Nonrelativistic Particle Research Articles

An artificial boundary approach for short-ranged interactions

Real physical systems are only understood, experimentally or theoretically, to a finite resolution so in their analysis there is generally an ignorance of possible short-range phenomena. It is also well-known that the boundary conditions of wavefunctions and fields can be used to model short-range interactions when those interactions are expected, a priori. Here, a real-space approach is described wherein an artificial boundary of ignorance is imposed to explicitly exclude from analysis the region of a system wherein short-distance effects may be obscure. The (artificial) boundary conditions encode those short-distance effects by parameterizing the possible UV completions of the wavefunction. Since measurable quantities, such as spectra and cross sections, must be independent of the position of the artificial boundary, the boundary conditions must evolve with the radius of the boundary in a particular way. As examples of this approach, an analysis is performed of the non-relativistic free particle, harmonic oscillator, and Coulomb potential, and some known results for point-like (contact) interactions are recovered, however from a novel perspective. Generically, observables differ from their canonical values and symmetries are anomalously broken compared to those of idealized models. Connections are made to well-studied physical systems, such as the binding of light nuclei and cold atomic systems. This method is arguably more physically transparent and mathematically easier to use than other techniques that require the regularization and renormalization of delta-function potentials, and may offer further generalizations of practical use.

Replacing the Singlet Spinor of the EPR-B Experiment in the Configuration Space with Two Single-Particle Spinors in Physical Space

Recently, for spinless non-relativistic particles, Norsen, Marian and Oriols show that in the de Broglie-Bohm interpretation it is possible to replace the wave function in the configuration space by single-particle wave functions in physical space. In this paper, we show that this replacment of the wave function in the configuration space by single-particle functions in the 3D-space is also possible for particles with spin, in particular for the particles of the EPR-B experiment, the Bohm version of the Einstein-Podolsky-Rosen experiment.

Double General Point Interactions: Symmetry and Tunneling Times

We consider the one dimensional problem of a non-relativistic quantum particle scattering off a double barrier built from two generalized point interactions (each one characterized as a member of the four parameter family of point interactions). The properties of the double point barrier under parity transformations are investigated, using the distributional approach, and the constraints on the parameters necessary for the interaction to have a well-defined parity are obtained. We show that the limit of zero interbarrier distance of a renormalized odd arrangement with two $\delta'$ is either trivial or does not exist as a generalized point interaction. Finally, we specialize to double barriers with defined parity, calculate the phase and Salecker-Wigner-Peres clock times and argue that the emergence of the generalized Hartman effect is an artifact of the extreme opaque limit.

Remarks about the thermodynamics of astrophysical systems in mutual interaction and related notions

.Aspects concerning the thermodynamics of astrophysical systems are discussed, generally, and also more specifically those relating to astrophysical systems in mutual interaction (or the so-called open astrophysical systems). A special interest is devoted in this paper to clarifying several misconceptions that are still common in the recent literature, such as the direct application to the astrophysical scenario of notions and theoretical frameworks that were originally conceived to deal with extensive systems of everyday practice (large systems with short-range interactions). This discussion starts by reviewing the current understanding of the notion of negative heat capacity. Beyond this, to clarify its physical relevance, the conciliation of this notion with classical fluctuation theory is discussed, as well as equilibrium conditions concerning systems with negative heat capacities. These results prompt a revision of our understanding about critical phenomena, phase transitions and the so-called zeroth law of thermodynamics. Afterwards, general features about the thermodynamics of astrophysical systems are presented through the consideration of simple models available in the literature. Particular attention is devoted to the influence of evaporation on the macroscopic behavior of these systems. These antecedents are then applied to a critical approach towards the thermodynamics of astrophysical systems in mutual interaction. It is discussed that the long-range character of gravitation leads to the incidence of long-range correlations. This peculiarity imposes a series of important consequences, such as the non-separability of a single astrophysical structure into independent subsystems, the breakdown of additivity and conventional thermodynamic limit, a great sensibility of the macroscopic behavior to the external conditions, the restricted applicability of the so-called thermal contact in astrophysics, and hence, the non-relevance of conventional statistical ensembles in this scenario. To clarify how some of conventional notions and theoretical frameworks could be extended to open astrophysical systems, an exploratory study of a paradigmatic situation is presented: a binary astrophysical system. This analysis is carried out in the framework of the quadrupole approximation, which represents the lowest coupling among internal and collective degrees of freedom. Apparently, collective motions are responsible for a non-linear energy interchange among the astrophysical systems. This mechanism introduces some modifications in stationary and stability conditions for the thermodynamic equilibrium such as a generalization of Thirring’s stability condition for systems with negative heat capacities (1970 Z. Phys. 235 339). Additionally, the stability of collective motions of this binary astrophysical system is also discussed, which is related to the low energy thermodynamic behavior of the model discussed by Votyakov and colleagues (2002 Phys. Rev. Lett. 89 031101). The thermodynamic limit for self-gravitating gas of identical non-relativistic point particles is then derived and compared with other different proposals. The astrophysical counterpart of the Gibbs–Duhem relation is obtained and compared with the recent proposal of Latella and colleagues (2015 Phys. Rev. Lett. 114 230601). Finally, the incidence of non-extensivity during the merger of two identical astrophysical systems is analyzed. Contrary to the situation considered in the Gibbs paradox, the merger is an irreversible process that crucially depends on the existence (or non-existence) of the external gravitational influence of other systems.

