Lectures on Super Analysis -- Why necessary and What's that?

TL;DR

This paper introduces super analysis, developing calculus on superspaces with Grassmann variables, and applies it to quantum mechanics, random matrix theory, and differential equations, highlighting its necessity and novel methods.

Contribution

It provides a comprehensive foundation of super analysis, including differential calculus, integral formulas, and applications to quantum mechanics and differential equations, with new constructions and proofs.

Findings

Developed super-smooth function spaces and calculus on superspaces.

Applied super analysis to derive solutions in quantum mechanics and random matrix theory.

Proved Berezin's change of variables formula and constructed Hamilton flows in superspace.

Abstract

Roughly speaking, RA(=real analysis) means to study properties of (smooth) functions defined on real space, and CA(=complex analysis) stands for studying properties of (holomorphic) functions defined on spaces with complex structure. But to treat boson and fermion on equal footing, we need to prepare as a "ground ring", Fr\'echet-Grassmann algebra having countably many Grassmann generators and we define so-called superspace over such algebra. On such superspaces, we introduce spaces of super-smooth functions and develop elementary differential and integral calculus. With a slight preparation of functional analysis, we explain the Efetov's method in RMT(=random matrix theory). The free Weyl equation is treated to answer the problem posed by Feynman in their famous book. Simple examples of SUSYQM(=supersymmetric quantum mechanics) are calculated from this point of view. In the final chapter, we give a precise proof of Berezin's formula for changing variables under integral sign, brief introduction of real analysis on superspace, another construction of fundamental solution of Qi's weakly hyperbolic equation, Bernardi's question for a system version of Egorov's theorem, etc. A little discussion of Funtional Derivative Equations which are candidates of new branch of mathematics, is given. And finally, we construct a Hamilton flow corresponding to the Weyl equation with external electro-magnetic potentials, where we need the countably infinite Grassmann generators and weak topology! I mention many open problems at least for me (alias ATLOM=a tiny little old mathematician).