### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 044, 15 pages      arXiv:2203.09296      https://doi.org/10.3842/SIGMA.2022.044
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

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

John E. Gough
Department of Physics, Aberystwyth University, SY23 3BZ, Wales, UK

Received March 18, 2022, in final form June 12, 2022; Published online June 14, 2022

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.

Key words: quantum field theory; Guichardet space; Feynman versus Schwinger; combinatorics.

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