Microscopic Origin of Page Curve

Abstract: Applications of the Ryu-Takayanagi formula to evaporating black holes are known to reproduce the Page curve, although leaving open the microscopic origin of the necessary black hole degrees of freedom responsible for the entanglement and microstates. Here, we fill this gap by utilizing a recently proven generalized Ryu-Takayanagi formula which keeps the contact to the theory's Hamiltonian phase space. Precisely, we show that in any diffeomorphism invariant field theory containing stationary black holes with bifurcate Killing horizons the phase space states distinguishable by their Hamiltonian surface charges over the bifurcation surface must provide the degrees of freedom responsible for the black hole's Wald entropy. This result is part of a larger program where measurable quantities of a quantum theory are expressed as weighted sums over paths in the theory's phase space. Suited reorganization tools for those sums make then physical phenomena evident which might be obscured in a naive summation relying on spacetime notions such as field configuration spaces and Feynman diagrams. Besides presenting this result, we provide here a conceptual overview of the program as an entry point for unfamiliar readers explaining the central notion of the program and why it reveals the results obtained so far. These include especially the gravitational entropy bound and the here applied generalized Ryu-Takayanagi prescription shown earlier within the program. Using the information paradox as a leitmotif, our proposed program uncovers black hole hair that may be hidden within a naive spacetime interpretation.

• The paper demonstrates that black hole microstates, captured by Hamiltonian phase space surface charges, account for the Page curve via a generalized Ryu-Takayanagi formula.
• It introduces the possifold method to reorganize the sum over histories in diffeomorphism-invariant field theories, revealing phase space configurations overlooked by standard approaches.
• The work reinterprets the information paradox by showing that large gauge transformations encode essential degrees of freedom for unitary black hole evaporation.

Microscopic Foundations of the Page Curve in Diffeomorphism-Invariant Field Theories

Introduction

This work addresses a core question in quantum gravity: the microscopic origin of the Page curve in black hole evaporation. While the Ryu-Takayanagi (RT) formula and its generalizations reproduce the Page curve and thus unitarity within gravitational path integrals, the explicit identification of microstates responsible for black hole entropy—and their role in emergent entanglement and information recovery—remains unresolved in conventional approaches. This paper applies a recently proven generalized Ryu-Takayanagi formula [Averin, (Averin, 20 Aug 2025)] that establishes a concrete link between the Hamiltonian phase space of diffeomorphism-invariant field theories and black hole entropy, directly addressing the problem of microstate identification.

Critique of Naive Spacetime Summation and Motivation for Possifolds

Standard perturbative quantum field theory is formulated as a sum over histories of fields on spacetime, typically organized via Feynman diagrams. This organization is efficient for some observables but can obscure nonperturbative phenomena, such as confinement or black hole entropy, which are highly sensitive to the entanglement structure and the gauge redundancies of the underlying theory.

The central methodological advance advocated is the "possifold" concept: organizing the quantum mechanical summation not over spacetime histories, but over appropriately bundled subsets (“possifolds”) in the Hamiltonian phase space. This abstraction enables identification of microphysical phenomena that can remain invisible in naive Feynman diagramatics, specifically in questions concerning internal structure and state-counting in gravitational systems.

The paper thus treats the sum-over-histories formula

$$O = \sum_{\text{possibilities}} (\text{weight factors})$$

as the foundational structure of quantum theory, and focuses on different possible organizations of "possibilities" to bring out the desired physical features. For gravity, it is argued that the standard spacetime-based organization omits key microstates intrinsic to the theory's phase space.

Generalized Ryu-Takayanagi Formula and Phase Space Structure

The critical technical input is a generalized Ryu-Takayanagi prescription rigorously derived for diffeomorphism-invariant theories [Averin, (Averin, 20 Aug 2025)]. Unlike the AdS/CFT-based RT prescriptions, this approach computes entanglement entropy as a sum over distinct Hamiltonian phase space paths, allowing for an explicit connection between canonical coordinates and entanglement.

