On time as a quantum-physical observable quantity

V. S. Olkhovsky

Analytical review of developments in researching $time$ as a quantum-physical observable, which is canonically conjugated to energy, is presented. A special attention is devoted to the duality of energy and time operators and also to the verification of time-energy uncertainty relations for continuous and discrete energy spectra.

Tadashi Fujimoto

R. Edward Nather

Giulia Rubino, Časlav Brukner, Gonzalo Manzano

In both classical and quantum thermodynamics, physical quantities are typically assigned objective values defined independently of our observations. We then refer to the “work performed by a gas,” or the “entropy of the gas,” regardless of how they are evaluated. Here, we question this conception in the context of quantum thermodynamics, estimating how the definition of pivotal thermodynamic quantities is affected by experimental instruments of limited precision. We find that the coarse-grained thermodynamic quantities frequently lead to different conclusions from those drawn in fine-grained scenarios. For instance, the irreversibility of a process, or its work payoff, can significantly vary with the instrument precision. We show nonetheless that coarse-grained thermodynamic quantities satisfy the relations (i.e., the second law inequality, the relation between dissipation and distinguishability of a process from its time reverse, and the quantum work fluctuation theorems) as their fine-grained counterparts. These results highlight the observation independence of relations linking thermodynamic quantities, which are themselves observation dependent.

Valia Allori

Roberto Carlos Moleirinho Batista

Abstract The physical status of time remains one of the most persistent open questions in fundamental physics. In quantum mechanics, time appears as an external parameter governing evolution but not as an observable, while in general relativity it is embedded in the geometric structure of spacetime and becomes observer-dependent. Despite their empirical success, neither framework explains the physical origin of time or its singular role among fundamental quantities. In this work, we propose an emergent interpretation of physical time grounded in quantum phase. Phase is treated as a pre-temporal structural feature of quantum systems, rather than as a quantity defined with respect to an underlying temporal parameter. Energy is interpreted as a relational characterisation of regularities in phase structure, and proper time emerges only in contexts where phase accumulation can be compared and synchronised between interacting systems. An isolated quantum system may possess phase, but it does not possess physical time in the operational sense; time arises only through relational compatibility. This approach does not introduce new fundamental dynamics nor posit a physical mechanism responsible for the generation of time. Instead, it establishes an ontological hierarchy in which phase is fundamental, energy is relational, and time is a phenomenological quantity that emerges from stable conditions of synchronisation. Within this framework, classical reality corresponds to domains of temporal compatibility, while loss of compatibility underlies the emergence of horizons and geometric structure. By reframing time as an emergent relational phenomenon rooted in quantum phase, this work provides a coherent interpretative framework in which the absence of a time operator in quantum mechanics and the difficulty of quantising time and gravity are understood as consequences of their non-fundamental status, rather than as shortcomings of existing formalisms.