A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity

The paper proposes a statistical fields theory of quantum reference frame underlying the Perelman's analogies between his formalism of the Ricci flow and the thermodynamics. The theory is based on a d=4-{\epsilon} quantum non-linear sigma model, interpreted as a quantum reference frame system which a to-be-studied quantum system is relative to. The statistic physics and thermodynamics of the quantum frame fields is studied by the density matrix obtained by the Gaussian approximation. The induced Ricci flow of the frame fields and the Ricci-DeTurck flow of the frame fields associated with the density matrix is deduced. In this framework, the diffeomorphism anomaly of the theory has a deep thermodynamic interpretation. The trace anomaly is related to a Shannon entropy in terms of the density matrix, which monotonically flows and achieves its maximal value at the flow limit, called the Gradient Shrinking Ricci Soliton (GSRS), corresponding to a thermal equilibrium state of spacetime. A relative Shannon entropy w.r.t. the maximal entropy gives a statistical interpretation to Perelman's partition function, which is also monotonic and gives an analogous H-theorem to the statistical frame fields system. A temporal static 3-space of a GSRS 4-spacetime is also a GSRS in lower 3-dimensional, we find that it is in a thermal equilibrium state, and Perelman's analogies between his formalism and the thermodynamics of the frame fields in equilibrium can be explicitly given in the framework. Extending the validity of the Equivalence Principle to the quantum level, the quantum frame fields theory at low energy gives an effective theory of gravity, a scale dependent Einstein-Hilbert action plus a cosmological constant is recovered. As a possible underlying microscopic theory of gravity, the theory is also applied to understand the thermodynamics of the Schwarzschild black hole.