An algebraic approach to quantum mechanics II

Istvan Montvay

В.В. Варламов

The 𝑆-matrix is defined in the framework of an algebraic formulation ofquantum theory with a binary structure.

Shun-Jin Wang

Contrary to the usual view that quantum mechanics in configuration space has, in general, only one algebaic representation( the Heisenberg algebraic representation), we have proved that quantum mechanics in configuration space has, in general, alternative algebraic representations: (i) In a finite domain of configuration space, it can be expressed in terms of su(2) algebra; (ii) in an infinite domain of configuration space, it can be expressed in terms of su(l,l) algebra.The above results open a new possibility to reformulate quantum mechanics and provide more mathematical tools to solve diverse physical problems.Nonlinear relations of different Lie algebas may imply a unification of Lie algebras and their physical implications ( the quantum motion modes ) in a deeper nonlinear domain.This observation raises challenging mathematical and physical questions.

Antonino Drago

Thomas Krüger

The overwhelming majority of scientists still takes it for granted that classical mechanics (ClM) is nothing but a limiting case of quantum mechanics (QM). Although some physicists restrict this belief to a generalized QM as represented, e. g., by the algebra of observables, it will be shown in this contribution that the view of ClM as a mere sub-set of QM is nevertheless unfounded. The usual attempts to derive the laws of ClM from QM are either insufficient or not universally applicable. The transition from traditional to algebraic QM does not add any further insight. It is demonstrated that typical constituents of the classical macroscopic world i) cannot be described reasonably in terms of QM and/or ii) do not show up the typical quantum behavior which manifests in the double-slit interference and in the Einstein-Podolsky-Rosen correlations. Moreover, both attempts to recover ClM from QM and approaches based on vacuum fluctuations are critically inspected, and we arrive at the conclusion: QM does not comprehend ClM, i. e., a wavefunction of the universe does not exist.

В. В. Варламов