On the Quadratic Heisenberg Group

Abstract

In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).

Normally Ordered Disentanglement of Multi-Dimensional Schrödinger Algebra Exponentials

“…□ Remark 3.10. Simplified versions of (3.51), for less general coefficients, can be found in [1] and [4].…”

Lévy processes through time shift on oscillator Weyl algebra

“…This domain will be called the principal domain of generalized Weyl operators, but here we will not discuss this choice. (see [9] for an example of principal domain of quadratic Weyl operator). In the spirit to deal with some properties of generalized Weyl operators, such as independence linearity, we assume, in the following, that a such domain D is fixed.…”

$$C^*$$ C ∗ -Quadratic Quantization

“…The renormalized square of white noise (RSWN) algebra was first introduced in [10]. The construction of the Fock representation of the RSWN algebra motivated a large number of papers extending it in different directions and exhibiting connections with almost all fields of Mathematics, see for example [21] for the case of free white noise; [14] for the connection with infinite divisibility and for the identification of the vacuum distributions of the generalized fields with the three nonstandard Meixner classes; [1] and [18] for finite temperature representations; [6], [7], [15], [16] for the construction of the Fock functor; the survey [3] and the paper [4] for the connections with conformal field theory and with the Virasoro-Zamolodchikov hierarchy; [5] for the connections between renormalization and central extensions; [11] for the construction of the quadratic Weyl operators and their associated quadratic Heisenberg group; [12], [13] for the study of the RSWN quantum time shift and quantum Markov semi-groups. The RSWN is the first and at the moment best understood example illustrating the program of non-linear quantization.…”

“…In the paper [11] a quadratic analogue of the usual Heisenberg group, naturally arising in first order quantization, was constructed starting from the realization of the one-mode quadratic Weyl C * -algebra in usual 1-mode Boson Fock space. This group is called the one-mode quadratic Heisenberg group and denoted by QHeis(1, F ).…”