Wick's theorem

Wick's theorem

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.

Definition of contraction

For two operators hat{A} and hat{B} we define their contraction to be:hat{A}^ullet, hat{B}^ullet 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^ullet ,hat{a}_j^ullet = hat{a}_i ,hat{a}_j ,- :,hat{a}_i, hat{a}_j,:, = 0:hat{a}_i^{daggerullet}, hat{a}_j^{daggerullet} = hat{a}_i^dagger, hat{a}_j^dagger ,-,:hat{a}_i^dagger,hat{a}_j^dagger,:, = 0:hat{a}_i^{daggerullet}, hat{a}_j^ullet = hat{a}_i^dagger, hat{a}_j ,- :,hat{a}_i^dagger ,hat{a}_j, :,= 0:hat{a}_i^ullet ,hat{a}_j^{daggerullet}= hat{a}_i, hat{a}_j^dagger ,- :,hat{a}_i,hat{a}_j^dagger ,:, = delta_{ij}where i,j = 1,ldots,N and delta_{ij} 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 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 ight] _- = 0 :left [hat{a}_i, hat{a}_j ight] _- = 0 :left [hat{a}_i, hat{a}_j^dagger ight ] _- = delta_{ij} where i,j = 1,ldots,N, left [ A, B ight] _i equiv AB - BA denotes the commutator and delta_{ij} 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^ullet ,hat{a}_j^{daggerullet} =,:,hat{a}_i, hat{a}_j^dagger ,: + hat{a}_i^ullet ,hat{a}_j^{daggerullet}

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^ullet ,hat{a}_j^{daggerullet}.

"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^ullet ,hat{a}_j^{daggerullet} hat{a}_k =,:,hat{a}_i, hat{a}_j^dagger hat{a}_k ,: + hat{a}_i^ullet ,hat{a}_j^{daggerullet} ,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^daggerhat{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^daggerhat{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^daggerhat{a}_k + delta_{ij} delta_{kl} := ,:hat{a}_i ,hat{a}_j^dagger , hat{a}_k ,hat{a}_l^dagger,: + :,hat{a}_i^ullet ,hat{a}_j^dagger , hat{a}_k ,hat{a}_l^{daggerullet},:+:,hat{a}_i ,hat{a}_j^dagger , hat{a}_k^ullet ,hat{a}_l^{daggerullet},:+:,hat{a}_i^ullet ,hat{a}_j^{daggerullet} , hat{a}_k ,hat{a}_l^dagger,:+ ,:hat{a}_i^ullet ,hat{a}_j^{daggerullet} , hat{a}_k^{ulletullet},hat{a}_l^{daggerulletullet},:

In the last line we have used different numbers of ^ullet 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.

tatement 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

: egin{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}^ullet hat{B}^{ullet} hat{C} hat{D} hat{E} hat{F}ldots ,: \ & + sum_{doubles} ,: hat{A}^ullet hat{B}^{ulletullet} hat{C}^{ulletullet} hat{D}^ullet 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

: egin{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^ullet ,hat{f}_2 , hat{f}_1^{daggerullet} ,hat{f}_2^dagger , : + ,: hat{f}_1^ullet ,hat{f}_2 , hat{f}_1^dagger ,hat{f}_2^{daggerullet} , : +,: hat{f}_1 ,hat{f}_2^ullet , hat{f}_1^{daggerullet} ,hat{f}_2^dagger , : + : hat{f}_1 ,hat{f}_2^ullet , hat{f}_1^dagger ,hat{f}_2^{daggerullet} , : \ & -: hat{f}_1^{ulletullet} ,hat{f}_2^ullet , hat{f}_1^{daggerulletullet} ,hat{f}_2^{daggerullet} , :+: hat{f}_1^{ulletullet} ,hat{f}_2^ullet , hat{f}_1^{daggerullet} ,hat{f}_2^{daggerulletullet}: end{array}

Wick's theorem applied to fields

:mathcal C(x_1, x_2)=left langle 0 |mathcal Tphi_i(x_1)phi_i(x_2)|0 ight angle=overline{phi_i(x_1)phi_i(x_2)}=iDelta_F(x_1-x_2)=iint{frac{d^4k}{(2pi)^4}frac{e^{-ik(x_1-x_2){(k^2-m^2)+iepsilon.

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 TPi_{k=1}^mphi(x_k)=:Piphi_i(x_k):+sum_{alpha,eta}overline{phi(x_alpha)phi(x_eta)}:Pi_{k ot=alpha,eta}phi_i(x_k):+

:mathcal+sum_{(alpha,eta),(gamma,delta)}overline{phi(x_alpha)phi(x_eta)};overline{phi(x_gamma)phi(x_delta)}:Pi_{k ot=alpha,eta,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 Tphi_i(x_1)phi_i(x_2)|0 ight angle=sum_mathrm{pairs}overline{phi(x_1)phi(x_2)}cdotsoverline{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 ight angle

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.

Bibliography

* G.C. Wick, [http://link.aps.org/abstract/PR/v80/p268 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)

ee also

*S matrix
*Normal order


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