a}_i^\dagger$ ( $i=1,\ldots,N$ ) are bosonic operators satisfying the commutation relations:

$\left [\hat{a}_i^\dagger, \hat{a}_j^\dagger \right]_- = 0$

$\left [\hat{a}_i, \hat{a}_j \right]_- = 0$

$\left [\hat{a}_i, \hat{a}_j^\dagger \right ]_- = \delta_{ij}$

where $i,j = 1,\ldots,N$ , $\left[ A, B \right]_i \equiv AB - BA$ denotes the commutator and $δ i j$ denotes the Kronecker delta.

We can use these relations, and the above definition of contraction, to express products of $\hat{a}_i$ and $\hat{a}_i^\dagger$ in other ways.

Example 1

$\hat{a}_i \,\hat{a}_j^\dagger = \hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij} = \hat{a}_j^\dagger \,\hat{a}_i + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} =\,:\,\hat{a}_i\, \hat{a}_j^\dagger \,: + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet}$

Note that we have not changed $\hat{a}_i \,\hat{a}_j^\dagger$ but merely re-expressed it in another form as $\,:\,\hat{a}_i\, \hat{a}_j^\dagger \,: + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet}$ .

Example 2

$\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k= (\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij})\hat{a}_k = \hat{a}_j^\dagger \,\hat{a}_i\, \hat{a}_k + \delta_{ij}\hat{a}_k = \hat{a}_j^\dagger \,\hat{a}_i\,\hat{a}_k + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \hat{a}_k =\,:\,\hat{a}_i\, \hat{a}_j^\dagger \hat{a}_k \,: + \hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \,\hat{a}_k$

Example 3

$\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^\dagger= (\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij})(\hat{a}_l^\dagger\,\hat{a}_k + \delta_{kl}) = \hat{a}_j^\dagger \,\hat{a}_i\, \hat{a}_l^\dagger\, \hat{a}_k + \delta_{kl}\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij}\hat{a}_l^\dagger\hat{a}_k + \delta_{ij} \delta_{kl}$

$= \hat{a}_j^\dagger (\hat{a}_l^\dagger\, \hat{a}_i + \delta_{il}) \hat{a}_k + \delta_{kl}\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij}\hat{a}_l^\dagger\hat{a}_k + \delta_{ij} \delta_{kl}$

$= \hat{a}_j^\dagger \hat{a}_l^\dagger\, \hat{a}_i \hat{a}_k + \delta_{il} \hat{a}_j^\dagger \, \hat{a}_k + \delta_{kl}\hat{a}_j^\dagger \,\hat{a}_i + \delta_{ij}\hat{a}_l^\dagger\hat{a}_k + \delta_{ij} \delta_{kl}$

$= \,:\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^\dagger\,: + :\,\hat{a}_i^\bullet \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^{\dagger\bullet}\,:+:\,\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k^\bullet \,\hat{a}_l^{\dagger\bullet}\,:+:\,\hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \, \hat{a}_k \,\hat{a}_l^\dagger\,:+ \,:\hat{a}_i^\bullet \,\hat{a}_j^{\dagger\bullet} \, \hat{a}_k^{\bullet\bullet}\,\hat{a}_l^{\dagger\bullet\bullet},:$

In the last line we have used different numbers of $^\bullet$ symbols to denote different contractions. By repeatedly applying the commutation relations it takes a lot of work, as you can see, to express $\hat{a}_i \,\hat{a}_j^\dagger \, \hat{a}_k \,\hat{a}_l^\dagger$ 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.

[edit] Statement of the theorem

For a product of creation and annihilation operators $\hat{A} \hat{B} \hat{C} \hat{D} \hat{E} \hat{F}\ldots$ we can express it as

$\begin{array}{ll} \hat{A} \hat{B} \hat{C} \hat{D} \hat{E} \hat{F}\ldots & = \,: \hat{A} \hat{B} \hat{C} \hat{D} \hat{E} \hat{F}\ldots \,: \\ & + \sum_{singles} \,: \hat{A}^\bullet \hat{B}^{\bullet} \hat{C} \hat{D} \hat{E} \hat{F}\ldots \,: \\ & + \sum_{doubles} \,: \hat{A}^\bullet \hat{B}^{\bullet\bullet} \hat{C}^{\bullet\bullet} \hat{D}^\bullet \hat{E} \hat{F}\ldots \,: \\ & +\ldots \end{array}$

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'' in Wick's paper).

Example:

If we have two fermions ( $N = 2$ ) with creation and annihilation operators $\hat{f}_i^\dagger$ and $\hat{f}_i$ ( $i = 1,2$ ) then

$\begin{array}{ll} \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \,&= \,: \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \, : \\ & + \,: \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, : + \,: \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, : +\,: \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, : + : \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, : \\ & -: \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet\bullet} \,\hat{f}_2^{\dagger\bullet} \, :+: \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^{\dagger\bullet\bullet}: \end{array}$

[edit] Wick's theorem applied to fields

$\mathcal C(x_1, x_2)=\left \langle 0 |\mathcal T\phi_i(x_1)\phi_i(x_2)|0\right \rangle=\overline{\phi_i(x_1)\phi_i(x_2)}=i\Delta_F(x_1-x_2) =i\int{\frac{d^4k}{(2\pi)^4}\frac{e^{-ik(x_1-x_2)}}{(k^2-m^2)+i\epsilon}}.$

Which means that $\overline{AB}=\mathcal TAB-:AB:$

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:

$\mathcal T\Pi_{k=1}^m\phi(x_k)=:\Pi\phi_i(x_k):+\sum_{\alpha,\beta}\overline{\phi(x_\alpha)\phi(x_\beta)}:\Pi_{k\not=\alpha,\beta}\phi_i(x_k):+$

$\mathcal +\sum_{(\alpha,\beta),(\gamma,\delta)}\overline{\phi(x_\alpha)\phi(x_\beta)}\;\overline{\phi(x_\gamma)\phi(x_\delta)}:\Pi_{k\not=\alpha,\beta,\gamma,\delta}\phi_i(x_k):+\cdots.$

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.

$F_m^i(x)=\left \langle 0 |\mathcal T\phi_i(x_1)\phi_i(x_2)|0\right \rangle=\sum_\mathrm{pairs}\overline{\phi(x_1)\phi(x_2)}\cdots \overline{\phi(x_{m-1})\phi(x_m})$

$G_p^{(n)}=\left \langle 0 |\mathcal T:v_i(y_1):\dots:v_i(y_n):\phi_i(x_1)\cdots \phi_i(x_p)|0\right \rangle$

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 $v=gy^4 \Rightarrow :v_i(y_1):=:\phi_i(y_1)\phi_i(y_1)\phi_i(y_1)\phi_i(y_1):$

This is analogous to the corresponding theorem in statistics for the moments of a Gaussian distribution.

[edit] Bibliography

G.C. Wick, The Evaluation of the Collision Matrix, Phys. Rev. 80, 268 - 272 (1950)

S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York (1962). (Chapter 13, Sec c)

M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books (1995). (§4.3)

Tony Philips (11 2001). "Finite-dimensional Feynman Diagrams". What's New In Math. American Mathematical Society. Retrieved on 2007-10-23.

Emilio San Fabian (2 2001). "Wick's theorem". Retrieved on 2008-07-29.

[edit] See also

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Wick's theorem

From Wikipedia, the free encyclopedia

Contents

[edit] Definition of contraction

[edit] Wick's theorem

[edit] Examples

[edit] Statement of the theorem

[edit] Wick's theorem applied to fields

[edit] Bibliography

[edit] See also

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