Finite-dimensional *-product and matrix algebras

*-product formulation of finite-dimensional quantum mechanics introduces an other multiplication law between square real matrices (which play a role of classical observables on discrete phase space). Then the analogy of classical mechanics on discrete space is done by multiplication law where the element of resulting matrix is the product of corresponding elements. The possible deformation is studied.Presented at the 4th Colloquium on Quantum Groups and Integratile Systems, Prague, 22–24 June 1995.The authors gratefully acknowledge the support of the Grant of Czech Technical University in Prague No. 10028154.     

Feynman path integral and ordering rules on discrete finite space

The Feynman path integral is constructed for systems whose configuration space is a discrete finite set. The construction is based on the operator formulation of quantum mechanics on a finite discrete space. We derive connections between operators and introduce the analogue of the^*-multiplication for discrete symbols.     

Polytopes vibrations within Coxeter group symmetries

We are considering polytopes with exact reflection symmetry group G in the real 3-dimensional Euclidean spaceR³. By changingone simple element of the polytope (position of one vertex or length of an edge), one canretain the exact symmetry of the polytope by simultaneously changing other correspondingelements of the polytope. A simple method of using the symmetry of polytopes in order todetermine several resonant frequencies is presented. Knowledge of these frequencies, or atleast their ratios can be used for control of some principal changes of the polytopes.     

Feynman path integral with underlying noncommutative “geometry” and quantum mechanics in external field

Noncommutative algebra of translations as a deforming commutative algebra induces a linear vector potential in the plane. This potential perturbs the generators of translations. Such potential we can identify, for example, with constant magnetic or with homogeneous electric fields. This potential introduces an extra phase factor in the Feynman formulation of quantum mechanic.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

The Exact Solution of the Eigenvalue Problem for the Parametric Down Conversion Process in the Kerr Medium

The eigenproblem for a class of Hamiltonians of the parametric down conversion process in the Kerr medium is solved. Some physical characteristics of the system are calculated.     

Para-Grassmann Star Product Calculation

In this Letter, we construct the star product for polynomials over the para-Grassmann variable and we present as an example a para-Grassmann version of the model of q–quantum mechanics. Moreover, using the structural relations of the q–deformed algebra generated by annihilation and creation operators, we decompose the Jacobi matrix in the product of three matrices: the diagonal matrix, the upper and lower diagonal matrices, mutually adjoint.