A note on the quantum of time

TL;DR

This paper explores how quantum fluctuations in real clocks introduce a fundamental quantum of time, leading to a delay-differential generalization of Heisenberg's evolution equation in quantum mechanics.

Contribution

It proposes a novel delay-differential equation for observable evolution that incorporates the quantum nature of time, extending traditional quantum mechanics.

Findings

Quantum fluctuations imply a nonzero uncertainty in time measurement.

The generalized Heisenberg equation becomes a delay-differential equation.

This approach modifies the standard evolution of quantum observables.

Abstract

Quantum mechanics rests on the assumption that time is a classical variable. As such, classical time is assumed to be measurable with infinite accuracy. However, all real clocks are subject to quantum fluctuations, which leads to the existence of a nonzero uncertainty in the time variable. The existence of a quantum of time modifies the Heisenberg evolution equation for observables. In this letter we propose and analyse a generalisation of Heisenberg's equation for observables evolving in real time (the time variable measured by real clocks), that takes the existence of a quantum of time into account. This generalisation of Heisenberg's equation turns out to be a delay-differential equation.