arXiv subjects

H. Falomir

Publications and source records attributed to H. Falomir.

At least 19 recordsLinked to original sources

arXiv2024

$\zeta$-function for a model with spectral dependent boundary conditions

We explore the meromorphic structure of the $\zeta$-function associated to the boundary eigenvalue problem of a modified Sturm-Liouville operator subject to spectral dependent boundary conditions at one end of a segment of length $l$. We find that it presents isolated simple poles which follow the general rule valid for second order differential operators subject to standard local boundary conditions. We employ our results to evaluate the determinant of the operator and the Casimir energy of the system it describes, and study its dependence on $l$ for both the massive and the massless cases.

hep-th↗

arXiv2018

Magnetic Seed and Cosmology as Quantum Hall Effect

In the framework of a bimetric model, we discuss a relation between the (modified) Friedmann equations and a mechanical system similar to the quantum Hall effect problem. Firstly, we show how these modified Friedmann equations are mapped to an anisotropic two-dimensional charged harmonic oscillator in the presence of a constant magnetic field, with the frequencies of the oscillator playing the role of the cosmological constants. This problem has two energy scales leading to the identification of two different regimes, namely, one dominated by the cosmological constants, with exponential expansions for the scale factors, and the other dominated by a magnetic seed, which would be responsible for both a component of dark energy and a primordial magnetic field. The latter regime would be described by a (nonperturbative) mapping between the cosmological evolution and the quantum Hall effect.

hep-th↗

arXiv2018

Classical Noncommutative Bicosmology Model

We propose a bicosmology model which is the classical analog of noncommutative quantum mechanics. From this point of view the sources of the modified FRW equations are dark energy ones governed by a Chapligyn's equation state. The parameters of noncommutativity $θ$ and $B$ are interpreted in terms of the Planck area and a like-magnetic field, presumably the magnetic seed of magnetogenesis.

hep-th↗

arXiv2017

Inflation without Inflaton: A Model for Dark Energy

The interaction between two initially causally disconnected regions of the universe is studied using analogies of non-commutative quantum mechanics and deformation of Poisson manifolds. These causally disconnect regions are governed by two independent Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) metrics with scale factors $a$ and $b$ and cosmological constants $Λ_a$ and $Λ_b$, respectively. The causality is turned on by positing a non-trivial Poisson bracket $[ {\cal P}_α, {\cal P}_β ] =ε_{αβ}\fracκ{G}$, where $G$ is Newton's gravitational constant and $κ$ is a dimensionless parameter. The posited deformed Poisson bracket has an interpretation in terms of 3-cocycles, anomalies and Poissonian manifolds. The modified FLRW equations acquire an energy-momentum tensor from which we explicitly obtain the equation of state parameter. The modified FLRW equations are solved numerically and the solutions are inflationary or oscillating depending on the values of $κ$. In this model the accelerating and decelerating regime may be periodic. The analysis of the equation of state clearly shows the presence of dark energy. By completeness, the perturbative solution for $κ\ll1 $ is also studied.

gr-qc↗

arXiv2016

Non-Abelian Monopoles as the Origin of Dark Matter

We suggest that dark matter may be partially constituted by a dilute 't Hooft-Polyakov monopoles gas. We reach this conclusion by using the Georgi-Glashow model coupled to a dual kinetic mixing term $ F{\tilde {\cal G}}$ where $F$ is the electromagnetic field and ${\cal G}$ the 't Hooft tensor. We show that these monopoles carry both (Maxwell) electric and (Georgi-Glashow) magnetic charges and the electric charge quantization condition is modified in terms of a dimensionless real parameter. This parameter could be determined from milli-charged particle experiments.

hep-th↗

arXiv2015

On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras $sl(2,\mathbb{R})$ or $su(2)$ according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential. PACS: 03.65.-w; 03.65.Fd MSC: 81R05; 20C35; 22E70

math-ph↗

arXiv2013

Boundaries in the Moyal plane

We study the oscillations of a scalar field on a noncommutative disc implementing the boundary as the limit case of an interaction with an appropriately chosen confining background. The space of quantum fluctuations of the field is finite dimensional and displays the rotational and parity symmetry of the disc. We perform a numerical evaluation of the (finite) Casimir energy and obtain similar results as for the fuzzy sphere and torus.

hep-th↗

arXiv2012

Noncommutativity in (2+1)-dimensions and the Lorentz group

In this article we considered models of particles living in a three-dimensional space-time with a nonstandard noncommutativity induced by shifting canonical coordinates and momenta with generators of a unitary irreducible representation of the Lorentz group. The Hilbert space gets the structure of a direct product with the representation space, where we are able to construct operators which realize the algebra of Lorentz transformations. We study the modified Landau problem for both Schrödinger and Dirac particles, whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of the dynamical variables from the ones of the usual problem in the normal space. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters. We find no constraint between the parameters referring to no-commutativity in coordinates and momenta but they rather play similar roles. Since the representation space of the unitary irreducible representations SL(2,R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.

hep-th↗

arXiv2012

Spectral functions of non essentially selfadjoint operators

One of the many problems to which J.S. Dowker devoted his attention is the effect of a conical singularity in the base manifold on the behavior of the quantum fields. In particular, he studied the small-$t$ asymptotic expansion of the heat-kernel trace on a cone and its effects on physical quantities, as the Casimir energy. In this article we review some peculiar results found in the last decade, regarding the appearance of non-standard powers of $t$, and even negative integer powers of $\log{t}$, in this asymptotic expansion for the selfadjoint extensions of some symmetric operators with singular coefficients. Similarly, we show that the $ζ$-function associated to these selfadjoint extensions presents an unusual analytic structure.

