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Infinite Dimensional Analysis Quantum Probability and Related Topics

Identification of the one-mode quadratic Heisenberg group with the projective group PSU(1,1) and holomorphic representation

Habib Rebei, Hafedh M. S. Rguigui, Anis Riahi, Ziyad A. Alhussain

Published Nov 28, 2020• 6 citations
Publisher

Abstract

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.

DOI
10.1142/s021902572050023x

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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.
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).
Quantum symmetries that leave invariant physical transition probabilities are described by projective representations of Lie groups. The mathematical theory of projected representations for topologically connected Lie groups is reviewed and applied to the inhomogeneous symplectic group ISp(2n). The projective representations are given in terms of the ordinary unitary representations of the central extension of the group with respect to its first homotopy group direct product with its second cohomology group. The second cohomology group of ISp(2n) is the one dimensional abelian group and the central extension turns the 2n dimension abelian normal subgroup of translations of ISp(2n) into the Weyl-Heisenberg group. This is the quantum symmetry origin of the Heisenberg commutation relations that are the Hermitian representation of the Lie algebra of the Weyl-Heisenberg group.
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl [Formula: see text]-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.
The kinematics of quantum systems, as is well known, are generally governed by the Heisenberg commutation rules. It was Hermann Weyl who suggested the consideration of the global versions of these rules which define a projective unitary representation of the additive group of the classical phase space. This prompted Weyl to formulate his principle that quantum kinematics is always governed by a projective unitary representation of some abelian group. The ideas of Weyl, and their natural development by his successors, notably Stone, Von Neumann, Mackey, Segal, Shale, Weil, and many others have had and still continue to have a great impact on many areas of mathematics and physics. This paper is an exposition of some of the themes that flow from these ideas.