Operator Calculus on Clifford Algebras: Combinatorics to Quantum Probability

G Stacey Staples, Southern Illinois University Edwardsville, USA

For nondegenerate quadratic form $Q$ on finite vector space $V$, multiplication in the Clifford algebra $\mathcal{C}\ell_Q(V)$ has a natural interpretation as the sum of lowering (annihilation) and raising (creation) operators. The study of these operators lies at the interface of algebraic combinatorics, operator theory and quantum probability. Their inherent combinatorial properties lead to applications in graph theory, coding theory, probability theory, and more. In many cases, they lend themselves to convenient symbolic computations using Mathematica. In this talk, I will discuss some of these properties, reveal some connections with Kravchuk polynomials, and discuss combinatorial properties and applications of induced and reduced operators on Clifford, Grassmann, and zeon algebras.