A process algebra model of QED

The process algebra approach to quantum mechanics posits a finite, discrete, determinate ontology of primitive events which are generated by processes (in the sense of Whitehead). In this ontology, primitive events serve as elements of an emergent space-time and of emergent fundamental particles and fields. Each process generates a set of primitive elements, using only local information, causally propagated as a discrete wave, forming a causal space termed a causal tapestry. Each causal tapestry forms a discrete and finite sampling of an emergent causal manifold (space-time) M and emergent wave function. Interactions between processes are described by a process algebra which possesses 8 commutative operations (sums and products) together with a non-commutative concatenation operator (transitions). The process algebra possesses a representation via nondeterministic combinatorial games. The process algebra connects to quantum mechanics through the set valued process and configuration space covering maps, which associate each causal tapestry with sets of wave functions over M. Probabilities emerge from interactions between processes. The process algebra model has been shown to reproduce many features of the theory of non-relativistic scalar particles to a high degree of accuracy, without paradox or divergences. This paper extends the approach to a semi-classical form of quantum electrodynamics.

Electromagnetic radiation of charged particles in stochastic motion

The study of the Brownian motion of a charged particle in electric and magnetic fields fields has many important applications in plasma and heavy ions physics, as well as in astrophysics. In the present paper we consider the electromagnetic radiation properties of a charged non-relativistic particle in the presence of electric and magnetic fields, of an exterior non-electromagnetic potential, and of a friction and stochastic force, respectively. We describe the motion of the charged particle by a Langevin and generalized Langevin type stochastic differential equation. We investigate in detail the cases of the Brownian motion with or without memory in a constant electric field, in the presence of an external harmonic potential, and of a constant magnetic field. In all cases the corresponding Langevin equations are solved numerically, and a full description of the spectrum of the emitted radiation and of the physical properties of the motion is obtained. The Power Spectral Density (PSD) of the emitted power is also obtained for each case, and, for all considered oscillating systems, it shows the presence of peaks, corresponding to certain intervals of the frequency.

Searching for modified gravity: scale and redshift dependent constraints from galaxy peculiar velocities