math-ph↗

arXiv2012

Graphene and non-Abelian quantization

In this article we employ a simple nonrelativistic model to describe the low energy excitation of graphene. The model is based on a deformation of the Heisenberg algebra which makes the commutator of momenta proportional to the pseudo-spin. We solve the Landau problem for the resulting Hamiltonian which reduces, in the large mass limit while keeping fixed the Fermi velocity, to the usual linear one employed to describe these excitations as massless Dirac fermions. This model, extended to negative mass, allows to reproduce the leading terms in the low energy expansion of the dispersion relation for both nearest and next-to-nearest neighbor interactions. Taking into account the contributions of both Dirac points, the resulting Hall conductivity, evaluated with a $ζ$-function approach, is consistent with the anomalous integer quantum Hall effect found in graphene. Moreover, when considered in first order perturbation theory, it is shown that the next-to-leading term in the interaction between nearest neighbor produces no modifications in the spectrum of the model while an electric field perpendicular to the magnetic field produces just a rigid shift of this spectrum. PACS: 03.65.-w, 81.05.ue, 73.43.-f

math-ph↗

arXiv2011

Spin Non-commutativity and the Three-Dimensional Harmonic Oscillator

A three-dimensional harmonic oscillator with spin non-commutativity in the phase space is considered. The system has a regular symplectic structure and by using supersymmetric quantum mechanics techniques, the ground state is calculated exactly. We find that this state is infinitely degenerate and it has explicit spontaneous broken symmetry. Analyzing the Heisenberg equations, we show that the total angular momentum is conserved.

hep-th↗

arXiv2011

Aharonov-Bohm effect in a Class of Noncommutative Theories

The Aharonov-Bohm effect including spin-noncommutative effects is considered. At linear order in $θ$, the magnetic field is gauge invariant although spatially strongly anisotropic. Despite this anisotropy, the Schrödinger-Pauli equation is separable through successive unitary transformations and the exact solution is found. The scattering amplitude is calculated and compared with the usual case. In the noncommutative Aharonov-Bohm case the differential cross section is independent of $θ$.

hep-th↗

arXiv2011

Observability of an induced electric dipole moment of the neutron from nonlinear QED

It has been shown recently that a neutron placed in an external quasistatic electric field develops an induced electric dipole moment $\mathbf{p}_{\mathrm{IND}}$ due to quantum fluctuations in the QED vacuum. A feasible experiment which could detect such an effect is proposed and described here. It is shown that the peculiar angular dependence of $\mathbf{p}_{\mathrm{IND}}$ on the orientation of the neutron spin leads to a characteristic asymmetry in polarized neutron scattering by heavy nuclei. This asymmetry can be of the order of $10^{-3}$ for neutrons with epithermal energies. For thermalized neutrons from a hot moderator one still expects experimentally accessible values of the order of $10^{-4}$. The contribution of the induced effect to the neutron scattering length is expected to be only one order of magnitude smaller than that due to the neutron polarizability from its quark substructure. The experimental observation of this scattering asymmetry would be the first ever signal of nonlinearity in electrodynamics due to quantum fluctuations in the QED vacuum.

hep-ph↗

arXiv2010

Perturbation Theory on the Non-Commutative Plane with a Singular Potential

In this article we study the problem of a non-relativistic particle in the presence of a singular potential in the noncommutative plane. The potential contains a term proportional to $1/R^2$, where $R^2$ is the squared distance to the origin in the noncommutative plane. We find that the spectrum of energies is non analytic in the noncommutativity parameter $θ$.

math-ph↗

arXiv2009

QED vacuum fluctuations and induced electric dipole moment of the neutron

Quantum fluctuations in the QED vacuum generate non-linear effects, such as peculiar induced electromagnetic fields. In particular, we show here that an electrically neutral particle, possessing a magnetic dipole moment, develops an induced electric dipole-type moment with unusual angular dependence, when immersed in a quasistatic, constant external electric field. The calculation of this effect is done in the framework of the Euler-Heisenberg effective QED Lagrangian, corresponding to the weak field asymptotic expansion of the effective action to one-loop order. It is argued that the neutron might be a good candidate to probe this signal of non-linearity in QED.

hep-ph↗

arXiv2009

Magnetic-Dipole Spin Effects in Noncommutative Quantum Mechanics

A general three-dimensional noncommutative quantum mechanical system mixing spatial and spin degrees of freedom is proposed. The analogous of the harmonic oscillator in this description contains a magnetic dipole interaction and the ground state is explicitly computed and we show that it is infinitely degenerated and implying a spontaneous symmetry breaking. The model can be straightforwardly extended to many particles and the main above properties are retained. Possible applications to the Bose-Einstein condensation with dipole-dipole interactions are briefly discussed.

hep-th↗

arXiv2008

Induced electromagnetic fields in non-linear QED

The Euler-Heisenberg effective Lagrangian is used to obtain general expressions for electric and magnetic fields induced by non-linearity, to leading order in the non-linear expansion parameter, and for quasistatic situations. These expressions are then used to compute the induced electromagnetic fields due to a spherical shell with uniform charge distribution on the surface, in the presence of an external constant magnetic field. The induced electric field contains several multipole terms with unusual angular dependences. Most importantly, the leading term of the induced magnetic field is due to an induced magnetic dipole moment.

hep-ph↗

arXiv2008

Non-commutative Supersymmetric Quantum Mechanics

General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two and three dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in physical systems with potentials involving spin and non-local interactions.

hep-th↗