Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations
Abstract
Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*algebra is presented for nonrelativistic particles at atomic scales. Without presupposing any quantization scheme, this algebra is inherently noncommutative and comprises a large set of dynamics. In contrast to other approaches, the generating elements of the algebra are not interpreted as observables, but as operations on the underlying system; they describe the impact of temporary perturbations caused by the surroundings. In accordance with the doctrine of Nils Bohr, the operations carry individual names of classical significance. Without stipulating from the outset their `quantization', their concrete implementation in the quantum world emerges from the inherent structure of the algebra. In particular, the Heisenberg commutation relations for position and velocity measurements are derived from it. Interacting systems can be described within the algebraic setting by a rigorous version of the interaction picture. It is shown that Hilbert space representations of the algebra lead to the conventional formalism of quantum mechanics, where operations on states are described by timeordered exponentials of interaction potentials. It is also discussed how the familiar statistical interpretation of quantum mechanics can be recovered from operations.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.02711
 Bibcode:
 2019arXiv190502711B
 Keywords:

 Quantum Physics;
 Mathematical Physics
 EPrint:
 21 pages, no figures