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Clock Quantum Mechanics

coppola, VALERIO

Published Dec 8, 2025• 0 citations
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Abstract

This work aims to introduce a novel framework than elevates time to an operator in Quantum Mechanics.This work is intended as a draft, all necessary considerations will be developed in time together with the further development of the theory.
DOI
10.5281/zenodo.17858092

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Structured Potential Expression Theory (SPET) is an ongoing research project exploring the relationships among structured possibility, probability, realized physical outcomes, history, and observer-specific experience. This upload contains Revision 3 of the working reference together with the previous Revision 2 reference for development history. These documents are working drafts intended to preserve the project's evolution. They are not peer reviewed and should not be interpreted as established physics or experimental confirmation. The framework distinguishes between mathematical definitions, candidate postulates, derived identities, interpretive principles, observer-framework concepts, and proposed experimental programs.
Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, this mechanism explored a stationary pure bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge and an equation of motion can be obtained for a quantum particle conditioned to the measurement of the state of the clock. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page–Wootters model. Connecting it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence. In a second step we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the systems initial conditions.
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