The main result states that in any diffeomorphism-invariant field theory with stationary black holes and bifurcate Killing horizons, the phase space states distinguishable by their Hamiltonian surface charges evaluated on the bifurcation surface must provide the degrees of freedom responsible for the black hole's Wald entropy. These surface charges are built from the theory's canonical structure (1-form $\Theta$), and generically include “soft hair” excitations and nontrivial edge modes.

Explicitly, the Wald entropy is derived as

$S_{\text{ent}} = \frac{1}{\hbar} K[\Phi^{(B)}]$

where $K$ is the Noether charge associated with the bifurcation surface $B$; $\Phi^{(B)}$ denotes the state in the relevant possifold submanifold of phase space. This quantity is then identified with the extremal value of the generalized RT functional for the relevant black hole solution.

Resolution of the Microstate Problem and Implications for the Information Paradox

The significance of this result is twofold:

1. No Need for Extra Microstates: Diffeomorphism invariance itself enforces the existence of sufficient Hilbert space structure to account for the entropy of stationary black holes. The necessary microstates are realized as those distinguishable by their bifurcation surface charges, i.e., those differing by “large” gauge or diffeomorphism transformations with support on the bifurcation surface but not trivialized by the Gauss law.
2. Reinterpretation of the Black Hole Interior and No-Hair Paradox: The construction clarifies that the interior region of a black hole, as conceptualized in the spacetime picture, does not correspond to independent canonical data in phase space. The information paradox and associated conundrums (e.g., the AMPS firewall) are thus reframed as artifacts of a naive spacetime interpretation. The interplay between gauge redundancy and physical degrees of freedom shows that large gauge transformations (including large diffeomorphisms) can change the physical state, even when they appear trivial in a spacetime description. This leads to a consistent picture of black hole “hair” encoded in the phase space rather than spacetime geometry.

The Page curve’s recovery via the RT prescription now has a concrete microscopic underpinning in the phase space structure: the black hole and its Hawking radiation are entangled in such a way that all diffeomorphism-invariant field theories automatically encode enough microstates to ensure unitary evaporation consistent with statistical mechanics.

Theoretical and Practical Consequences

The identification of black hole microstates as phase space configurations distinguished by surface charges establishes a concrete microphysical account of the Wald and Bekenstein-Hawking entropies within gravitational theories. This result has several ramifications:

• It strongly constrains the allowable configuration space for any candidate quantum gravity theory: the presence and structure of surface charges must be consistent with the gravitational entropy bound [Averin, (Averin, 2024)].
• The framework implies that microstate counting and entropy calculations in quantum gravity should focus not on geometric excitations (e.g., fuzzballs or stringy microstate constructions per se), but on the spectrum and organization of phase space surface charges.
• The “possifold” approach could be extended to analogous problems such as QCD confinement, mass gap, and UV finiteness in quantum gravity, as the program identifies how unitarity and entropy bounds reflect phase space geometries and the distribution of quantum distinguishable states.
• Apparent UV sensitivity in standard spacetime-based loop expansions may be an artifact of an inappropriate organization of the sum over configurations; the gravitational entropy bound restricts phase space contributions, potentially ameliorating UV divergences.

Future Developments

The analysis sets the stage for explicit computations and classifications of black hole microstates: a concrete list of physical states could in principle be constructed by evaluating surface charges for all physically distinct solutions in a given theory. It suggests that similar methods may resolve entropy puzzles in cosmological horizons and de Sitter space, and invites a reformulation of observables in quantum gravity in terms of phase space geometry.

Ongoing work is necessary to detail the enumeration of physically distinct states and to relate them with explicit semiclassical or quantum solutions in various settings.

Conclusion

This paper rigorously establishes the microscopic origin of the Page curve and black hole entropy within diffeomorphism-invariant field theories by leveraging a generalized Ryu-Takayanagi formula and the "possifold" organization of phase space. The identification of black hole microstates as distinguishable by bifurcation surface charges provides a concrete realization of entropic degrees of freedom in the gravitational context, resolves ambiguities inherent in the spacetime-based description, and clarifies the link between bulk unitarity and quantum statistical mechanics. This approach fundamentally grounds the statistical mechanics of horizons in canonical phase space and provides a blueprint for exploring similar phenomena in other gauge and gravitational contexts.