W. J. Zakrzewski — Page 5

arXiv1993

Classical and Quantum Mechanics of Free \k Relativistic Systems

We consider the Hamiltonian and Lagrangian formalism describing free \k-relativistic particles with their four-momenta constrained to the \k-deformed mass shell. We study the modifications of the formalism which follow from the introduction of space coordinates with nonvanishing Poisson brackets and from the redefinitions of the energy operator. The quantum mechanics of free \k-relativistic particles and of the free \k-relativistic oscillator is also presented. It is shown that the \k-relativistic oscillator describes a quantum statistical ensemble with finite Hagedorn temperature. The relation to a \k-deformed Schrödinger quantum mechanics in which the time derivative is replaced by a finite difference derivative is also discussed.

hep-th↗

arXiv1992

Quantum Group $σ$ Models

Field-theoretic models for fields taking values in quantum groups are investigated. First we consider $SU_q(2)$ $σ$ model ($q$ real) expressed in terms of basic notions of noncommutative differential geometry. We discuss the case in which the $σ$ models fields are represented as products of conventional $σ$ fields and of the coordinate-independent algebra. An explicit example is provided by the $U_q(2)$ $σ$ model with $q\sp{N}=1$, in which case quantum matrices $U_q(2)$ are realised as $2N\times 2N$ unitary matrices. Open problems are pointed out.

hep-th↗