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P. Fernandez de Cordoba

Publications and source records attributed to P. Fernandez de Cordoba.

At least 37 records · Page 2Linked to original sources
arXiv2014

Schroedinger vs. Navier-Stokes

Quantum mechanics has been argued to be a coarse-graining of some underlying deterministic theory. Here we support this view by establishing a map between certain solutions of the Schroedinger equation, and the corresponding solutions of the irrotational Navier-Stokes equation for viscous fluid flow. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state-dependent) viscosity of this fluid is proportional to Planck's constant, while the volume density of entropy is proportional to Boltzmann's constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information-loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier-Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation).

math-ph↗
arXiv2013

The irreversible quantum

We elaborate on the existing notion that quantum mechanics is an emergent phenomenon, by presenting a thermodynamical theory that is dual to quantum mechanics. This dual theory is that of classical irreversible thermodynamics. The linear regime of irreversibility considered here corresponds to the semiclassical approximation in quantum mechanics. An important issue we address is how the irreversibility of time evolution in thermodynamics is mapped onto the quantum-mechanical side of the correspondence.

quant-ph↗
arXiv2013

Emergent quantum mechanics as a thermal ensemble

It has been argued that gravity acts dissipatively on quantum-mechanical systems, inducing thermal fluctuations that become indistinguishable from quantum fluctuations. This has led some authors to demand that some form of time irreversibility be incorporated into the formalism of quantum mechanics. As a tool towards this goal we propose a thermodynamical approach to quantum mechanics, based on Onsager's classical theory of irreversible processes and on Prigogine's nonunitary transformation theory. An entropy operator replaces the Hamiltonian as the generator of evolution. The canonically conjugate variable corresponding to the entropy is a dimensionless evolution parameter. Contrary to the Hamiltonian, the entropy operator is not a conserved Noether charge. Our construction succeeds in implementing gravitationally-induced irreversibility in the quantum theory.

math-ph↗
arXiv2012

Emergence from irreversibility

The emergent nature of quantum mechanics is shown to follow from a precise correspondence with the classical theory of irreversible thermodynamics.

math-ph↗
arXiv2012

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.

math-ph↗
arXiv2011

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) Planck's constant, while the entropy must be measured in units of Boltzmann's constant. In this way we are led to an intriguing relation between the second law of thermodynamics and the uncertainty principle of quantum mechanics.

hep-th↗
arXiv2011

An entropic picture of emergent quantum mechanics

Quantum mechanics emerges a la Verlinde from a foliation of space by holographic screens, when regarding the latter as entropy reservoirs that a particle can exchange entropy with. This entropy is quantised in units of Boltzmann's constant k. 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 space. 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 Planck's constant h from Boltzmann's constant k.

hep-th↗
arXiv2010

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.

hep-th↗
arXiv2009

Positive curvature can mimic a quantum

We elaborate on the existing idea that quantum mechanics is an emergent phenomenon, in the form of a coarse-grained description of some underlying deterministic theory. We apply the Ricci flow as a technical tool to implement dissipation, or information loss, in the passage from an underlying deterministic theory to its emergent quantum counterpart. A key ingedient in this construction is the fact that the space of physically inequivalent quantum states (either pure or mixed) has positive Ricci curvature. This leads us to an interesting thermodynamical analogy of emergent quantum mechanics.

hep-th↗
arXiv2009

A note on the quantum-mechanical Ricci flow

We obtain Schroedinger quantum mechanics from Perelman's functional and from the Ricci flow equations of a conformally flat Riemannian metric on a closed 2-dimensional configuration space. We explore links with the recently discussed "emergent quantum mechanics".

hep-th↗
arXiv2009

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.

hep-th↗
arXiv2009

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 Schroedinger quantum mechanics from Perelman's 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.

hep-th↗
arXiv2008

The Ricci flow on Riemann surfaces

We establish a 1-to-1 relation between metrics on compact Riemann surfaces without boundary, and mechanical systems having those surfaces as configuration spaces.

hep-th↗
arXiv2008

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 prequantisation, 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 CP^{d-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 CP^{d-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.

hep-th↗
arXiv2008

Noncommutative space and the low-energy physics of quasicrystals

We prove that the effective low-energy, nonlinear Schroedinger equation for a particle in the presence of a quasiperiodic potential is the potential-free, nonlinear Schroedinger equation on noncommutative space. Thus quasiperiodicity of the potential can be traded for space noncommutativity when describing the envelope wave of the initial quasiperiodic wave.

math-ph↗
arXiv2008

A note on the quantum of time

Quantum mechanics rests on the assumption that time is a classical variable. As such, classical time is assumed to be measurable with infinite accuracy. However, all real clocks are subject to quantum fluctuations, which leads to the existence of a nonzero uncertainty in the time variable. The existence of a quantum of time modifies the Heisenberg evolution equation for observables. In this letter we propose and analyse a generalisation of Heisenberg's equation for observables evolving in real time (the time variable measured by real clocks), that takes the existence of a quantum of time into account. This generalisation of Heisenberg's equation turns out to be a delay-differential equation.

gr-qc↗
arXiv2007

Dirichlet branes and a cohomological definition of time flow

Dirichlet branes are objects whose transverse coordinates in space are matrix-valued functions. This leads to considering a matrix algebra or, more generally, a Lie algebra, as the classical phase space of a certain dynamics where the multiplication of coordinates, being given by matrix multiplication, is nonabelian. Further quantising this dynamics by means of a star-product introduces noncommutativity (besides nonabelianity) as a quantum h-deformation. The algebra of functions on a standard Poisson manifold is replaced with the universal enveloping algebra of the given Lie algebra. We define generalised Poisson brackets on this universal enveloping algebra, examine their properties, and conclude that they provide a natural framework for dynamical setups (such as coincident Dirichlet branes) where coordinates are matrix-valued, rather than number-valued, functions.

math-ph↗