Grassmann Calculus, Pseudoclassical Mechanics and Geometric Algebra

We present a reformulation of Grassmann calculus in terms of geometric algebra – a unified language for physics based on Clifford algebra. In this reformulation, Grassmann generators are replaced by vectors, so that every product of generators has a natural geometric interpretation. The calculus introduced by Berezin is shown to be unnecessary, amounting to no more than an algebraic contraction. Our approach is not only conceptually clearer, but it is computationally more efficient, which we demonstrate by treatments of the “Grauss” integral and the Grassmann Fourier Transform. Our reformulation is applied to pseudoclassical mechanics, where it is shown to lead to a new concept, the multivector Lagrangian. To illustrate this idea, the 3-dimensional Fermi oscillator is reformulated and solved, and its symmetry properties discussed. As a result, a new and highly compact formula for generating super-Lie algebras is revealed. We finish with a discussion of quantization, outlining a new approach to fermionic path integrals.

A. N. Lasenby, C. J. L. Doran and S. F. Gull, Grassmann Calculus, Pseudoclassical Mechanics and Geometric Algebra, J. Math. Phys. 34(8), 3683-3712 (1993)