Time, classical and quantum

TL;DR

This paper introduces a novel approach to the concept of time in quantum mechanics by replacing the traditional time operator with a state-dependent real-valued function, linking it to system dynamics rather than Hamiltonian generators.

Contribution

It extends the classical geometric approach to quantum mechanics, defining a new notion of simultaneity based on dynamical evolution, not associated with any self-adjoint operator.

Findings

Time as a state-dependent function varies with dynamics.

Different evolutions imply different notions of simultaneity.

The approach applies to finite-level quantum systems.

Abstract

We propose a new point of view regarding the problem of time in quantum mechanics, based on the idea of replacing the usual time operator $\mathbf{T}$ with a suitable real-valued function $T$ on the space of physical states. The proper characterization of the function $T$ relies on a particular relation with the dynamical evolution of the system rather than with the infinitesimal generator of the dynamics (Hamiltonian). We first consider the case of classical Hamiltonian mechanics, where observables are functions on phase space and the tools of differential geometry can be applied. The idea is then extended to the case of the unitary evolution of pure states of finite-level quantum systems by means of the geometric formulation of quantum mechanics. It is found that $T$ is a function on the space of pure states which is not associated to any self-adjoint operator. The link between $T$ and the dynamical evolution is interpreted as defining a simultaneity relation for the states of the system with respect to the dynamical evolution itself. It turns out that different dynamical evolutions lead to different notions of simultaneity, i.e., the notion of simultaneity is a dynamical notion.