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Non-Hermitian realization of a Lie-deformed Heisenberg algebra

A. D. Jannussis, A. Leodaris, R. Mignani

Physics Letters AJan 1, 199510 citations

Anyons and Deformed Lie Algebras

M. Frau, Alberto Lerda A, S. Sciuto

We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.

arXiv:hep-th/9407161Jul 25, 19942 citations

Local currents for a stable algebra of quantum mechanics

E M Chabab, Gerald A. Goldin, Sarben Sarkar

Jan 1, 20050 citations

Local currents for a stable algebra of quantum mechanics

Gerald A. Goldin, S Sarkar

Jan 1, 20050 citations

A universal kinematical group for quantum mechanics

Gerald A. Goldin, David H. Sharp

In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities – notably, anyons and nonabelian anyons in two space dimensions. We present here foundational reasons why this semidirect product group serves as a universal kinematical group for quantum mechanics. We obtain thus a unified account of all possible quantum kinematics for systems with mass in an arbitrary physical space, and clarify the role played by topology in quantum mechanics. Our development does not require quantization of classical phase space; rather, the classical limit follows from the quantum mechanics. We also consider the relationship of our development to Heisenberg quantization.In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities – notably, anyons and nonabelian anyons in two space dimensions. We present here foundational reasons why this semidirect product group serves as a universal kinematical group for quantum mechanics. We obtain thus a unified account of all possible quantum kinematics for systems with mass in an arbitrary physical space, and clarify the role played by topology in quantum mechanics. Our development does not require quantization of classical phase space; rather, the classical limit follows from the quantum mechanics. We also consider the relationship of our development to Heisenberg quantization.

arXiv:2404.18274Journal of Mathematical PhysicsAug 1, 20252 citations

Local Current Algebras: Early Ideas and New Quantum-Theoretical Directions

Gerald A. Goldin

Abstract. Local currents combine the ideas of space-time locality and symme-try by means of an infinite-dimensional Lie algebra of operators. We review some early ideas about relativistic and nonrelativistic local current algebras, and discuss their relation to quantum kinematics. We then summarize three directions of recent research: a model for quantized vortex filaments (with R. Owczarek and D. H. Sharp); a prescription for nonlinear modification of quan-tum mechanics (with H.-D. Doebner); and local currents for a deformed algebra of quantum mechanics (with S. Sarkar). PACS number: 02.20.TwAbstract. Local currents combine the ideas of space-time locality and symme-try by means of an infinite-dimensional Lie algebra of operators. We review some early ideas about relativistic and nonrelativistic local current algebras, and discuss their relation to quantum kinematics. We then summarize three directions of recent research: a model for quantized vortex filaments (with R. Owczarek and D. H. Sharp); a prescription for nonlinear modification of quan-tum mechanics (with H.-D. Doebner); and local currents for a deformed algebra of quantum mechanics (with S. Sarkar). PACS number: 02.20.Tw

Jan 1, 20061 citation

Local Currents for a Deformed Heisenberg–Poincaré Lie Algebra of Quantum Mechanics, and Anyon Statistics

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