Grassmannvalued

Grassmann-valued describes quantities whose values lie in a Grassmann algebra, an associative algebra over a field (typically the real or complex numbers) generated by elements that anticommute: θi θj = - θj θi, with each generator satisfying θi^2 = 0. Because of nilpotency, any Grassmann-valued expression is a finite sum of basis monomials θi1…θik, with k ranging from 0 up to the number of generators. The algebra is Z2-graded into even and odd parts: even elements commute with all generators, while odd elements anticommute with other odd elements.

In mathematics, Grassmann-valued maps or sections assign a Grassmann algebra element to each point of a space or manifold. They appear in contexts such as Grassmann-valued differential forms and fields, and in the exterior algebra framework where wedge products relate to anticommuting variables.

In physics, Grassmann-valued quantities model fermionic degrees of freedom. They underpin formalisms used in quantum field theory, notably path integrals with Berezin integration and the use of superfields in supersymmetry, where component fields can be Grassmann-valued. Grassmann variables provide a way to encode anti-commutation relations of fermions at a classical level before quantization.

Example: a Grassmann-valued quantity a + bθ, where a and b are real or complex numbers and θ^2 = 0, is a typical simple case. More generally, elements of a Grassmann algebra are finite sums of products of distinct generators.

Notes: The term highlights anticommutation and nilpotency; practical calculations require careful tracking of signs when permuting Grassmann variables.