Perturbation theory of transformed quantum fields

Abstract

We consider a scalar quantum field $\varphi$ with arbitrary polynomial self-interaction in perturbation theory. If the field variable $\varphi$ is repaced by a local diffeomorphism $\varphi (x)=\rho (x)+a_1{\rho }^2(x)+\dots$, this field $\rho$ obtains infinitely many additional interaction vertices. We show that the $S$-matrix of $\rho$ coincides with the one of $\varphi$ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. If tadpole diagrams vanish, the diffeomorphism can be tuned to cancel all contributions of an underlying ${\varphi }^s$-type self interaction at one fixed external offshell momentum, rendering $\rho$ a free theory at this momentum. Finally, we propose one way to extend the diffeomorphism to a non-local transformation involving derivatives without spoiling the combinatoric structure of the local diffeomorphism.

28 pages, 9 figures