Stefan Giller

More about the q-deformed Poincaré algebra

Abstract The recently proposed deformed Poincare algebra is considered. Its structure is clarified and “unitary” representations are found. The contraction to the deformed Galilei algebra is performed. The three-dimensional counterpart is discussed.

Space–time symmetry of noncommutative field theory

We consider the deformed Poincare group describing the space-time symmetry of noncommutative field theory. It is shown how the deformed symmetry is related to the explicit symmetry breaking.

Sphalerons and normal modes in the (1 + 1)-dimensional abelian Higgs model on the circle

Abstract The abelian Higgs model is considered on the circle. The periodic sphaleron solutions are constructed explicitly. The equations for the normal modes about these solutions resemble Lame equations. For special values of the Higgs field mass a number of modes are obtained analytically, including in particular the negative modes.

On the consistency of twisted gauge theory

It is argued that the twisted gauge theory is consistent provided it exhibits also the standard noncommutative gauge symmetry.

Global symmetries of noncommutative space-time

The global counterpart of infinitesimal symmetries of noncommutative space-time is discussed.

THE QUANTUM GALILEI GROUP

The quantum Galilei group Gκ is defined. The bicross-product structure of Gκ and the corresponding Lie algebra is revealed. The projective representations for two-dimensional quantum Galilei group are constructed.

Linear operators with invariant polynomial space and graded algebra

The irreducible, finite-dimensional representations of the graded algebras osp(j,2) (j=1,2,3) are expressed in terms of differential operators. Some quantum deformations of these algebras are shown to admit similar kinds of representations. These are formulated in terms of finite difference operators. The results are discussed in the framework of the quasi-exactly solvable equations.

The structure of quasi‐exactly solvable systems

The linear, differential operators that preserve the N‐dimensional vector space whose entries are polynomials of fixed, but arbitrary, degrees in one variable are considered herein. They are relevant for the classification of quasi‐exactly solvable systems of equations. Generating elements are explicitly constructed and rules for ordering their products are derived. Some examples of equations that can be reduced to such systems are discussed.

Fundamental solution method applied to time evolution of two-energy-level systems: Exact and adiabatic limit results

A method of fundamental solutions has been used to investigate transitions in two-energy-level-systems with no level crossing in a real time. Compact formulas for transition probabilities have been found in their exact form as well as in their adiabatic limit. No interference effects resulting from many-level complex crossings as announced by Joye, Mileti, and Pfister [Phys. Rev. A 44, 4280 (1991)] have been detected in either case. It is argued that these results of this work are incorrect. However, some effects of Berrys phases are confirmed.

Borel summable solutions to one-dimensional Schrödinger equation

It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the one-dimensional (1D) Schrodinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be considered as the canonical one. The latter means that any Borel summable solution can be obtained by the Borel transformation of this unique canonical Borel function multiplied by some ℏ-dependent and Borel summable constant. This justifies the exceptional role the fundamental solutions play in 1D quantum mechanics and completes the relevant semiclassical theory relied on the Borel resummation technique and developed in our other papers.