Spin effects on the semiclassical trajectories of Dirac electrons

The relativistic semiclassical evolution of the position of an electron in the presence of an external electromagnetic field is studied in terms of a Newton equation that incorporates spin effects directly. This equation emerges from the Dirac equation and allows the identification of scenarios where spin effects are necessary to understand the main characteristics of the electron trajectories. It involves the eigenvalues of the non-Hermitian operator $\Sigma_{\mu\nu}F^{\mu\nu}$ with $\Sigma_{\mu\nu}$ and $F^{\mu\nu}$ as the spin and electromagnetic tensors. The formalism allows a deeper understanding on the physics behind known analytical solutions of the Dirac equation when translational dynamics decouples from spin evolution. As an illustrative example, it is applied to an electron immersed in an electromagnetic field which exhibits chiral symmetry and optical vortices. It is shown that the polarization of intense structured light beams can be used to suppress or enhance spin effects on the electron semiclassical trajectory; the latter case yields a realization of a Stern-Gerlach apparatus for an electronComment: 18 pages, 3 figure

Control of atomic transition rates via laser light shaping

A modular systematic analysis of the feasibility of modifying atomic transition rates by tailoring the electromagnetic field of an external coherent light source is presented. The formalism considers both the center of mass and internal degrees of freedom of the atom, and all properties of the field: frequency, angular spectrum, and polarization. General features of recoil effects for internal forbidden transitions are discussed. A comparative analysis of different structured light sources is explicitly worked out. It includes spherical waves, Gaussian beams, Laguerre-Gaussian beams, and propagation invariant beams with closed analytical expressions. It is shown that increments in the order of magnitude of the transition rates for Gaussian and Laguerre-Gaussian beams, with respect to those obtained in the paraxial limit, requires waists of the order of the wavelength, while propagation invariant modes may considerably enhance transition rates under more favorable conditions. For transitions that can be naturally described as modifications of the atomic angular momentum, this enhancement is maximal (within propagation invariant beams) for Bessel modes, Mathieu modes can be used to entangle the internal and center of mass involved states, and Weber beams suppress this kind of transitions unless they have a significant component of odd modes. However, if a recoil effect of the transition with an adequate symmetry is allowed, the global transition rate (center of mass and internal motion) can also be enhanced using Weber modes. The global analysis presented reinforces the idea that a better control of the transitions between internal atomic states requires both a proper control of the available states of the atomic center of mass, and shaping of the background electromagnetic field.Comment: 23 pages, 3 figure

Implications of maximal Jarlskog invariant and maximal CP violation

We argue here why CP violating phase Phi in the quark mixing matrix is maximal, that is, Phi=90 degrees. In the Standard Model CP violation is related to the Jarlskog invariant J, which can be obtained from non commuting Hermitian mass matrices. In this article we derive the conditions to have Hermitian mass matrices which give maximal Jarlskog invariant J and maximal CP violating phase Phi. We find that all squared moduli of the quark mixing elements have a singular point when the CP violation phase Phi takes the value Phi=90 degrees. This special feature of the Jarlskog invariant J and the quark mixing matrix is a clear and precise indication that CP violating Phase Phi is maximal in order to let nature treat democratically all of the quark mixing matrix moduli.Comment: 25 pages, 3 figures, revte

On the lower semicontinuity of the ADM mass

The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetry assumptions) for $n=3$, and second, assuming rotational symmetry, for weak convergence of the associated canonical embeddings into Euclidean space, for $n \geq 3$. We also apply recent results of LeFloch and Sormani to deal with the rotationally symmetric case, with respect to a pointed type of intrinsic flat convergence. We provide several examples, one of which demonstrates that the positive mass theorem is implied by a statement of the lower semicontinuity of the ADM mass.Comment: 18 pages, 4 figure

Invariants of the harmonic conformal class of an asymptotically flat manifold

Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that every metric in $[g]_h$ has the same pointwise sign of scalar curvature. For this reason, the harmonic conformal class appears in the study of general relativity, where scalar curvature is related to energy density. Our purpose is to introduce and study invariants of the harmonic conformal class. These invariants are closely related to constrained geometric optimization problems involving hypersurface area-minimizers and the ADM mass. In the final section, we discuss possible applications of the invariants and their relationship with zero area singularities and the positive mass theorem.Comment: 26 pages, 2 figure

Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass

Quite a number of distinct versions of Bartnik's definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary conditions (of which there are three variants) and the non-degeneracy or "no-horizon" condition (at least six variants). To address the boundary conditions, we show that given a 3-dimensional region $\Omega$ of nonnegative scalar curvature ($R \geq 0$) extended in a Lipschitz fashion across $\partial \Omega$ to an asymptotically flat 3-manifold with $R \geq 0$ (also holding distributionally along $\partial \Omega$), there exists a smoothing, arbitrarily small in $C^0$ norm, such that $R \geq 0$ and the geometry of $\Omega$ are preserved, and the ADM mass changes only by a small amount. With this we are able to show that the three boundary conditions yield equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also discuss subtleties pertaining to the various non-degeneracy conditions and produce a nontrivial inequality between a no-horizon version of the Bartnik mass and Bray's replacement of this with the outward-minimizing condition.Comment: 25 pages, 3 figure

Lower semicontinuity of the ADM mass in dimensions two through seven

The semicontinuity phenomenon of the ADM mass under pointed (i.e., local) convergence of asymptotically flat metrics is of interest because of its connections to nonnegative scalar curvature, the positive mass theorem, and Bartnik's mass-minimization problem in general relativity. In this paper, we extend a previously known semicontinuity result in dimension three for $C^2$ pointed convergence to higher dimensions, up through seven, using recent work of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality of H. Bray and D. Lee). For a technical reason, we restrict to the case in which the limit space is asymptotically Schwarzschild. In a separate result, we show that semicontinuity holds under weighted, rather than pointed, $C^2$ convergence, in all dimensions $n \geq 3$, with a simpler proof independent of the positive mass theorem. Finally, we also address the two-dimensional case for pointed convergence, in which the asymptotic cone angle assumes the role of the ADM mass.Comment: 23 pages. Comments welcome

Convergence of the probability of large deviations in a model of correlated random variables having compact-support QQ-Gaussians as limiting distributions

We consider correlated random variables $X_1,\dots,X_n$ taking values in $\{0,1\}$ such that, for any permutation $\pi$ of $\{1,\dots,n\}$, the random vectors $(X_1,\dots,X_n)$ and $(X_{\pi(1)},\dots,X_{\pi(n)})$ have the same distribution. This distribution, which was introduced by Rodr\'iguez et al (2008) and then generalized by Hanel et al (2009), is scale-invariant and depends on a real parameter $\nu>0$ ($\nu\to\infty$ implies independence). Putting $S_n=X_1+\cdots+X_n$, the distribution of $S_n-n/2$ approaches a $Q$-Gaussian distribution with compact support ($Q=1-1/(\nu-1)<1$) as $n$ increases, after appropriate scaling. In the present article, we show that the distribution of $S_n/n$ converges, as $n\to\infty$, to a beta distribution with both parameters equal to $\nu$. In particular, the law of large numbers does not hold since, if $0\le x<1/2$, then $\mathbb{P}(S_n/n\le x)$, which is the probability of the event $\{S_n/n\le x\}$ (large deviation), does not converges to zero as $n\to\infty$. For $x=0$ and every real $\nu>0$, we show that $\mathbb{P}(S_n=0)$ decays to zero like a power law of the form $1/n^\nu$ with a subdominant term of the form $1/n^{\nu+1}$. If $00$ is an integer, we show that we can analytically find upper and lower bounds for the difference between $\mathbb{P}(S_n/n\le x)$ and its ($n\to\infty$) limit. We also show that these bounds vanish like a power law of the form $1/n$ with a subdominant term of the form $1/n^2$.Comment: 8 pages, 6 figures. Accepted for publication in Journal of Mathematical Physic

Paradoxical probabilistic behavior for strongly correlated many-body classical systems

Using a simple probabilistic model, we illustrate that a small part of a strongly correlated many-body classical system can show a paradoxical behavior, namely asymptotic stochastic independence. We consider a triangular array such that each row is a list of $n$ strongly correlated random variables. The correlations are preserved even when $n\to\infty$, since the standard central limit theorem does not hold for this array. We show that, if we choose a fixed number $m<n$ of random variables of the $n$th row and trace over the other $n-m$ variables, and then consider $n\to\infty$, the $m$ chosen ones can, paradoxically, turn out to be independent. However, the scenario can be different if $m$ increases with $n$. Finally, we suggest a possible experimental verification of our results near criticality of a second-order phase transition.Comment: 5 pages, 7 figure

Electromagnetic waves in uniaxial crystals: General formalism with an application to Bessel beams

We present a mathematical formalism describing the propagation of a completely general electromagnetic wave in a birefringent medium. Analytic formulas for the refraction and reflection from a plane interface are obtained. As a particular example, a Bessel beam impinging at an arbitrary angle is analyzed in detail. Some numerical results showing the formation and destruction of optical vortices are presented.Comment: 18 pages, 6 figure