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.
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.
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).
We construct an inductive system of C*-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previous paper (Accardi and Dhahri in Open Syst Inf Dyn 22(3):1550001, 2015). We prove that the inductive limit C*-algebra is factorizable and has a natural localization given by a family of C*-sub-algebras each of which is localized on a bounded Borel subset of R. Finally, we prove that the corresponding family of Fock states, defined on the inductive family of C*-algebras, is projective if and only if n = 1. This is a weak form of the no-go theorems which emerge in the study of representations of current algebras over Lie algebras.
summary:In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the d-dimensional quantum harmonic oscillator are considered. In this case the usual set of the creation and annihilation operators is used as basis of the Weyl algebra under consideration. From this model are deduced the underlying symplectic structure and some of its consequences. In particular, the transitions from the initial $\text{Sp}(d,\bbfC)$-invariant model to the $\text{GL}(d,\bbfC)$- and, at last, U(d)-invariant models are performed.
We examine the possibility of a direct Fock representation of the recently obtained non-trivial central extensions CEHeis of the Heisenberg algebra, generated by elements a, a†, h and E satisfying the commutation relations [a, a†]CEHeis = h, [h, a†]CEHeis = z E and [a, h]CEHeis = ̄z E, where a and a† are dual, h is self-adjoint, E is the non-zero selfadjoint central element and z 2 C \ {0}. We define the exponential vectors associated with the CEHeis Fock space, we compute their Leibniz function (inner product), we describe the action of a, a† and h on the exponential vectors and we compute the moment generating and characteristic functions of the classical random variable corresponding to the self-adjoint operator X = a + a† + h.
The polynomial Heisenberg algebras (PHA) are deformations of the Heisenberg- Weyl algebra characterizing the underlying symmetry of the supersymmetric partners of the Harmonic oscillator. When looking for the simplest system ruled by PHA, however, we end up with the harmonic oscillator. In this paper we are going to realize the first-order PHA through the harmonic oscillator. The associated coherent states will be also constructed, which turn out to be the well known even and odd coherent states.
Couples are wholes and not wholes, what agrees disagrees, the concordant is discordant. From all things one and from one all things. ( Heraclitus , On the Universe 59.) This chapter introduces the notion of joint (Wigner) density of random variables, for future use in quantum Malliavin calculus. For this we will rely on functional calculus on general Lie algebras, starting with the Heisenberg– Weyl algebra. We also consider some applications to quantum optics and timefrequency analysis. Joint moments of noncommuting random variables The notion of moment ⟨ ϕ , P n ψ ⟩ of order n of P in the state ⟨ ϕ , · ψ ⟩ is well understood as the noncommutative analogue of E [ X n ]. Indeed, in the single variable case, various functional calculi are available for elements of an algebra. In particular, • polynomials can be applied to any element in an algebra; • holomorphic functions can be applied to elements of a Banach algebra provided the function is holomorphic on a neighborhood of the spectrum of the element; • continuous functions can be applied to normal elements of C*-algebras provided they are continuous on the spectrum of the element; • measurable ( i.e. , Borel) functions can be applied to normal elements in a von Neumann algebra ( e.g. , B (h ) ), provided they are measurable on the spectrum of the operator. In addition, the Borel functional calculus can be extended to unbounded normal operators (where normal means that XX ∗ and X ∗ X have the same domain and coincide on this domain). All the above functional calculi work due to a restriction to a commutative subalgebra. In that sense, Weyl calculus is fundamentally different because it applies to functions of noncommuting operators. We now need to focus on the meaning of joint moments. For example, which is the noncommutative analogue of E [ XY ]? A possibility is to choose ⟨ ϕ , PQψ ⟩ as the the noncommutative analogue of E [ XY ], however this would lead to ⟨ ϕ , QPψ ⟩ as the the noncommutative analogue of E [ YX ] and we have E [ XY ] = E [ YX ] while due to noncommutativity. A way to solve this contradiction can be to define the first joint moment as Next, how can we define the analogue of E [ XY 2 ]? Proceeding similarly, a natural extension could be Then what is the correct way to construct E[ X 2 Y 3 ]? Clearly there is a combinatorial pattern which should be made explicit.