Space of events and the time observable

TL;DR

This paper introduces a quantum time operator within an extended phase space framework, revealing that the unique properties of the time observable originate from classical constraints, bridging classical and quantum perspectives.

Contribution

It proposes a novel quantum time operator based on classical extended phase space, linking classical constraints to quantum time measurement properties.

Findings

Time observable properties are classical in origin.

Quantum constraint defines the space of physical events.

Extended phase space approach unifies classical and quantum time concepts.

Abstract

Quantum mechanical time operator is introduced following the parametric formulation of classical mechanics in the extended phase space. Quantum constraint on the extended quantum system is defined in analogy to the constraint of the classical extended system, and is interpreted as the condition defining the space of physical events. It is seen that the peculiar properties of the time observable, otherwise obtained in the models of time measurement, are of the classical origin, (i.e.), due to the quantized classical constraint of the parametric Hamiltonian dynamics.