We present measurements of both scale- and time-dependent deviations from the standard gravitational field equations. These late-time modifications are introduced separately for relativistic and non-relativistic particles, by way of the parameters $G_{\rm matter}(k,z)$ and $G_{\rm light}(k,z)$ using two bins in both scale and time, with transition wavenumber $0.01$ Mpc$^{-1}$ and redshift 1. We emphasize the use of two dynamical probes to constrain this set of parameters, galaxy power spectrum multipoles and the direct peculiar velocity power spectrum, which probe fluctuations on different scales. The multipole measurements are derived from the WiggleZ and BOSS Data Release 11 CMASS galaxy redshift surveys and the velocity power spectrum is measured from the velocity sub-sample of the 6-degree Field Galaxy Survey. We combine with additional cosmological probes including baryon acoustic oscillations, Type Ia SNe, the cosmic microwave background (CMB), lensing of the CMB, and the temperature--galaxy cross-correlation. Using a Markov Chain Monte Carlo likelihood analysis, we find the inferred best-fit parameter values of $G_{\rm matter}(k,z)$ and $G_{\rm light}(k,z)$ to be consistent with the standard model at the $95\%$ confidence level. Furthermore, accounting for the Alcock-Paczynski effect, we perform joint fits for the expansion history and growth index gamma; we measure $\gamma = 0.665 \pm 0.0669$ ($68\%$ C.L) for a fixed expansion history, and $\gamma = 0.73^{+0.08}_{-0.10}$ ($68\%$ C.L) when the expansion history is allowed to deviate from $\Lambda$CDM. With a fixed expansion history the inferred value is consistent with GR at the $95\%$ C.L; alternatively, a $2\sigma$ tension is observed when the expansion history is not fixed, this tension is worsened by the combination of growth and SNe data.

Dirac operator on the sphere with attached wires**Project partially financially supported by the Funds from the Government of the Russian Federation (Grant No. 074-U01), the Funds from the Ministry of Education and Science of the Russian Federation (GOSZADANIE 2014/190) (Grant Nos. 14.Z50.31.0031 and 1.754.2014/K), and the President Foundation of the Russian Federation (Grant No. MK-5001.2015.1).

An explicitly solvable model for tunnelling of relativistic spinless particles through a sphere is suggested. The model operator is constructed by an operator extensions theory method from the orthogonal sum of the Dirac operators on a semi-axis and on the sphere. The transmission coefficient is obtained. The dependence of the transmission coefficient on the particle energy has a resonant character. One observes pairs of the Breit–Wigner and the Fano resonances. It correlates with the corresponding results for a non-relativistic particle.

Leading QCD corrections for indirect dark matter searches: A fresh look

The annihilation of non-relativistic dark matter particles at tree level can be strongly enhanced by the radiation of an additional gauge boson. This is particularly true for the helicity-suppressed annihilation of Majorana particles, like neutralinos, into fermion pairs. Surprisingly, and despite the potentially large effect due to the strong coupling, this has so far been studied in much less detail for the internal bremsstrahlung of gluons than for photons or electroweak gauge bosons. Here, we aim at bridging that gap by presenting a general analysis of neutralino annihilation into quark anti-quark pairs and a gluon, allowing e.g. for arbitrary neutralino compositions and keeping the leading quark mass dependence at all stages in the calculation. We find in some cases largely enhanced annihilation rates, especially for scenarios with squarks being close to degenerate in mass with the lightest neutralino, but also notable distortions in the associated antiproton and gamma-ray spectra. Both effects significantly impact limits from indirect searches for dark matter and are thus important to be taken into account in, e.g., global scans. For extensive scans, on the other hand, full calculations of QCD corrections are numerically typically too expensive to perform for each point in parameter space. We present here for the first time an efficient, numerically fast implementation of QCD corrections, extendable in a straight-forward way to non-supersymmetric models, which avoids computationally demanding full one-loop calculations or event generator runs and yet fully captures the leading effects relevant for indirect dark matter searches. In this context, we also present updated constraints on dark matter annihilation from cosmic-ray antiproton data. Finally, we comment on the impact of our results on relic density calculations.

Discreteness of space from GUP in a weak gravitational field

Quantum gravity effects modify the Heisenberg's uncertainty principle to a generalized uncertainty principle (GUP). Earlier work showed that the GUP-induced corrections to the Schr\"odinger equation, when applied to a non-relativistic particle in a one-dimensional box, led to the quantization of length. Similarly, corrections to the Klein-Gordon and the Dirac equations, gave rise to length, area and volume quantizations. These results suggest a fundamental granular structure of space. In this work, it is investigated how spacetime curvature and gravity might influence this discreteness of space. In particular, by adding a weak gravitational background field to the above three quantum equations, it is shown that quantization of lengths, areas and volumes continue to hold. However, it should be noted that the nature of this new quantization is quite complex and under proper limits, it reduces to cases without gravity. These results suggest that quantum gravity effects are universal.

