Zeon Algebra, Fock Space, and Markov Chains

Fock spaces over zeons are introduced. Trace identities and a noncommutative "integration-by-parts" formula are developed. As an application, we find a new criterion, without involving powers of the transition matrix, for a Markov chain to be ergodic

Krawtchouk-Griffiths systems II: as Bernoulli systems

We call Krawtchouk-Griffiths systems, KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. Here we present a Fock space construction with raising and lowering operators. The operators of "multiplication by X" are found in terms of boson operators and corresponding recurrence relations presented. The Riccati partial differential equations for the differentiation operators, Berezin transform and associated partial differential equations are found. These features provide the specifications for a Bernoulli system as a quantization formulation of multivariate Krawtchouk polynomials.Comment: 20 page

Krawtchouk-Griffiths systems I: matrix approach

We call Krawtchouk-Griffiths systems, or KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. Our approach is to work directly with matrices comprising the values of the polynomials at points of a discrete grid based on the possible counting values of the underlying multinomial distribution. The starting point for the construction of a KG-system is a generating matrix satisfying the K-condition, orthogonality with respect to the basic probability distribution associated to an individual step of the multinomial process. The variables of the polynomials corresponding to matrices may be interpreted as quantum observables in the real case, or quantum variables in the complex case. The structure of the recurrence relations for the orthogonal polynomials is presented with multiplication operators as the matrices corresponding to the quantum variables. An interesting feature is that the associated random walks correspond to the Lie algebra of the representation of symmetric tensor powers of matrices.Comment: 25 pages Fixed typo and added several reference