Quantum Field Perturbation Theory Revisited

TL;DR

This paper revisits quantum field perturbation theory for scalar theories with exponential interactions, introducing a new formulation that simplifies calculations by absorbing normal ordering terms and replacing functional derivatives with ordinary derivatives.

Contribution

It develops a novel perturbation theory formulation for exponential and polynomial potentials in scalar quantum field theories, simplifying the treatment of normal ordering and renormalization.

Findings

New perturbation theory formulation for exponential interactions

Absorption of normal ordering terms at the outset

Simplification by replacing functional derivatives with ordinary derivatives

Abstract

Schwinger's formalism in quantum field theory can be easily implemented in the case of scalar theories in $D$ dimension with exponential interactions, such as $\mu^D\exp(\alpha\phi)$. In particular, we use the relation $$ \exp\big(\alpha{\delta\over \delta J(x)}\big)\exp(-Z_0[J])=\exp(-Z_0[J+\alpha_x]) $$ with $J$ the external source, and $\alpha_x(y)=\alpha\delta(y-x)$. Such a shift is strictly related to the normal ordering of $\exp(\alpha\phi)$ and to a scaling relation which follows by renormalizing $\mu$. Next, we derive a new formulation of perturbation theory for the potentials $V(\phi)={\lambda\over n!}:\phi^n:$, using the generating functional associated to $:\exp(\alpha\phi):$. The $\Delta(0)$-terms related to the normal ordering are absorbed at once. The functional derivatives with respect to $J$ to compute the generating functional are replaced by ordinary derivatives with respect to auxiliary parameters. We focus on scalar theories, but the method is general and similar investigations extend to other theories.