Postmodern Quantum Mechanics

Abstract

Postmodern movements are well known in the arts. After a major artistic revolution, and after the “modern” innovations have been assimilated, the threads of premodern thought are always reconsidered. Much of value may be rediscovered and put to new use. The modern context casts new light on premodern thought, which in turn shades perspectives on modernism.

Chaotic properties of the truncated elliptical billiards

“…Rigorous investigations were concentrating on methods for producing fully chaotic billiards and on specific properties (Bernoulli, K-property, mixing and hyperbolicity) expressing differences between chaotic systems [5,6,7,8,9]. Various aspects of billiard dynamics have been extensively examined during last decades [10,11,12,13,14,15,16,17,18,19]. In recent years, properties of classical billiards and their quantum-mechanical counterparts were used to explain and improve performances of devices in microelectronics and nanotechnology, especially of optical microresonators in dielectrical and polymer lasers [20,21,22,23,24,25].…”

Ergodicity in Umbrella Billiards

“…On a bounded region Q ⊂ R 2 (the billiard table), an infinitesimal particle moves along segments at unit speed, changing direction according to the law of specular reflection upon collisions at boundaries. The essential link in billiards between the geometry of the table and the dynamics of the system facilitates a robust model which has proved useful in approaching problems ranging from the foundations of the Boltzmann's ergodic hypothesis [9], to the description of shell effects in semiclassical physics [5], to the design of microwave resonators in quantum chaos [33], and many other other applications [1,17,23,25,26]. In particular, ergodic properties are determined by the shape of the table, producing a spectrum of behaviors from completely integrable to strongly chaotic.…”

“…In [20] the ovals were generalized to a two-parameter family encompassing seven varieties, including special cases of lemon, moon, and a particular example of a class which in this paper we will designate as umbrella billiards, while [4] gives an alternate generalization of [7] to squash billiard tables, on which the elementary defocusing mechanism does not take place. In [25] symmetric lemon billiards were considered, and recently a class of asymmetric lemon-shaped convex billiard tables were constructed in [13], obtained by intersection of two disks in the plane. It was also proved that a subclass of these billiards are indeed hyperbolic using continued fraction techniques [12].…”

Parasupersymmetries and Non-Lie Constants of Motion for Two-Particle Equations

“…It is not surprising that such an approximation becomes bad at times so large that the true orbits have begun to intersect each other (caustics). Many other researchers have had things to say about this problem much more profound than I have to offer [e.g., 6,9,11,14,17,31]. I would, however, like to emphasize a point that evidently is not generally appreciated.…”