<p>In canonical quantum gravity, time does not appear as a fundamental coordinate, posing the longstanding problem of how dynamical evolution arises in a fundamentally timeless universe. In this work, we propose that entropy—interpreted as a coarse-grained, monotonically increasing measure of system complexity—can serve as an emergent internal clock. We unify three complementary mechanisms underpinning this idea: (i) the monotonic growth of entanglement entropy under unitary dynamics, (ii) thermal modular flow associated with Kubo-Martin-Schwinger states, and (iii) relational time from the Page–Wootters framework. These mechanisms jointly define a physical arrow and parametrization of time grounded in the informational structure of the quantum state. From this foundation, we derive explicit entropic time laws of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\tau (\Delta S)=\left( \Delta S/\lambda \right) ^{1/\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced close=")" open="("> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo stretchy="false">/</mo> <mi>λ</mi> </mfenced> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>γ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, showing how the parameters <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(q,N_{0},\lambda ,\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <msub> <mi>N</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> emerge from microscopic statistical properties such as non-extensive correlations, phase space growth, and entropy production rates. We apply this framework to cosmological epochs, identifying entropy increase across inflation, radiation and matter domination as a natural proxy for internal time progression. This entropic approach provides a unified view linking quantum foundations, thermodynamic irreversibility, and cosmological evolution. We also discuss interpretational subtleties, clarifying how entropic time differs from coordinate time and under what conditions it defines a meaningful temporal structure. We emphasize that the entropic time, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\tau ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> provides an arrow and parametrization of change in relational regimes (Wheeler–DeWitt,Page–Wootters, KMS/modular frameworks), but it is not proposed as a universal bijective substitute for <i>coordinate time, t.</i></p>

Entropy as a Clock: Foundations and Parametrizations of Emergent Time

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2026年1月22日

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José Weberszpil,
Oscar Sotolongo-Costa

摘要

In canonical quantum gravity, time does not appear as a fundamental coordinate, posing the longstanding problem of how dynamical evolution arises in a fundamentally timeless universe. In this work, we propose that entropy—interpreted as a coarse-grained, monotonically increasing measure of system complexity—can serve as an emergent internal clock. We unify three complementary mechanisms underpinning this idea: (i) the monotonic growth of entanglement entropy under unitary dynamics, (ii) thermal modular flow associated with Kubo-Martin-Schwinger states, and (iii) relational time from the Page–Wootters framework. These mechanisms jointly define a physical arrow and parametrization of time grounded in the informational structure of the quantum state. From this foundation, we derive explicit entropic time laws of the form $\tau (\Delta S)=\left( \Delta S/\lambda \right) ^{1/\gamma }$ $τ (Δ S) = {(Δ S / λ)}^{1 / γ}$ , showing how the parameters $(q,N_{0},\lambda ,\gamma )$ $(q, N_{0}, λ, γ)$ emerge from microscopic statistical properties such as non-extensive correlations, phase space growth, and entropy production rates. We apply this framework to cosmological epochs, identifying entropy increase across inflation, radiation and matter domination as a natural proxy for internal time progression. This entropic approach provides a unified view linking quantum foundations, thermodynamic irreversibility, and cosmological evolution. We also discuss interpretational subtleties, clarifying how entropic time differs from coordinate time and under what conditions it defines a meaningful temporal structure. We emphasize that the entropic time, $\tau ,$ $τ,$ provides an arrow and parametrization of change in relational regimes (Wheeler–DeWitt,Page–Wootters, KMS/modular frameworks), but it is not proposed as a universal bijective substitute for coordinate time, t.