Stable compact motions of a particle driven by a central force in six-dimensional spacetime

We give a counterexample to the well-known Ehrenfest’s assertion that the existence of stable electromagnetic bound systems is impossible in spaces of more than three dimensions. If we require that the Maxwellian laws of electromagnetism be preserved for any even spacetime dimension, and that the dynamics as a whole be consistent, then the laws of mechanics must be amended by the addition of terms with higher derivatives. We consider a nonrelativistic particle with an acceleration-dependent Lagrangian which moves in an attractive 1/r3 potential in five-dimensional space. There are compactly supported motions whose projections on the SO(5)-reduced Hamiltonian system are Poisson equilibrium points. The nonlinearly stable equilibria correspond to physically stable motions over the direct product of two three-spheres in configuration space. The Energy-Casimir method turns out to be not appropriate for checking the stability. The studied system is shown to be stable through an analysis of numerical solutions to the equations of motion for small perturbations on the reduced phase space. This implies that falling to the center is prevented.

Novel Quantum Criticality in Two Dimensional Topological Phase transitions.

Topological quantum phase transitions intrinsically intertwine self-similarity and topology of many-electron wave-functions, and divining them is one of the most significant ways to advance understanding in condensed matter physics. Our focus is to investigate an unconventional class of the transitions between insulators and Dirac semimetals whose description is beyond conventional pseudo relativistic Dirac Hamiltonian. At the transition without the long-range Coulomb interaction, the electronic energy dispersion along one direction behaves like a relativistic particle, linear in momentum, but along the other direction it behaves like a non-relativistic particle, quadratic in momentum. Various physical systems ranging from TiO2-VO2 heterostructure to organic material α-(BEDT-TTF)2I3 under pressure have been proposed to have such anisotropic dispersion relation. Here, we discover a novel quantum criticality at the phase transition by incorporating the long range Coulomb interaction. Unique interplay between the Coulomb interaction and electronic critical modes enforces not only the anisotropic renormalization of the Coulomb interaction but also marginally modified electronic excitation. In connection with experiments, we investigate several striking effects in physical observables of our novel criticality.

Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle

We consider the Hamiltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition which is a version of the "Fermi Golden Rule". We prove that in the large time approximation any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.

Regularization of ultraviolet divergence for a particle interacting with a scalar quantum field

When a nonrelativistic particle interacts with a scalar quantum field, the standard perturbation theory leads to a dependence of the energy of its ground state on an undefined parameter---"momentum cutoff"---due to the ultraviolet divergence. We show that the use of nonasymptotic states of the system results in a calculation scheme in which all observable quantities remain finite and continuously depend on the coupling constant without any additional parameters. It is furthermore demonstrated that the divergence of traditional perturbation series is caused by the energy being a function with a logarithmic singularity for small values of the coupling constant.

Comparison between two models of absorption of matter waves by a thin time-dependent barrier

We report a quantitative, analytical and numerical, comparison between two models of the interaction of a non-relativistic quantum particle with a thin time-dependent absorbing barrier. The first model represents the barrier by a set of time-dependent discontinuous matching conditions, which are closely related to Kottler boundary conditions used in stationary wave optics as a mathematical basis for Kirchhoff diffraction theory. The second model mimics the absorbing barrier with an off-diagonal $\delta$-potential with a time-dependent amplitude. We show that the two models of absorption agree in their predictions in a semiclassical regime -- the regime readily accessible in modern experiments with ultracold atoms.

