In quantum field theory, Wick's theorem is a powerful tool that allows for the systematic calculation of expectation values of products of field operators in terms of contractions. However, Wick's theorem, as conventionally formulated, does not directly apply to the contraction of a field with its four-derivative.
Wick's theorem is based on the normal ordering of field operators, which orders creation and annihilation operators in a specific way. It is derived using the commutation or anticommutation relations of the field operators and relies on time-ordering. The standard version of Wick's theorem allows for the expansion of products of fields in terms of normal-ordered products plus contractions.
When it comes to the contraction of a field with its four-derivative, one typically encounters terms involving higher derivatives of field operators. These terms do not follow the usual pattern of Wick contractions and cannot be straightforwardly accounted for using Wick's theorem.
However, there are techniques that can be employed to handle such terms. One approach is to use the equations of motion for the field operators to rewrite higher derivatives in terms of lower derivatives and field operators. By manipulating the resulting expressions, one can often simplify calculations and express the contracted terms in terms of Wick contractions and known commutation or anticommutation relations.
Another approach is to use Feynman diagrams, which provide a graphical representation of terms in quantum field theory calculations. Feynman diagrams naturally account for the propagators (Green's functions) associated with the derivatives of field operators, allowing for the systematic calculation of higher derivative contractions.
In summary, while Wick's theorem is not directly applicable to the contraction of a field with its four-derivative, there are techniques available to handle such terms. These techniques involve manipulating the equations of motion, utilizing Feynman diagrams, and expressing the contractions in terms of known commutation or anticommutation relations.