# Internal clock formulation of quantum mechanics

## TL;DR

This paper explores how quantum mechanics can be formulated using internal clocks instead of external time, revealing that the choice of clock affects the perceived dynamics and spectrum of observables in timeless systems.

## Contribution

It provides a rigorous mathematical framework for internal clock-based quantum mechanics, showing how different clocks influence the system's dynamical interpretation and observable spectra.

## Key findings

- Internal clocks can alter the perceived spectra of quantum observables.
- The dynamical interpretation of quantum states depends on the chosen internal clock.
- Discrete spectra may emerge from continuous ones when measured with unusual clocks.

## Abstract

The basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom.