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).

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We extend the Lie-algebra time shift technique, introduced in [2], from the usual Weyl algebra (associated to the additive group of a Hilbert space H) to the generalized Weyl algebra (oscillator algebra), associated to the semi-direct product of the additive group H with the unitary group on H (the Euclidean group of H, in the terminology of [14]). While in the usual Weyl algebra the possible quantum extensions of the time shift are essentially reduced to isomorphic copies of the Wiener process, in the case of the oscillator algebra a larger class of Lévy process arises.Our main result is the proof of the fact that the generators of the quantum Markov semigroups, associated to these time shifts, share with the quantum extensions of the Laplacian the important property that the generalized Weyl operators are eigenoperators for them and the corresponding eigenvalues are explicitly computed in terms of the Lévy-Khintchin factor of the underlying classical Lévy process.

In this paper, we provide a new reformulation of the quadratic analogue of the Weyl relations. Especially, we offer some adjustments [10] on these relations and the corresponding group law, i.e., the quadratic Heisenberg group law. We provide a much more transparent description of the underlying manifold and we give a connection with the projective group PSU(1,1). Finally, we deduce such a holomorphic realization of the quadratic Heisenberg group.