Fermions and the Calculus for Grassmann Variables

Abstract

In 1844, Hermann Grassmann (1809-1877) emphasized the importance of the wedge product (Grassmann product) for geometry in higher dimensions. But his contemporaries did not understand him. Nowadays the wedge product is fundamental for modern mathematics (cohomology) and physics (fermions and supersymmetry). Folklore Recall that we distinguish between bosons (elementary particles with integer spin like photons or mesons) and fermions (elementary particles with half-integer spin like electrons and quarks). The rigorous finite-dimensional approach from the preceding Chap. 7 refers to bosons. However, it is possible to extend this approach to fermions by replacing complex numbers by Grassmann variables. In this chapter, we are going to discuss this. 9.1 The Grassmann Product Vectors. Let X be a complex linear space. For two elements ϕ and ψ of X, we define the Grassmann product ϕ ∧ ψ by setting (ϕ ∧ ψ)(f, g) := f (ϕ)g(ψ) − f (ψ)g(ϕ) for all f, g ∈ X d. Recall that the dual space X d consists of all linear functionals f : X → C. The map ϕ ∧ ψ : X d × X d → C is bilinear and antisymmetric. Explicitly, for all f, g, h ∈ X d and all complex numbers α, β, we have • (ϕ ∧ ψ)(f, g) = −(ϕ ∧ ψ)(g, f); • (ϕ ∧ ψ)(f, αg + βh) = α(ϕ ∧ ψ)(f, g) + β(ϕ ∧ ψ)(f, h). The two crucial properties of the Grassmann product are • ϕ ∧ ψ = −ψ ∧ ϕ (anticommutativity), and • (αϕ + βχ) ∧ ψ = αϕ ∧ ψ + βχ ∧ ψ (distributivity) for all ϕ, ψ, χ ∈ X and all complex numbers α, β. If we write briefly ϕψ instead of the wedge product ϕ ∧ ψ, then • ϕψ = −ψϕ, and • (αϕ + βχ)ψ = αϕψ + βχψ.