J. L. G. Santander

AN ENTROPIC PICTURE OF EMERGENT QUANTUM MECHANICS

Quantum mechanics emerges a la Verlinde from a foliation of ℝ3 by holographic screens, when regarding the latter as entropy reservoirs that a particle can exchange entropy with. This entropy is quantized in units of Boltzmanns constant kB. The holographic screens can be treated thermodynamically as stretched membranes. On that side of a holographic screen where spacetime has already emerged, the energy representation of thermodynamics gives rise to the usual quantum mechanics. A knowledge of the different surface densities of entropy flow across all screens is equivalent to a knowledge of the quantum-mechanical wavefunction on ℝ3. The entropy representation of thermodynamics, as applied to a screen, can be used to describe quantum mechanics in the absence of spacetime, that is, quantum mechanics beyond a holographic screen, where spacetime has not yet emerged. Our approach can be regarded as a formal derivation of Plancks constant ℏ from Boltzmanns constant kB.

EMERGENT QUANTUM MECHANICS AS A CLASSICAL, IRREVERSIBLE THERMODYNAMICS

We present an explicit correspondence between quantum mechanics and the classical theory of irreversible thermodynamics as developed by Onsager, Prigogine et al. Our correspondence maps irreversible Gaussian Markov processes into the semiclassical approximation of quantum mechanics. Quantum-mechanical propagators are mapped into thermodynamical probability distributions. The Feynman path integral also arises naturally in this setup. The fact that quantum mechanics can be translated into thermodynamical language provides additional support for the conjecture that quantum mechanics is not a fundamental theory but rather an emergent phenomenon, i.e., an effective description of some underlying degrees of freedom.

RICCI FLOW, QUANTUM MECHANICS AND GRAVITY

It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantization, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space ℂℙd-1 of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base ℂℙd-1. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.

A Holographic Map of Action onto Entropy

We propose a holographic correspondence between the action integral I describing the mechanics of a finite number of degrees of freedom in the bulk, and the entropy S of the boundary (a holographic screen) enclosing that same volume. The action integral must be measured in units of (i times) Plancks constant, while the entropy must be measured in units of Boltzmanns constant. In this way we are led to an intriguing relation between the second law of thermodynamics and the uncertainty principle of quantum mechanics.

On the noncommutative eikonal

We study the Eikonal approximation to quantum mechanics on the Moyal plane. Instead of using a star product, the analysis is carried out in terms of operator-valued wavefunctions depending on noncommuting, operator-valued coordinates.

On the Ricci flow and emergent quantum mechanics

The Ricci flow equation of a conformally flat Riemannian metric on a closed 2–dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton–Jacobi equation for a point particle subject to a potential function that is proportional to the Ricci scalar curvature of configuration space. This allows one to obtain Schrodinger quantum mechanics from Perelmans action functional: the quantum–mechanical wavefunction is the exponential of i times the conformal factor of the metric on configuration space. We explore links with the recently discussed emergent quantum mechanics.

A MECHANICS FOR THE RICCI FLOW

We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent Hamilton–Jacobi equation of the mechanics so defined.

Remarks on the Representation Theory of the Moyal Plane

We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.The authors would like to thank the referee for constructive suggestions. J. M. Isidro thanks Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut) Golm, Germany, for hospitality. This paper has been supported by Universidad Politecnica de Valencia under Grant PAID-06-09, and by Generalitat Valenciana Spain.

Heat transfer between a gas and an ultralow thermal conductivity porous structure

The analytical solution of a one-dimensional problem of heat transfer between an inert gas and an ultralow thermal conductivity porous medium is obtained by means of the Laplace transform method. The solution shows a sharp heat front propagation through the medium during its heating or cooling depending on its initial temperature.