Postulates for time evolution in quantum mechanics

Edward Parker

Abstract Discussions of quantum mechanics often loosely claim that time evolution logically must be unitary, in order for the probabilistic interpretation of the amplitudes of the state vector to make sense at all times. We discuss from first principles whether this claim is true: if we assume only that the time-evolution operator is linear, then does the stronger requirement that it be unitary follow from the other axioms of quantum mechanics? The answer is subtle. We discuss two mathematically distinct but physically equivalent formulations of the axioms of quantum mechanics, and consider generalizing each to postulate only that time evolution is linear. Within one formulation, the unitarity of time evolution follows logically from the other axioms – but within the other formulation, it does not. Allowing the time-evolution operator to be (a priori) arbitrarily linear does not change the physical observables in one formulation of quantum mechanics, but changes the other formulation to a distinct (internally consistent) physical theory that allows new phenomenology like (e.g.) faster-than-light communication. Therefore, the unitarity of time evolution is arguably better thought of as a logically independent and experimentally falsifiable axiom of quantum mechanics, not as a tautological consequence of the other axioms.

Abani K. Bhuyan

Chapter 2, Postulates of Quantum Mechanics, points to the complementarity and inseparability of quantum mechanics and spectroscopy. To elucidate the close relationship of spectroscopy and quantum mechanics and to arouse interest in the subject, the collaborative evolution of the theoretical and experimental ideas of the quantization of magnetic moment of particle spin is briefed first. The necessary quantum mechanical background is introduced by discussing the physical and statistical meaning of the postulates. The physical meaning of the observable of a measurement is also explained. The principle of superposition of eigenstates and the time evolution of a state function during system-radiation interactions are analyzed. It is mentioned that a system weakly perturbed by an impulse can often be analyzed by perturbation theory. The chapter ends with a discussion of perturbation approaches to degenerate and nondegenerate systems.

Léon Brenig, Marc Vincke

A unique postulate is shown to underly the whole quantum mechanics theory: the invariance of the Heisenberg uncertainty inequality under a group of special nonlinear gauge transformations (NLGT). With this postulate, the quantum mechanics of a free particle is derived from classical mechanics, including the statements of the postulates of quantum mechanics, except for the wave-function-collapse postulate. An explanatory mechanism for the latter postulate is derived by performing an analytical continuation of the NLGTs. This extension results in a Schrödinger-bridge process, intertwined under the NLGT with the standard unitary quantum evolution, and revealing non-quantum (or beyond-quantum) phenomena. Mechanisms of that latter kind, like the ones associated to the quantum measurement process, occur in a new space-like dimension and hence are non causal in nature, in opposition to a time evolution. The present exercice focusses on the free particle in order to highlight the features of the performed derivation in the simplest possible way. Work is in progress to extend the performed derivation beyond that simple case.

Bernhard Baumgartner

Brian P. Anderson