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Journal of Mathematical Physics

The q-deformation of quantum mechanics of one degree of freedom

V. Maximov, Anatol Odzijewicz

Published Apr 1, 1995• 14 citations
Publisher

Abstract

The q-quantum mechanics of the one degree of freedom is studied. Among others the holomorphic representation of q-deformed Heisenberg–Weyl algebra and its realization by covariant Berezin symbols is described.
DOI
10.1063/1.531080

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On a one-dimensional lattice with a geometric sequence spacing, the Hermitian conjugation of a (p,q)-derivative operator is discussed by means of (p,q)-integration. Then a (p,q)-deformation of both the Heisenberg algebra for the canonical coordinates and the Heisenberg–Weyl algebra for the harmonic oscillator is presented. It is shown that although in the algebraic aspect the (p,q)-deformation discussed here is identical with q-deformation given by Truong, the (p,q)-deformed Schrodinger picture is in fact different from the q-deformed one.
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. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory
On a one-dimensional lattice without uniform interval, the Hermitian conjugation of a q-differential operator is discussed. Then a deformation of quantum mechanics in one dimension is presented. As an application, the harmonic oscillator is discussed. The energy spectrum and the eigenfunctions are shown to depend on an arbitrary deformation function. The deformed coherent states are also discussed. It is found that the completeness relation of coherent states holds for the case of q-coherent states, i.e. the deformation of the Heisenberg-Weyl algebra is a q-analogue Hopf algebra.
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.