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

Abstract

Let ν be a finite measure on R whose Laplace transform is analytic in a neighborhood of zero. An anyon Lévy white noise on (R d , dx) is a certain family of noncommuting operators ω, ϕ in the anyon Fock space over L 2 (R d × R, dx ⊗ ν). Here ϕ = ϕ(x) runs over a space of test functions on R d , while ω = ω(x) is interpreted as an operator-valued distribution on R d . Let L 2 (τ ) be the noncommutative L 2 -space generated by the algebra of polynomials in variables ω, ϕ , where τ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in L 2 (τ ) of the form P n (ω), f (n) , where f (n) is a test function on (R d ) n . Using these orthogonal polynomials, we derive a unitary isomorphism U between L 2 (τ ) and an extended anyon Fock space over dx)) if and only if the measure ν is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism U , we realize the operators ω, ϕ as a Jacobi (i.e., tridiagonal) field in F(L 2 (R d , dx)). We derive a Meixner-type class of anyon Lévy white noise for which the respective Jacobi field in F(L 2 (R d , dx)) has a relatively simple structure. Each anyon Lévy white noise of the Meixner type is characterized by two parameters: λ ∈ R and η ≥ 0. Furthermore, we get the representation ω(x) = ∂ † x + λ∂ † x ∂ x + η∂ † x ∂ x ∂ x + ∂ x . Here ∂ x and ∂ † x are annihilation and creation operators at point x.

Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations

“…, x n ) := 1≤i<j≤n π(i)>π (j) Q(x i , x j ). (9) Note that, in the case Q ≡ 1, we get Q π ≡ 1, while in the case Q ≡ −1, we get Q π ≡ (−1) |π| = sgn π. Here |π| is the number of inversions of π, i.e., the number of i < j such that π(i) > π(j).…”

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

“…Note that, for any generalized statistics, the operator T given by (6) is unitary. In fact, for any operator T that is additionally unitary, the corresponding operator P n on H ⊗n is a multiple of an orthogonal projection.…”

“…Bożejko, Lytvynov and Wysoczański [8] discussed Fock representations of the deformed commutation relations in the case where the operator T is given by formula (6) in which the function Q satisfies Q(x, y) = Q(y, x) and |Q(x, y)| ≤ 1. In this work, the n-particle subspaces F n (H) were described explicitly, and it was proved that the corresponding creation and annihilation operators satisfy the commutation relation (7).…”

Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

“…For other studies of the free Meixner-type Lévy processes and the free Meixner polynomials on R we refer to [2-8, 15, 18, 19, 35]. Furthermore, in [14], the notion of a non-commutative Lévy noise was introduced for the anyon statistics, and in [13], the corresponding Meixner class of non-commutative orthogonal polynomials was studied. Quite unexpectedly, this class was again fully described by a formula similar to (4), albeit its meaning was quite different.…”