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

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.