Ziemowit Domański

Many theories of quantum gravity suggest that there should exist a minimal scale with which it is possible to measure distances and time. We present a formalism of quantum mechanics exchibiting a minimal length and time scale. The work is based on the Page-Wootters formalism which gives a possible solution to the problem of time in quantum mechanics. This formalism is an example of a relational approach to quantum dynamics. It is based on the idea that reference frames with respect to which motion is described are themselves physical systems interacting with the degrees of freedom they wish to describe. Quantum dynamics is described in terms of quantum correlations between a clock and a system. In our work we modify the commutation relations between the time and frequency operators leading to a minimal uncertainty of time measurement. This results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We further investigate the e ects of such modi cation on a couple simple quantum systems.

The existence of a minimum measurable length scale was suggested by various theories of quantum gravity, string theory and black hole physics. Motivated by this, we examine a quantum theory exhibiting a minimum measurable time scale. We use the Page–Wootters formalism to describe time evolution of a quantum system with the modified commutation relations between the time and frequency operator. Such modification leads to a minimal uncertainty in the measurement of time. This causes breaking of the time-translation symmetry and results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We demonstrate the developed theory on a couple simple quantum systems.

This is the second article in a suite of articles investigating relations between Stäckel‐type systems and Painlevé‐type systems. In this paper, we construct isomonodromic Lax representations for Painlevé‐type systems found in the previous paper by Frobenius integrable deformations of Stäckel‐type systems. We first construct isomonodromic Lax representations for Painlevé‐type systems in the so‐called magnetic representation and then, using a multitime‐dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé‐type systems in the nonmagnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painlevé‐type. We also present isomonodromic Lax representations for all one‐, two‐, and three‐dimensional Painlevé‐type systems originating in our scheme. Based on these results we propose complete hierarchies of that follow from our construction.

This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel type and of Painlevé type. In this article, we present a deformation of autonomous Stäckel‐type systems to nonautonomous Frobenius integrable systems. First, we consider quasi‐Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time‐dependent coefficients, and then, we present a procedure of deforming these equations to nonautonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi‐Stäckel systems with so‐called magnetic separable potentials to nonautonomous Frobenius integrable systems. We also provide a complete list of all two‐ and three‐dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, which originate in our construction. Further, we prove the equivalence between both classes of systems. Finally, we show how Painlevé equationscan be derived from our scheme.

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Fréchet algebra as well as a pre-C^*-algebra. Basic properties of the star-product are proved, and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C^*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Frechet algebra as well as a pre-C*-algebra. Basic properties of the star-product are proved and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.