Perturbation Theory of Transformed Quantum Fields

Abstract

AbstractWe consider a scalar quantum field ϕ with arbitrary polynomial self-interaction in perturbation theory. If the field variable ϕ is repaced by a global diffeomorphism ϕ(x) = ρ(x) + a1ρ2(x) + …, this field ρ obtains infinitely many additional interaction vertices. We propose a systematic way to compute connected amplitudes for theories involving vertices which are able to cancel adjacent edges. Assuming tadpole graphs vanish, we show that the S-matrix of ρ coincides with the one of ϕ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. The diffeomorphism can be tuned to cancel all contributions of an underlying ϕt-type self interaction at one fixed external offshell momentum, rendering ρ a free theory at this momentum. Finally, we mention one way to extend the diffeomorphism to a non-diffeomorphism transformation involving derivatives without spoiling the combinatoric structure of the global diffeomorphism.