Complete normal ordering 1: Foundations

We introduce a new prescription for quantising scalar field theories (in generic spacetime dimension and background) 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 σ-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.