A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity
arXiv:2302.08651 · doi:10.1142/S0218271823500220
Abstract
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-ε 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.
arXiv comment26 pages, 2 tables. Version to appear in Int. J. Mod. Phys. D. arXiv admin note: text overlap with arXiv:2112.00218
References in corpus (26)
- Reference frames, superselection rules, and quantum information
- Hawking Radiation and Black Hole Thermodynamics
- Quantum mechanics and the covariance of physical laws in quantum reference frames
- Fundamental decoherence from quantum gravity: a pedagogical review
- Ricci flow and black holes
- Physics within a quantum reference frame
- Fixed Points of Nonlinear Sigma Models in d>2
- A Renormalization Group Approach to Relativistic Cosmology
- Toy Model for a Relational Formulation of Quantum Theory
- Field Theory Entropy, the -theorem and the Renormalization Group
- The Averaging Problem in Cosmology and Macroscopic Gravity
- On Relativistic Generalization of Perelman's W-entropy and Statistical Thermodynamic Description of Gravitational Fields
- On the geometry of Riemannian manifolds with density
- On Cosmological Implications of Gravitational Trace Anomaly
- Non-compact nonlinear sigma models
- Geometric flows and black hole entropy
- Energy, entropy and the Ricci flow
- The Cosmological Constant Problem and Re-interpretation of Time
- Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems
- The Cosmological Constant Problem and Quantum Spacetime Reference Frame
- Entropy functionals and thermodynamics of relativistic geometric flows, stationary quasi-periodic Ricci solitons, and gravity
- Ricci Flow Approach to The Cosmological Constant Problem
- Ricci flow deformation of cosmological initial data sets
- Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime
- Trace Anomaly, Perelman's Functionals and the Cosmological Constant
- Implementation of the Quantum Equivalence Principle