Complete Normal Ordering 1: Foundations

TL;DR

The paper introduces a 'complete normal ordering' method for scalar field theories that simplifies perturbative calculations by automatically canceling certain diagrams and reducing the number of diagrams needed, especially around trivial vacua.

Contribution

It presents a new prescription for quantising scalar field theories that extends normal ordering, leading to automatic diagram cancellations and computational simplifications.

Findings

Automatic cancellation of tadpole and cephalopod diagrams at finite order.

Significant reduction in the number of Feynman diagrams to compute.

Extension of the method to theories with derivative interactions.

Abstract

We introduce a new prescription for quantising scalar field theories perturbatively around a true minimum of the full quantum effective action, which is to `complete normal order' the bare action of interest. When the true vacuum of the theory is located at zero field value, the key property of this prescription is the automatic cancellation, to any finite order in perturbation theory, of all tadpole and, more generally, all `cephalopod' Feynman diagrams. The latter are connected diagrams that can be disconnected into two pieces by cutting one internal vertex, with either one or both pieces free from external lines. In addition, this procedure of `complete normal ordering' (which is an extension of the standard field theory definition of normal ordering) reduces by a substantial factor the number of Feynman diagrams to be calculated at any given loop order. We illustrate explicitly the complete normal ordering procedure and the cancellation of cephalopod diagrams in scalar field theories with non-derivative interactions, and by using a point splitting `trick' we extend this result to theories with derivative interactions, such as those appearing as non-linear sigma-models in the world-sheet formulation of string theory. We focus here on theories with trivial vacua, generalising the discussion to non-trivial vacua in a follow-up paper.