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

TL;DR

This paper explicitly derives the map relating on-shell and off-shell formalisms in perturbative quantum field theory for various fields and dimensions, clarifying their connection beyond recursive solutions.

Contribution

It provides a closed-form expression for the sigma map connecting on-shell and off-shell fields for scalar, Dirac, and gauge fields in arbitrary dimensions.

Findings

Explicit sigma map formulas for scalar, Dirac, and gauge fields

Generalization to arbitrary spacetime dimensions

Clarification of on-shell and off-shell formalism relation

Abstract

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 formalism correctly, a map sigma from on-shell fields to off-shell fields was introduced axiomatically by Duetsch and Fredenhagen. In that paper it was 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 was given there only recursively. 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.