First class models from linear and nonlinear second class constraints

Two models with linear and nonlinear second class constraints are considered and gauged by embedding in an extended phase space. These models are studied by considering a free non-relativistic particle on the hyperplane and hypersphere in the configuration space. The gauged theory of the first model is obtained by converting the very second class system to the first class one directly. In contrast, the first class system related to the free particle on the hypersphere is derived with the help of the infinite Batalin–Fradkin–Tyutin (BFT) embedding procedure. We propose a practical formula, based on the simplified BFT method, which is practical in gauging linear and some nonlinear second class systems. As a result of gauging these two models, we show that in the conversion of second class constraints to the first class ones, the minimum number of phase space degrees of freedom for both systems is a pair of phase space coordinates. This pair is made up of a coordinate and its conjugate momentum for the first model, but the corresponding Poisson structure of the embedded non-relativistic particle on hypersphere is a nontrivial one. We derive infinite correction terms for the Hamiltonian of the nonlinear constraints and an interacting gauged Hamiltonian is constructed by summing over them. At the end, we find an open algebra for three first class objects of the embedded nonlinear system.

Semi-spectral method for the Wigner equation

We propose a numerical method to solve the Wigner equation in quantum systems of spinless, non-relativistic particles. The method uses a spectral decomposition into $L^2(\mathbb{R}^d)$ basis functions in momentum-space to obtain a system of first-order advection-reaction equations. The resulting equations are solved by splitting the reaction and advection steps so as to allow the combination of numerical techniques from quantum mechanics and computational fluid dynamics by identifying the skew-hermitian reaction matrix as a generator of unitary rotations. The method is validated for the case of particles subject to a one-dimensional (an-)harmonic potential using finite-differences for the advection part. Thereby, we verify the second order of convergence and observe non-classical behavior in the evolution of the Wigner function.

Polymer quantization and the saddle point approximation of partition functions

The saddle point approximation of the path integral partition functions is an important way of deriving the thermodynamical properties of black holes. However, there are certain black hole models and some mathematically analog mechanical models for which this method cannot be applied directly. This is due to the fact that their action evaluated on a classical solution is not finite and its first variation does not vanish for all consistent boundary conditions. These problems can be dealt with by adding a counterterm to the classical action, which is a solution of the corresponding Hamilton-Jacobi equation. In this work we study the effects of polymer quantization on a mechanical model presenting the aforementioned difficulties and contrast it with the above counterterm method. This type of quantization for mechanical models is motivated by the loop quantization of gravity which is known to play a role in the thermodynamics of black hole systems. The model we consider is a nonrelativistic particle in an inverse square potential, and analyze two polarizations of the polymer quantization in which either the position or the momentum is discrete. In the former case, Thiemann's regularization is applied to represent the inverse power potential but we still need to incorporate the Hamilton-Jacobi counterterm which is now modified by polymer corrections. In the latter, momentum discrete case however, such regularization could not be implemented. Yet, remarkably, owing to the fact that the position is bounded, we do not need a Hamilton-Jacobi counterterm in order to have a well-defined saddle point approximation. Further developments and extensions are commented upon in the discussion.

Scattering of wave packets on atoms in the Born approximation

Laser photons carrying non-zero orbital angular momentum are known and exploited during the last twenty years. Recently it has been demonstrated experimentally that such (twisted) electrons can be produced and even focused to a subnanometer scale. Thus, twisted electrons emerge as a new tool in atomic physics. The state of a twisted electron can be considered as a specific wave packet of plane waves. In the present paper-I we consider elastic scattering of the wave packets of fast non-relativistic particles on a potential field. We obtain simple and convenient formulae for a number of events in such a scattering. The equations derived represent, in fact, generalization of the well-known Born approximation for the case when finite sizes and inhomogeneity of the initial packet should be taken into account. To illustrate the obtained results, we consider two simple models corresponding to scattering of a Gaussian wave packet on the Gaussian potential and on the hydrogen atom. The scattering of twisted electrons on atoms will be considered in the next paper-II.

Exactly solvable -symmetric models in two dimensions

Non-Hermitian, -symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, potentials for a non-relativistic particle confined in a circular geometry. We show that the -symmetry threshold can be tuned by introducing a second gain-loss potential or its Hermitian counterpart. Our results explicitly demonstrate that breaking in two dimensions has a rich phase diagram, with multiple re-entrant -symmetric phases.