The equivalent emergence of time dependence in classical and quantum mechanics

TL;DR

This paper explores how time emerges in classical and quantum mechanics from a timeless closed system, highlighting the parallels and conditions needed for defining a classical time variable, including the role of environment classicality and entanglement removal.

Contribution

It demonstrates a unified approach to deriving time-dependent equations in classical and quantum mechanics from a timeless framework using Hamilton-Jacobi theory.

Findings

Time can be defined by changes in a part of a closed system.

Quantum environment must become classical to define real time.

Emergence of time parallels quantum gravity scenarios.

Abstract

Beginning with the principle that a closed mechanical composite system is timeless, time can be defined by the regular changes in a suitable position coordinate (clock) in the observing part, when one part of the closed composite observes another part. Translating this scenario into both classical and quantum mechanics allows a transition to be made from a time-independent mechanics for the closed composite to a time-dependent description of the observed part alone. The use of Hamilton- Jacobi theory yields a very close parallel between the derivations in classical and quantum mechanics. The time-dependent equations, Hamilton-Jacobi or Schrodinger, appear as approximations since no observed system is truly closed. The quantum case has an additional feature in the condition that the observing environment must become classical in order to define a real classical time variable. This condition leads to a removal of entanglement engendered by the interaction between the observed system and the observing environment. Comparison is made to the similar emergence of time in quantum gravity theory