An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Abstract

Let be a finite measure on whose Laplace transform is analytic in a neighbourhood of zero. An anyon Lévy white noise on is a certain family of noncommuting operators on the anyon Fock space over , where runs over a space of test functions on , while is interpreted as an operator-valued distribution on . Let be the noncommutative -space generated by the algebra of polynomials in the variables , where is the vacuum expectation state. Noncommutative orthogonal polynomials in of the form are constructed, where is a test function on , and are then used to derive a unitary isomorphism between and an extended anyon Fock space over . The usual anyon Fock space over is a subspace of . Furthermore, the equality holds if and only if the measure is concentrated at a single point, that is, in the Gaussian or Poisson case. With use of the unitary isomorphism , the operators are realized as a Jacobi (that is, tridiagonal) field in . A Meixner-type class of anyon Lévy white noise is derived for which the corresponding Jacobi field in has a relatively simple structure. Each anyon Lévy white noise of Meixner type is characterized by two parameters, and . In conclusion, the representation is obtained, where and are the annihilation and creation operators at the point . Bibliography: 57 titles.