We discuss the q-deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in quantum mechanics. We show that the q-deformation parameter labels the Weyl systems in quantum mechanics and the unitarily inequivalent representations of the canonical commutation relations in quantum field theory.

arXiv:math-ph/0009035Modern Physics Letters BFeb 20, 199417 citations

Kähler Representation Theory

Bryan Wilson Roberts, Nicholas J. Teh

We show that Jordan-Lie-Banach algebras, which provide an abstract characterization of quantum theory equivalent to C^* algebras, can always be canonically represented in terms of smooth functions on a Kähler manifold.

arXiv:1602.05979Feb 18, 20161 citation

arXiv:hep-th/9403042 · Modern Physics Letters A

q-OSCILLATORS, NON-KÄHLER MANIFOLDS AND CONSTRAINED DYNAMICS

SERGEI V. SHABANOV

Published Apr 20, 1995• 4 citations

arXiv HTML PDF

Abstract

DOI: 10.1142/s0217732395001034

Related papers

q-oscillators, (non-)Kähler manifolds and constrained dynamics

Sergei V. Shabanov

It is shown that q-deformed quantum mechanics (systems with q-deformed Heisenberg commutation relations) can be interpreted as an ordinary quantum mechanics on Kähler manifolds, or as a quantum theory with second (or first)-class constraints. 1. The q-deformed Heisenberg-Weyl algebras [1], [2] exhibiting the quantum group symmetries [3],[4] have attracted much attention of physicists and mathematicians. In particular, some works have been devoted to establishing potentials for ordinary quantum systems which exhibit a q-deformed energy spectrum (see, for example, [5] and references therein) in order to obtain classical dynamical systems associated with the q-deformed Heisenberg-Weyl algebras. In the present paper, we analyze the correspondence limit, when ¯h → 0 (rather than q → 1), of multimode q-deformed Heisenberg-Weyl algebras [1], [2] regardless of the specific form of a Hamiltonian. We show that classical systems associated with the q-deformed Heisenberg-Weyl algebras in this way possess a non-trivial symplectic structure which, in turn, is related to the Kähler symplectic structure [6]. From the other hand, a natural (physical) reason for a dynamical system to have a nontrivial symplectic structure is the existence of ”frozen ” , non-dynamical degrees of freedom or, in other words, constraints [7]. Dirac pointed out that a nontrivial symplectic structure might naturally occur in dynamical systems through second-class constraints [7]. We shall demonstrate below how constrained dynamics can be associated with the q-deformed Heisenberg-Weyl algebra. As was shown in [8], a non-commutative phase space does not necessarily emerges in the formal classical limit of q-deformed quantum mechanics, provided the deformation parameter q is a function of the Planck constant. In the latter case, q-deformed Heisenberg-Weyl algebras yield quadratic symplectic structures on a commutative phase space. For example, the one-mode q-deformed Heisenberg-Weyl algebra defined by the commutation relation ˆ b ˆ b − qIt is shown that q-deformed quantum mechanics (systems with q-deformed Heisenberg commutation relations) can be interpreted as an ordinary quantum mechanics on Kähler manifolds, or as a quantum theory with second (or first)-class constraints. 1. The q-deformed Heisenberg-Weyl algebras [1], [2] exhibiting the quantum group symmetries [3],[4] have attracted much attention of physicists and mathematicians. In particular, some works have been devoted to establishing potentials for ordinary quantum systems which exhibit a q-deformed energy spectrum (see, for example, [5] and references therein) in order to obtain classical dynamical systems associated with the q-deformed Heisenberg-Weyl algebras. In the present paper, we analyze the correspondence limit, when ¯h → 0 (rather than q → 1), of multimode q-deformed Heisenberg-Weyl algebras [1], [2] regardless of the specific form of a Hamiltonian. We show that classical systems associated with the q-deformed Heisenberg-Weyl algebras in this way possess a non-trivial symplectic structure which, in turn, is related to the Kähler symplectic structure [6]. From the other hand, a natural (physical) reason for a dynamical system to have a nontrivial symplectic structure is the existence of ”frozen ” , non-dynamical degrees of freedom or, in other words, constraints [7]. Dirac pointed out that a nontrivial symplectic structure might naturally occur in dynamical systems through second-class constraints [7]. We shall demonstrate below how constrained dynamics can be associated with the q-deformed Heisenberg-Weyl algebra. As was shown in [8], a non-commutative phase space does not necessarily emerges in the formal classical limit of q-deformed quantum mechanics, provided the deformation parameter q is a function of the Planck constant. In the latter case, q-deformed Heisenberg-Weyl algebras yield quadratic symplectic structures on a commutative phase space. For example, the one-mode q-deformed Heisenberg-Weyl algebra defined by the commutation relation ˆ b ˆ b − q

Jan 1, 19944 citations

A q-deformed Uncertainty Relation

Zhang, J

Within the formulation of a q-deformed Quantum Mechanics a qualitative undercut of the q-deformed uncertainty relation from the Heisenberg uncertainty relation is revealed. When $q$ is some fixed value not equal to one, recovering of ordinary quantum mechanics and the corresponding recovering condition are discussed.Within the formulation of a q-deformed Quantum Mechanics a qualitative undercut of the q-deformed uncertainty relation from the Heisenberg uncertainty relation is revealed. When $q$ is some fixed value not equal to one, recovering of ordinary quantum mechanics and the corresponding recovering condition are discussed.

Jan 1, 199922 citations

Quantum and classical mechanics of q-deformed systems

Sergei V. Shabanov

Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.

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Quantum Mechanics and Operator algebras on the Hilbert ball

Katsunori Kawamura

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C $^{*}$ -algebra is represented as a function algebra on the set of pure states with a noncommutative $*$ -product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$ . The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$ . The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$ . We introduce the quantum mechanics on $B_{\cal H}$ . As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$ .Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.

arXiv:funct-an/9710002Oct 21, 19971 citation

QUANTUM GROUPS AND VON NEUMANN THEOREM

Alfredo Iorio, Giuseppe Vitiello

arXiv:math-ph/0009035Modern Physics Letters BFeb 20, 199417 citations

Kähler Representation Theory

Bryan Wilson Roberts, Nicholas J. Teh

arXiv:1602.05979Feb 18, 20161 citation

Within the formulation of a q-deformed Quantum Mechanics a qualitative undercut of the q-deformed uncertainty relation from the Heisenberg uncertainty relation is revealed. When

q

is some fixed value not equal to one, recovering of ordinary quantum mechanics and the corresponding recovering condition are discussed.

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C

^{*}

-algebra is represented as a function algebra on the set of pure states with a noncommutative

*

-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space

{\cal P}({\cal H})

of a Hilbert space

{\cal H}

. The space

{\cal P}({\cal H})

is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature

c

. The Hilbert ball

B_{\cal H}

is defined by the open unit ball in

{\cal H}

and it is a Kähler manifold with

c<0

. We introduce the quantum mechanics on

B_{\cal H}

. As an application, we show the structure of the noncommutative function algebra on

B_{\cal H}