Nonrelativistic Particle Research Articles

A product picture for quantum electrodynamics

We present a short account of our work to provide quantum electrodynamics (QED) with a product picture. We aim to complement the longer exposition in a recent paper in Foundations of Physics and to help to make that work more accessible. The product picture is a formulation of QED, equivalent to standard Coulomb gauge QED, in which the Hilbert space arises as (a certain physical subspace of) a product of a Hilbert space for the electromagnetic field and a Hilbert space for charged matter (i.e., the Dirac field) and the Hamiltonian arises as the sum of an electromagnetic Hamiltonian, a charged matter Hamiltonian, and an interaction term. (The Coulomb gauge formulation of QED is not a product picture because, in it, the longitudinal part of the electromagnetic field is made out of charged matter operators.) We also recall a “Contradictory Commutator Theorem” for QED, which exposes flaws in previous attempts at temporal gauge quantization of QED, and we explain how our product picture appears to offer a way to overcome those flaws. Additionally, we discuss the extent to which that theorem may be generalized to Yang–Mills fields. We also develop a product picture for nonrelativistic charged particles in interaction with the electromagnetic field and point out how this leads to a novel way of thinking about the theory of many nonrelativistic electrically charged particles with Coulomb interactions. In an afterword, we explain how the provision of a product picture for QED gives hope that one will be able likewise to have a product picture for (Yang Mills and for) quantum gravity—the latter being needed to make sense of the author's matter-gravity entanglement hypothesis. Also, we briefly discuss some similarities and differences between that hypothesis and its predictions and ideas of Roger Penrose related to a possible role of gravity in quantum state reduction and related to cosmological entropy.

The modified fundamental equations of quantum mechanics

The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’s equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac’s equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrödinger equation, and the other is the NKE Schrödinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author’s opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.

Angular momentum eigenstates of the isotropic 3-D harmonic oscillator: Phase-space distributions and coalescence probabilities

The isotropic 3-dimensional harmonic oscillator potential can serve as an approximate description of many systems in atomic, solid state, nuclear, and particle physics. In particular, the question of 2 particles binding (or coalescing) into angular momentum eigenstates in such a potential has interesting applications. We compute the probabilities for coalescence of two distinguishable, non-relativistic particles into such a bound state, where the initial particles are represented by generic wave packets of given average positions and momenta. We use a phase-space formulation and hence need the Wigner distribution functions of angular momentum eigenstates in isotropic 3-dimensional harmonic oscillators. These distribution functions have been discussed in the literature before but we utilize an alternative approach to obtain these functions. Along the way, we derive a general formula that expands angular momentum eigenstates in terms of products of 1-dimensional harmonic oscillator eigenstates.

Neutrino forces and the Sommerfeld enhancement

The Sommerfeld enhancement plays an important role in dark matter (DM) physics, and can significantly enhance the annihilation cross section of non-relativistic DM particles. In this paper, we study the effect of neutrino forces, which are generated by the exchange of a pair of light neutrinos, on the Sommerfeld enhancement. We demonstrate that in certain cases, a neutrino force can cause a significant correction to the Sommerfeld enhancement. Models that can realise DM-neutrino interactions and sizeable Sommerfeld enhancement are also briefly discussed, together with the impacts on DM phenomenology of neutrino forces.

Gamma convergence and renormalization group: Two sides of a coin?

We discuss, both from the point of view of Gamma convergence and from the point of view of the renormalization Group, the zero range strong contact interaction of three non-relativistic massive particles. Formally, the potential term is g (delta (x_3-x_1) + delta (x_3 -x_2)), ;, g < 0 and is the limit epsilon rightarrow 0 of approximating potentials V_epsilon (|x_i -x_3|) = epsilon ^{-3} V ( frac{|x_i - x_3|}{epsilon }) , V( x) in L^1(R^3) cap L^2 (R^3) . The presence of a delta function in the limit does not allow the use of standard tools of functional analysis. In the first approach (European Phys. J. Plus 136-363, 2021), (European Phys. J. Plus 1136-1161, 2021), we introduced a map mathcal{K}, called Krein Map , from L^2 (R^9) to a space (Minlos space) mathcal{M}) of more singular functions. In { mathcal M}, the system is represented by a one parameter family of self-adjoint operators. In the topology of L^2 (R^9), the system is an ordered family of weakly closed quadratic forms. By Gamma convergence, the infimum is a self-adjoint operator, the Hamiltonian H of the system. Gamma convergence implies resolvent convergence (An Introduction to Gamma Convergence Springer 1993) but not operator convergence!. This approach is variational and non-perturbative. In the second approach, perturbation theory is used. At each order of perturbation theory, divergences occur when epsilon rightarrow 0. A finite renormalized Hamiltonian H_R is obtained by redefining mass and coupling constant at each order of perturbation theory. In this approach, no distinction is made between self-adjoint operators and quadratic forms. One expects that H = H_R , i.e., that “renormalization” amounts to the difference between the Hamiltonian obtained by quadratic form convergence and the one obtained by Gamma convergence. We give some hints, but a formal proof is missing. For completeness, we discuss briefly other types of zero-range interactions.

