Relating on-shell and off-shell formalisms in perturbative quantum field theory

In the on-shell formalism (mostly used in perturbative quantum field theory), the entries of the time-ordered product T are on-shell fields (i.e., the basic fields satisfy the free field equations). With that, (multi)linearity of T is incompatible with the action Ward identity. This can be circumvented by using the off-shell formalism in which the entries of T are off-shell fields. To relate on- and off-shell formalisms correctly, a map sigma from on-shell fields to off-shell fields was axiomatically introduced by Dutsch and Fredenhagen [Common. Math. Phys. 243, 275 (2003)]. In that paper it is shown that, in the case of one real scalar field in N=4 dimensional Minkowski space, these axioms have a unique solution. However, this solution is only recursively given there. We solve this recurrence relation and give a fully explicit expression for sigma in the cases of the scalar, Dirac, and gauge fields for arbitrary values of the dimension N. (C) 2008 American Institute of Physics.