# Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations

May 07, 2019

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 non-relativistic particles at atomic scales. Without presupposing
any quantization scheme, this algebra is inherently non-commutative 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 time-ordered exponentials of interaction potentials. It is also
discussed how the familiar statistical interpretation of quantum mechanics can
be recovered from operations.

Keywords:

operations; arrow of time; dynamical C*-algebra