We have developed a quantum field theory of spinors based on the algebra canonical anticommutation relations (CAR algebra) Grassmann densities in momentum space. proven existence two spinor vacua. Operators C and T transform normal vacuum into an alternative one, which leads to breaking symmetries. The CPT is real structure operator; it preserves vacuum. that, Dirac Sea, formula for charge conj...

The maximal dimension of a commutative subalgebra of the Grassmann algebra is determined. It is shown that for any commutative subalgebra there exists an equidimensional commutative subalgebra spanned by monomials. It follows that the maximal dimension of a commutative subalgebra can be expressed in terms of the maximal size of an intersecting system of subsets of odd size in a finite set. 2010...

We deal with the definition of superdistributions on superspace over a Grassmann-Banach algebra and the extension of the Hörmander's description of the singularity structure (wavefront set) of a distribution to include the supersymmetric case.

A supermanifold, Mm/n, can be caracterired by its smooth superfunctions which constitute an algebra A (Leites, Kostant). We associate canonically ~a Ia Gelfand, certain fibred manifolds on which the automorphisms (the Jordan automorphisms) of A actas diffeomorphisms. For example, the kernelsof all homomorphisms from the algebra of superfunctions onto the Grassmann algebra of dimension n form na...

Hermann Grassmann is cast as a pivotal figure in the historical development of a universal geometric calculus for mathematics and physics which continues to this day. He formulated most of the basic ideas and, to a remarkable extent, anticipates later developments. His influence is far more potent and pervasive than generally recognized. After nearly a century on the brink of obscurity, Hermann...

After a brief introduction of lhe Grassmann-Cayley OI' double algebra we proceed to demonstrate its use for modelling systems of cameras. In the case of three cameras, we give a new interpretation of the trifocal tensors and study in detail some of the constraints that they satisfy. In particular we prove that simple subsets of those constraints characterize the trifocal tensors, in other words...

We have developed a quantum field theory of spinors based on the algebra canonical anticommutation relations (CAR algebra). The proposed approach is use Grassmann densities in momentum space and their derivatives with respect to construction from these both basis Clifford vectors spacetime spinor vacuum. shown existence two vacua: normal alternative. proven that CPT real structure operator Krei...

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E.Cartan. Especially, the E.Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and CartanEhresmann connection theory on fibered spaces. General structu...

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincaré-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-alg...

The classic Cayley identity states that det(∂) (detX) = s(s+ 1) · · · (s+ n− 1) (detX) where X = (xij) is an n× n matrix of indeterminates and ∂ = (∂/∂xij) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterio...