Proper Time in Classical and Quantum Mechanics

Philippe Droz-Vincent

Bernd Thaller

Igor Pikovski, Magdalena Zych, Fabio Costa +1 more

M. Ichiyanagi

We develop a formal theory which rests upon Nambu's description of the quantum state and especially on the use of the proper time. We show that the introduction of the proper time in quantum mechanics in understandable from the viewpoint that eligible trajectries in the well-known variational treatment of classical mechanics can be parametrized by an auxiliary variable. A new theory developed here permits a formulation of barrier penetrations in terms of the so-called Euclidean Lagrangian.

Valia Allori

Hrvoje Nikolić

Bohmian mechanics can be generalized to a relativistic theory without preferred foliation, with a price of introducing a puzzling concept of spacetime probability conserved in a scalar time. We explain how analogous concept appears naturally in classical statistical mechanics of relativistic particles, with scalar time being identified with the proper time along particle trajectories. The conceptual understanding of relativistic Bohmian mechanics is significantly enriched by this classical insight. In particular, the analogy between classical and Bohmian mechanics suggests the interpretation of Bohmian scalar time as a quantum proper time different from the classical one, the two being related by a nonlocal scale factor calculated from the wave function. In many cases of practical interest, including the macroscopic measuring apparatus, the fundamental spacetime probability explains the more familiar space probability as an emergent approximate description. Requiring that the quantum proper time in the classical limit should reduce to the classical proper time, we propose that only massive particles have Bohmian trajectories. An analysis of the macroscopic measuring apparatus made up of massive particles restores agreement with the predictions of standard quantum theory.