Introduction

In this chapter we apply temporal discretization to mechanical systems described by anticommuting variables rather than the commuting variables normally used in classical mechanics (CM). The anticommuting numbers representing such variables are called Grassmann(ian) numbers by mathematicians, after Hermann G. Grassmann (1809–1877), who did some pioneering work in linear algebra. On that account we shall refer to anticommuting numbers as g-numbers, in contrast to c-numbers (c for commuting1) when we refer to ordinary real or complex variables.

The significant difference between g-numbers and c-numbers is that, where as any two c numbers x and y satisfy the commutative multiplication rule xy = +yx, any two g-numbers θ and ϕ satisfy the anticommutation rule θϕ = –ϕθ. G-numbers and c-numbers can be multiplied together in any order, i.e., g-numbers and c-numbers commute. The product zθ of a c-number z and a g-number θ is a g-number.

The values of classical variables in ordinary CM are c-numbers, so we may refer to such variables as c-variables. It is possible to construct forms of CM where the classical variables are g-numbers, in which case we shall refer to such variables as g-variables.

G-variables should not be thought of as quantized versions of c-variables. They are just as classical as c-variables but much less familiar in terms of their applications to the real world. G-variables should not be confused with Dirac's q-numbers either.