Actualization, Records, and the Emergence of Entropic Time

The Emergence of Internal Time

This paper proposes a relational approach to time in quantum mechanics, moving away from the standard view of time as an external parameter. Instead, the author constructs an internal clock based on the formation and accumulation of physical records. By interpreting quantum states as distributions of potentiality, the paper frames measurement as the conditioning of these potentialities on actualized records. This framework allows for the definition of an internal chronology based on the inclusion of record algebras, where later records contain the information of earlier ones.

The Surprisal Clock

To move from an ordinal sequence of events to a metric of duration, the author imposes three conditions on the increments of the clock: dependence only on conditional Born probabilities, additive composition under sequential conditioning, and continuity. These requirements uniquely identify the surprisal of an outcome—the negative logarithm of its probability—as the fundamental unit of internal duration. Consequently, a certain outcome contributes no duration, while rare outcomes contribute larger temporal increments. This construction provides a rigorous link between quantum measurement and the flow of time.

Statistical Properties and Doob Decomposition

The paper further analyzes the statistical behavior of this clock. The ensemble mean of the accumulated clock is shown to be the Shannon entropy of the record process, while its fluctuations are governed by the Rényi entropy spectrum. A key result is the Doob decomposition of the clock into a predictable entropic compensator and a martingale of fluctuations. This reveals that internal time is essentially the information gain realized through actualization, with the entropic compensator representing the systematic growth of information and the martingale capturing the stochastic fluctuations of the clock.

This work offers a formal, operational account of how time can emerge from the structure of quantum actualization without requiring an external clock. By grounding time in the information-theoretic properties of records, the paper provides a bridge between quantum foundations and thermodynamics, suggesting that the directionality and flow of time are consequences of the accumulation of distinguishable, stable records in a quantum system.