Quantum clock and Newtonian time

TL;DR

This paper proposes a quantum clock model replacing classical Newtonian time, deriving modified quantum evolution equations including corrections to the von Neumann and Lindblad equations, with bounds established via atomic clock precision.

Contribution

It introduces a quantum clock framework that generalizes standard quantum evolution, deriving new correction terms to the von Neumann equation and analyzing their implications.

Findings

Leading term recovers von Neumann equation

Higher-order corrections form a generalized Lindblad equation

Lower bounds on clock parameters from atomic clock precision

Abstract

An extension of standard quantum mechanics is proposed in which the Newtonian time appearing as a parameter in the unitary evolution operator is replaced with the time shown by a `quantum clock'. Such a clock is defined by the following properties: (a) the time that the clock shows is non-decreasing, (b) the clock ticks at random Newtonian times with random tick sizes, and (c) on average the clock shows the Newtonian time. We show that the leading term in the evolution equation for the density matrix associated with any quantum clock gives the von Neumann equation. The leading correction to the von Neumann equation is given by the Lindblad equation generated by the Hamiltonian, but there are higher-order terms that generalize the von Neumann equation and the Lindblad equation. Modifications to the von Neumann equation are worked out in detail in a parametric family of models for which the tick sizes are gamma distributed. Lower bounds on the parameters of these quantum clock models are derived using the precision limit of an atomic clock.