Complete Normal Ordering 1: Foundations

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.

40 pages; a significantly condensed version compared to v1; matches published article. Refer to v1 for a pedagogical exposition

References in corpus (6)

• Universality of Gauge Thresholds in Non-Supersymmetric Heterotic Vacua
• Constraining the effective action by a method of external sources
• The Super Period Matrix With Ramond Punctures
• Symmetry Improved 2PI Effective Action and the Infrared Divergences of the Standard Model
• Highly Excited Strings I: Generating Function
• Complete Normal Ordering 1: Foundations

Cited by in corpus (7)

• Visualising quantum effective action calculations in zero dimensions
• On Evolution of Coherent States as Quantum Counterpart of Classical Dynamics
• Highly Excited Strings I: Generating Function
• Exact RG Invariance and Symmetry Improved 2PI Effective Potential
• Complete Normal Ordering 1: Foundations
• Handle Operators in String Theory
• Quantization due to the time evolution, with applications to Quantum Yang-Mills theory, Quantum Gravity and Classical Statistical Field Theory