We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3+1)dimensional spacetimes. We ascribe this entropy to the non-trivial topology of the space-like sections Σ of the horizon. This is not forbidden by topological censorship, since all the known energy inequalities needed to prove the spherical topology of Σ are violated in quantum theory. We choose the systoles of Σ to encode its complexity, which gives rise to the black hole entropy. We present hand-waving reasons why the entropy of the black hole can be considered as a function of the volume entropy of Σ. We focus on the limiting case of Σ having a large genus.

The issue of how time reversible microscopic dynamics gives rise to macroscopic irreversible processes has been a recurrent issue in Physics since the time of Boltzmann whose ideas shaped, and essentially resolved, such an apparent contradiction. Following Boltzmann's spirit and ideas, but employing Gibbs's approach, we advance the view that macroscopic irreversibility of Hamiltonian systems of many degrees of freedom can be also seen as a result of the symplectic non-squeezing theorem.Comment: 20 pages. No figures. Standard LaTeX2e. Version 2: clarification of a point and correction of typos. To be published in Physica

Abstract:We address the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.

We present an argument whose goal is to trace the origin of the macroscopically irreversible behavior of Hamitonian systems of many degrees of freedom. We use recent flexibility and rigidity results of symplectic embeddings, quantified via the (stabilized) Fibonacci and Pell staircases, to encode the underlying breadth of the possible initial conditions, which alongside the multitude of degrees of freeedom of the underlying system give rise to time-irreversibility.

We propose an alternative definition for a Tsallis entropy composition-inspired Fourier transform, which we call "τ q -Fourier transform". We comment about the underlying "covariance" on the set of algebraic fields that motivates its introduction. We see that the definition of the τ q -Fourier transform is automatically invertible in the proper context. Based on recent results in Fourier analysis, it turns that the τ q -Fourier transform is essentially unique under the assumption of the exchange of the point-wise product of functions with their convolution.