Evolution of energy and magnetic moment of a quantum charged particle in power-decaying magnetic fields

We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a homogeneous time-dependent magnetic field B(t)=B0(1+t/t0)−1−g, directed along the z-axis and described by means of the vector potential A(t)=B(t)[−y,x]/2. Assuming that the particle was initially in the thermal equilibrium state with a negligible coupling to a reservoir, we obtain exact formulas for the time-dependent mean values of the energy and magnetic moment in terms of the Bessel functions. Different regimes of the evolution are discovered and illustrated in several figures. The energy goes asymptotically to a finite value if g>0 (“fast” decay), while it goes asymptotically to zero if g≤0 (“slow” decay). The dependence on parameter t0 practically disappears when 1+g is close to zero value (“superslow” decay). The mean magnetic moment goes to zero for g>1, while it grows unlimitedly if g<1.

Relativistic spin-0 particle in a box: Bound states, wave packets, and the disappearance of the Klein paradox

The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.

On a Strong Increase in the Amplitude of the Wave Function of a Massive Nonrelativistic Particle Incident on a Crystal (One-Dimensional Approximation)

On a Strong Increase in the Amplitude of the Wave Function of a Massive Nonrelativistic Particle Incident on a Crystal (One-Dimensional Approximation)

Taking approximations seriously: The cases of the Chew and Nambu-Jona-Lasinio models

Physical theories, for the most part, should be understood as modelling isolated subsystems of a larger universe; doing so, among other benefits, greatly clarifies the interpretation of the dynamical symmetries of those theories. I provide a detailed framework for analysing the subsystem structure of physical theories and applying it to the interpretation of their symmetries: the core concept is subsystem recursivity, whereby interpretative conclusions about a sector of a theory can be deduced from considering subsystems of other models of the same theory. I illustrate the framework by extensive examples from nonrelativistic particle mechanics, and in particular from Newtonian theories of gravity. A sequel to the paper will apply the framework to the local and global symmetries of classical field theory.

Isolated systems and their symmetries, part I: General framework and particle-mechanics examples

Physical theories, for the most part, should be understood as modelling isolated subsystems of a larger universe; doing so, among other benefits, greatly clarifies the interpretation of the dynamical symmetries of those theories. I provide a detailed framework for analysing the subsystem structure of physical theories and applying it to the interpretation of their symmetries: the core concept is subsystem recursivity, whereby interpretative conclusions about a sector of a theory can be deduced from considering subsystems of other models of the same theory. I illustrate the framework by extensive examples from nonrelativistic particle mechanics, and in particular from Newtonian theories of gravity. A sequel to the paper will apply the framework to the local and global symmetries of classical field theory.

Torsional string Newton-Cartan geometry for non-relativistic strings

We revisit the formulation of non-relativistic (NR) string theory and its target space geometry. We obtain a new formulation in which the geometry contains a two-form field that couples to the tension current and that transforms under string Galilei boosts. This parallels the Newton-Cartan one-form that couples to the mass current of a non-relativistic point particle. We show how this formulation of the NR string arises both from an infinite speed of light limit and a null reduction of the relativistic closed bosonic string. In both cases, the two-form originates from a combination of metric quantities and the Kalb-Ramond field. The target space geometry of the NR string is seen to arise from the gauging of a new algebra that is obtained by an İnönü-Wigner contraction of the Poincaré algebra extended by the symmetries of the Kalb-Ramond field. In this new formulation, there are no superfluous target space fields that can be removed by fixing a Stückelberg symmetry. Classically, there are no foliation/torsion constraints imposed on the target space geometry.

