Wick theorem

The Wick theorem , after the physicist Gian-Carlo Wick , is a statement in quantum field theory . It allows the vacuum expectation value of a product of time-ordered field operators to be written as the sum of the vacuum expectation value of a product of two field operators. The importance of the Wick theorem lies in the fact that such products occur in the calculation of scatter amplitudes and that they can be translated using the Wick theorem in the form of Feynman diagrams .

Wick contraction

The wick contraction of two bosonic field operators is as ${\displaystyle \phi }$

${\displaystyle \phi ^{\bullet }(x)\phi ^{\bullet }(y)={\begin{cases}\left[\phi ^{+}(x),\phi ^{-}(y)\right]&{\text{wenn }}x^{0}>y^{0}\\\left[\phi ^{+}(y),\phi ^{-}(x)\right]&{\text{wenn }}x^{0}<y^{0}\end{cases}}}$

Are defined. The square brackets are the commutator and denote the proportions of positive or negative frequency of the field, i.e. ${\displaystyle \phi ^{\pm }}$

${\displaystyle \phi ^{+}(x)=\int {\frac {\mathrm {d} ^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega }}}a(p)e^{-\mathrm {i} px}\ ,\quad }$ such as ${\displaystyle \quad \phi ^{-}(x)=\int {\frac {\mathrm {d} ^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega }}}a^{\dagger }(p)e^{\mathrm {i} px}\ ,}$

where is the annihilation operator and the creation operator . ${\displaystyle a}$ ${\displaystyle a^{\dagger }}$

In the case of fermionic field operators and , the Wick contraction contains an additional minus sign and the anti-commutator instead of the commutator: ${\displaystyle \psi }$ ${\displaystyle {\bar {\psi }}}$

${\displaystyle \psi ^{\bullet }(x){\bar {\psi }}^{\bullet }(y)={\begin{cases}\ \ \{\psi ^{+}(x),{\bar {\psi }}^{-}(y)\}&{\text{wenn }}x^{0}>y^{0}\ ,\\-\{{\bar {\psi }}^{+}(y),\psi ^{-}(x)\}&{\text{wenn }}x^{0}<y^{0}\ .\end{cases}}}$

With this definition for the contraction of fermionic fields, all of the following statements apply to both fermions and bosons.

The vacuum expectation value of a contraction of two field operators is equal to the Feynman propagator of a particle between these two spacetime points. So it applies ${\displaystyle D_{F}(x-y)}$

${\displaystyle \left\langle \phi ^{\bullet }(x)\phi ^{\bullet }(y)\right\rangle =D_{F}(x-y)\ .}$

Key message

The key message of the Wick theorem is:

${\displaystyle T\left(\prod _{i}\phi _{i}\right)\ =\ {\frac {1}{0!}}:\!\prod _{i}\phi _{i}\!:+{\frac {1}{1!}}\sum _{i<j}(\phi _{i}^{\bullet }\phi _{j}^{\bullet })\ :\!\prod _{k\not \in \{i,j\}}\phi _{k}\!:+{\frac {1}{2!}}\sum _{i,j,k,l{\text{ paarweise verschieden}} \atop {\text{und }}i<j,\ k<l}\left((\phi _{i}^{\bullet }\phi _{j}^{\bullet })(\phi _{k}^{\bullet }\phi _{l}^{\bullet })\ :\!\prod _{m\not \in \{i,j,k,l\}}\phi _{m}\!:\right)+{\text{ entsprechende Terme mit 3 oder mehr Kontraktionen}}}$

The time order operator is here and the notation denotes the normal order, which means that in this expression all creation operators are to the left of the annihilation operators. The shorthand was also used. The faculties in the expressions are statistical factors, since the sums are summed up over different identical configurations. In particular, the order of the field operators in all terms is not important, since this is either determined by the time order or the definition of the contraction, or since the creation and annihilation operators are interchanged. ${\displaystyle T}$ ${\displaystyle :\!O\!:}$ ${\displaystyle \phi _{i}=\phi (x_{i})}$

The simplifications by Wick's theorem are based on the fact that the vacuum expectation value of every normally ordered product of field operators vanishes, since the effect of the annihilation operator on the vacuum vanishes as well:

${\displaystyle \langle :\!\prod \phi _{i}\!:\rangle =0}$

Therefore, the Wick theorem leads to the fact that only fully contracted terms in the vacuum expectation value do not differ from zero. It therefore results directly for an odd number of field operators for which there can be no fully contracted expressions

${\displaystyle \left\langle T\left(\prod _{i=1}^{2n+1}\phi _{i}\right)\right\rangle =0}$.

The vacuum expectation value over a product of an even number of field operators is transformed by means of the Wick theorem into a sum over a product of Feynman propagators, in which every combination of spacetime points is connected exactly once with a propagator.

example

For four field operators we get

${\displaystyle {\begin{aligned}T\left(\phi _{1}\phi _{2}\phi _{3}\phi _{4}\right)&=:\!\phi _{1}\phi _{2}\phi _{3}\phi _{4}\!:+(\phi _{1}^{\bullet }\phi _{2}^{\bullet }):\!\phi _{3}\phi _{4}\!:+(\phi _{1}^{\bullet }\phi _{3}^{\bullet }):\!\phi _{2}\phi _{4}\!:+(\phi _{1}^{\bullet }\phi _{4}^{\bullet }):\!\phi _{2}\phi _{3}\!:+(\phi _{2}^{\bullet }\phi _{3}^{\bullet }):\!\phi _{1}\phi _{4}\!:+(\phi _{2}^{\bullet }\phi _{4}^{\bullet }):\!\phi _{1}\phi _{3}\!:+(\phi _{3}^{\bullet }\phi _{4}^{\bullet }):\!\phi _{1}\phi _{2}\!:\\&+(\phi _{1}^{\bullet }\phi _{2}^{\bullet })(\phi _{3}^{\bullet }\phi _{4}^{\bullet })+(\phi _{1}^{\bullet }\phi _{3}^{\bullet })(\phi _{2}^{\bullet }\phi _{4}^{\bullet })+(\phi _{1}^{\bullet }\phi _{4}^{\bullet })(\phi _{2}^{\bullet }\phi _{3}^{\bullet })\end{aligned}}}$

and when the vacuum expectation value is formed, all terms that are not fully contracted are omitted, i.e. in this example the first line: ${\displaystyle \left\langle T\left(\phi _{1}\phi _{2}\phi _{3}\phi _{4}\right)\right\rangle =D_{F}(x_{1}-x_{2})D_{F}(x_{3}-x_{4})+D_{F}(x_{1}-x_{3})D_{F}(x_{2}-x_{4})+D_{F}(x_{1}-x_{4})D_{F}(x_{2}-x_{3})}$

literature

• Michael E. Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory . Perseus Books, Reading 1995, ISBN 0-201-50397-2 (English).