What is supermathematics?

All supersymmetric theories are based on the use of anticommuting classical variables first introduced by Grassmann in the last century. At first glance, these objects look very artificial and seem to have no relation to the real world. There is a certain threshold for physicists to start using the Grassmann anticommuting variables for calculations because one expects the game to have very unusual rules. Surprisingly, it is not true, and provided proper definitions are given, one can simply generalize conventional mathematical constructions so that it is possible to treat both commuting and anticommuting variables on an equal footing. Sometimes the corresponding branch of mathematics is called supermathematics.

Of course, the main purpose of this book is to consider different physical results obtained with the use of the Grassmann variables, and therefore one could try to demonstrate how these variables work while making some concrete calculations. However, it seems to be more reasonable to present the basic formulae of supermathematics in one place, first, because it may be the best way to get used to the anticommuting variables, and, second, because one can see that practically all the rules of operating with “superobjects” are quite standard.

Today the mathematical analysis and algebra of functions of both commuting and anticommuting variables are very well developed.