Fermionic King model: critical masses resulting from the competition of quantum and evaporation effects

We study the influence of evaporation (the escape of constituents) on the thermodynamics of a self-gravitating non-relativistic gas of fermions in the framework of Newtonian gravitation. For this purpose, it is reconsidered as the so-called fermionic King model introduced by Ruffini and Stella in the context of dark matter halos problems (1983 Astron. Astrophys. 119 3). As the usual King model, the present model predicts density profiles that drop to zero at a certain finite radius R, which is referred to as the tidal radius. Our interest is focused on the thermodynamics of this model at constant total mass M and constant tidal radius R. A special interest is devoted to study the incidence of mass on the collective phenomena of gravitational collapse and evaporation disruption. The underlying physics is driven by the Fermi mass M F ∼ ℏ 6/G 3 m 8 R 3, which corresponds to the total mass of a self-gravitating degenerate Fermi gas of non-relativistic point particles with mass m, whose linear size is equal to the tidal radius R of evaporation. Numerical calculations predict three critical values for the total mass, M 1 > M 2 > M 3, which hallmark the thermodynamic stability. The lower bound is a critical value of the total mass below the which the system gravitation is unable to confine its constituents, so that the system with total mass M < M 3 undergoes an evaporation disruption. The other bounds, M 2 ≃ 90.9M F and M 1 ≃ 8.9 × 106 M F, are critical values that hallmark the continuous (or discontinuous) character of the microcanonical phase transition associated with gravothermal collapse and the presence of states with negative heat capacities.

Investigation of two-electron quantum dot in a magnetic field by using of quasi-exact-solvable method

We study the non-relativistic particles of a two dimensional two-electron quantum dot. We consider the system in the presence of harmonic plus linear terms as well as a term related to spin interaction. The solution of the system is obtained by using of Quasi-Exactly-Solvable (QES) method. Next, we calculate the energy eigenstates and wave functions of the system in the presence of an external magnetic field.

Testing modified (Horndeski) gravity by combining intrinsic galaxy alignments with cosmic shear

ABSTRACT We study the impact of modified gravity of the Horndeski class on intrinsic shape correlations in cosmic shear surveys. As these correlations intrinsic alignments are caused by tidal gravitational fields acting on galaxies as a collection of massive non-relativistic test particles, they are only sensitive to the gravitational potential, which forms in conjunction with the curvature perturbation. In contrast, the cosmic shear signal probes the sum of these two, i.e. both Bardeen potentials. Combining these probes therefore constitutes a test of gravity, derived from a single measurement. Focusing on linear scales and alignments of elliptical galaxies, we study the impact on inference of the braiding $\hat{\alpha }_{\rm B}$ and the time evolution of the Planck mass $\hat{\alpha }_{\rm M}$ by treating IAs as a genuine signal contributing to the overall ellipticity correlation. We find that for euclid, IAs can help to improve constraints on modified gravity of the Horndeski-class by 10 per cent if the alignment parameter needed for the linear alignment model is provided by simulations. If, however, the IA needs to be self-calibrated, all of the sensitivity is put into the inference of the alignment strength D since there is a very strong correlation with the evolution of the Planck mass. Thus, diminishing the benefit of IA for probing modified gravitational theories. While this paper shows results mainly for modified gravity parameters, similar deductions can be drawn for the investigation of anisotropic stresses, parametrized modifications to the Poisson equation, the phenomenology of gravitational slip and to breaking degeneracies in a standard cosmology.

Measurement of the Dark Matter Velocity Dispersion with Dwarf Galaxy Rotation Curves

Warm dark matter has, by definition, a velocity dispersion. Let vhms(a)=vhms(1)/a be the root-mean-square velocity of non-relativistic warm dark matter particles in the early universe at expansion parameter a. vhms(1) is an adiabatic invariant. We obtain vhms(1) in the core of 11 dwarf galaxies dominated by dark matter, from their observed rotation curves, up to a rotation and relaxation correction. We obtain a mean 0.490 km/s and standard deviation 0.160 km/s, with a distribution peaked at the lower end. We apply a mild, data driven, rotation and relaxation correction that obtains the adiabatic invariant in the core of the galaxies: vhms(1)=0.406 ± 0.069 km/s. These two small relative standard deviations justify the prediction that the adiabatic invariant vhms(1) in the core of the galaxies is of cosmological origin if dark matter is warm. This result is in agreement with measurements of vhms(1) based on spiral galaxy rotation curves, galaxy ultra-violet luminosity distributions, galaxy stellar mass distributions, the formation of first galaxies, reionization, and the velocity dispersion cut-off mass.

