Quantum Uncertainty as an Intrinsic Clock

TL;DR

This paper explores how quantum uncertainty can serve as an intrinsic clock by linking the evolution of wave-packet width to classical invariants, with implications for quantum gravity and cosmology.

Contribution

It demonstrates that quantum uncertainty functions as a natural intrinsic time parameter, extending classical invariants to quantum systems and providing new insights into quantum dynamics.

Findings

Quantum uncertainty acts as an intrinsic clock in quantum systems.

The Ermakov-Lewis invariant corresponds to quantum uncertainty of a Gaussian wave-packet.

The approach has potential applications in quantum gravity and cosmology.

Abstract

In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrodinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.