Field Calculus: Quantum and Statistical Field Theory without the Feynman Diagrams

TL;DR

This paper introduces a novel 'field calculus' framework based on the Guichardet integral, enabling a differential approach to quantum and statistical field theories without Feynman diagrams, simplifying Green function relations for Boson systems.

Contribution

It develops a differential (local field) calculus using Guichardet integrals, providing an alternative to Feynman diagrams for Boson systems and recasting Dyson-Schwinger equations in this new language.

Findings

Framework simplifies Green function relations for Bosons.

Reformulates Dyson-Schwinger equations without Feynman diagrams.

Provides combinatorial tree-expansion methods.

Abstract

For a given base space $M$ (spacetime), we consider the Guichardet space over the Guichardet space over $M$. Here we develop a ''field calculus'' based on the Guichardet integral. This is the natural setting in which to describe Green function relations for Boson systems. Here we can follow the suggestion of Schwinger and develop a differential (local field) approach rather than the integral one pioneered by Feynman. This is helped by a DEFG (Dyson-Einstein-Feynman-Guichardet) shorthand which greatly simplifies expressions. This gives a convenient framework for the formal approach of Schwinger and Tomonaga as opposed to Feynman diagrams. The Dyson-Schwinger is recast in this language with the help of bosonic creation/annihilation operators. We also give the combinatorial approach to tree-expansions.