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

John E. Gough*

*Corresponding author for this work

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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 (lo-cal field) approach rather than the integral one pioneered by Feynman. This is helped by a DEFG (Dyson–Einstein–Feynman–Guichardet) shorthand which greatly simplifies expres-sions. This gives a convenient framework for the formal approach of Schwinger and Tomon-aga 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.

Original languageEnglish
Article number044
Number of pages15
JournalSymmetry, Integrability and Geometry: Methods and Applications
Publication statusPublished - 14 Jun 2022


  • quantum field theory
  • Guichardet space
  • Feynman versus Schwinger
  • combinatorics


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