Method of extended $S$-matrix in quantum field theory

A general theory of extending the $S$-matrix off the mass shell is developed by introducing a dependence on functions that describe an arbitrary external classical object. A space-time description is introduced in terms of a classical functional argument of this kind; the description is needed to formulate causality, which is adopted in Bogolyubov's form. Systematic exploitation of this principle makes it possible to construct a detailed dynamical theory. Three well-known special cases of the classical object are considered: the interaction switching-on function, a classical addition to the quantized out-field, and a classical current source. The relationship between the two last methods of extension is established by the introduction of an $S$-matrix that is extended twice – with respect to the field and the current. This provides a basis for elucidating the relationship between the consequences of the Bogolyubov and the Lehmann–Symanzik–Zimmermann axiomatics.