Chapter 5 - Grassmann Variables

Abstract

The need for Grassmann variables is related to the mathematical problem of writing a second-order differential operator as the product of two first-order differential operators. These variables are necessary when such an operator is represented in a path integral or a functional. In this chapter, the basic properties of real and complex Grassmann variables are introduced and differentiation and integration are defined. The Gaussian integrals of Grassmann variables are evaluated for both the real and complex cases in the chapter. The chapter explains the development of classical mechanics for the case of Grassmann variables and presents the equivalent forms for the Euler–Lagrange and Hamilton's equations. The fermionic coherent state and the rudiments of supersymmetric quantum mechanics are also discussed in the chapter.