Complex Numbers and Grassmann Numbers
Bryan J. Dalton, John R. Jeffers, Stephen Mark Barnett · 2014
Abstract This chapter sets out the basic algebraic properties of both complex numbers (c-numbers) and Grassmann numbers (g-numbers). The formal rules of addition and multiplication together with the associative and distributive rules are stated, highlighting the anticommuting multiplication property of g-numbers and the consequences that all powers of g-numbers are zero and they do not have inverses. Complex conjugation rules are also stated, highlighting a second major difference between c- and g-numbers. The concepts of monomials and their order are introduced for Grassmann variables. Grassmann functions are defined, their division into even and odd functions noted, and commutation properties of even and odd Grassmann functions are obtained. Important results for exponential functions involving Grassmann variables and functions are derived, including the Baker–Hausdorff theorem. Linear transformations of Grassmann variables are defined.