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This paper proposes a theoretical framework linking the emergence of internal time in quantum mechanics to information theory, specifically deriving duration increments based on surprisal (Shannon entropy) rather than external clock mechanisms. By modeling record formation through decoherence and stability criteria, it connects fundamental temporal evolution directly to the accumulation of distinguishable outcomes within quantum histories.
Actualization, Records, and the Emergence of Entropic Time
arXiv:2607.25307v1 Announce Type: new Abstract: We develop a record-based account of internal time in quantum mechanics, where the formation of a stable record is represented as conditioning on actualized information, and along a history the accumulated record algebras are ordered by inclusion. If the duration of a realized outcome depends only on its conditional Born probability, composes additively under sequential conditioning, and is continuous and calibrated, then the actualization of each outcome contributes an internal duration equal to its surprisal, the negative logarithm of that probability, so that a certain outcome contributes no duration, whereas less likely outcomes contribute larger increments. The ensemble mean of the accumulated clock is the Shannon entropy of the record process, its moment-generating function is fixed by the R\'enyi entropy spectrum, and the realized clock admits a Doob decomposition into a predictable entropic compensator and a martingale of clock fluctuations, so that each increment is the information gain of the corresponding actualization. Records are characterized by graded criteria of distinguishability, decoherence, and stability. We also clarify the multiple-clock problem: in one fixed context, additivity of two surprisal clocks is equivalent to factorization of the Born distribution in that context, whereas for a pure bipartite state, additivity in every pair of local contexts is equivalent to rank-one factorization of the joint state and to the vanishing of all its $2\times2$ minors.
Abstract:We develop a record-based account of internal time in quantum mechanics, where the formation of a stable record is represented as conditioning on actualized information, and along a history the accumulated record algebras are ordered by inclusion. If the duration of a realized outcome depends only on its conditional Born probability, composes additively under sequential conditioning, and is continuous and calibrated, then the actualization of each outcome contributes an internal duration equal to its surprisal, the negative logarithm of that probability, so that a certain outcome contributes no duration, whereas less likely outcomes contribute larger increments. The ensemble mean of the accumulated clock is the Shannon entropy of the record process, its moment-generating function is fixed by the Rényi entropy spectrum, and the realized clock admits a Doob decomposition into a predictable entropic compensator and a martingale of clock fluctuations, so that each increment is the information gain of the corresponding actualization. Records are characterized by graded criteria of distinguishability, decoherence, and stability. We also clarify the multiple-clock problem: in one fixed context, additivity of two surprisal clocks is equivalent to factorization of the Born distribution in that context, whereas for a pure bipartite state, additivity in every pair of local contexts is equivalent to rank-one factorization of the joint state and to the vanishing of all its $2\times2$ minors.
arXiv:2607.25307 [quant-ph]
(or arXiv:2607.25307v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2607.25307 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 311 words