Evaluation of cross section of elastic scattering for non-relativistic and relativistic particles by means of fundamental scattering formulas

In evaluating differential cross section of elastic scattering, different theories were applied to low-momentum and relativistic particles. For low-momentum motion, Lippmann-Schwinger scattering equation was applied, called fundamental formula; while for relativistic particles, a general scattering theory was used which calculates S matrix. In this paper, Lippmann-Schwinger equation is applied uniformly to both low-momentum and relativistic particles. The cross sections are valuated to the first order of Born approximation. One-body time-independent Green's functions for relativistic free particles are given. Compared to the general scattering theory, the fundamental theory has a clearer physical picture and the approximations made are more explicit.

Measurement of the Dark Matter Velocity Dispersion with Galaxy Stellar Masses, UV Luminosities, and Reionization

The root-mean-square of non-relativistic warm dark matter particle velocities scales as vhrms(a)=vhrms(1)/a , where a is the expansion parameter of the universe. This velocity dispersion results in a cut-off of the power spectrum of density fluctuations due to dark matter free-streaming. Let kfs (teq) be the free-streaming comoving cut-off wavenumber at the time of equal densities of radiation and matter. We obtain , and , at 68% confidence, from the observed distributions of galaxy stellar masses and rest frame ultra-violet luminosities. This result is consistent with reionization. From the velocity dispersion cut-off mass we obtain the limits vhrms(1)kfs (teq) >1.5 Mpc-1. These results are in agreement with previous measurements based on spiral galaxy rotation curves, and on the formation of first galaxies and reionization. These measured parameters determine the temperature-to-mass ratio of warm dark matter. This ratio happens to be in agreement with the no freeze-in and no freeze-out warm dark matter scenario of spin 0 dark matter particles decoupling early on from the standard model sector. Spin 1/2 and spin 1 dark matter are disfavored if nature has chosen the no freeze-in and no freeze-out scenario. An extension of the standard model of quarks and leptons, with scalar dark matter that couples to the Higgs boson that is in agreement with all current measurements, is briefly reviewed. Discrepancies with limits on dark matter particle mass that can be found in the literature are addressed.

Disguised electromagnetic connections in classical electron theory

In the first quarter of the 20th century, physicists were not aware of the existence of classical electromagnetic zero-point radiation nor of the importance of special relativity. Inclusion of these aspects allows classical electron theory to be extended beyond its 19th century successes. Here we review spherical electromagnetic radiation modes in a conducting-walled spherical cavity and connect these modes to classical electromagnetic zero-point radiation and to electromagnetic scale invariance. Then we turn to the scattering of radiation in classical electron theory within a simple approximation. We emphasize that, in steady-state, the interaction between matter and radiation is disguised so that the mechanical motion appears to occur without the emission of radiation, even though the particle motion is actually driven by classical electromagnetic radiation. It is pointed out that, for nonrelativistic particles, only the harmonic oscillator potential taken in the low-velocity limit allows a consistent equilibrium with classical electromagnetic zero-point radiation. For relativistic particles, only the Coulomb potential is consistent with electrodynamics. The classical analysis places restrictions on the value of e 2/ℏc.

Pauli oscillator in noncommutative space

In this study, we investigate the Pauli oscillator in a noncommutative space. In other words, we derive wave function and energy spectrum of a spin half non-relativistic charged particle that is moving under a constant magnetic field with an oscillator potential in noncommutative space. We obtain critical values of the deformation parameter and the magnetic field, which they counteract the normal and anomalous Zeeman effects. Moreover, we find that the deformation parameter has to be smaller than [Formula: see text]. Then, we derive the Helmholtz free energy, internal energy, specific heat and entropy functions of the Pauli oscillator in the non commutative space. With graphical methods, at first, we compare these functions with the ordinary ones, and then, we demonstrate the effects of magnetic field on these thermodynamic functions in the commutative and noncommutative space, respectively.

Fundamental Gravity and Gravitational Waves

While being as old as general relativity itself, the gravitational two-body problem has never been under so intense investigation as it is today, spurred by both phenomenological and theoretical motivations. The observations of gravitational waves emitted by compact binary coalescences bear the imprint of the source dynamics, and as the sensitivity of detectors improve over years, more accurate modeling is being required. The analytic modeling of classical gravitational dynamics has been enriched in this century by powerful methods borrowed from field theory. Despite being originally developed in the context of fundamental particle quantum scatterings, their applications to classical, bound system problems have shown that many features usually associated with quantum field theory, such as, e.g., divergences and counterterms, renormalization group, loop expansion, and Feynman diagrams, have only to do with field theory, be it quantum or classical. The aim of this work is to present an overview of this approach, which models massive astrophysical objects as nonrelativistic particles and their gravitational interactions via classical field theory, being well aware that while the introductory material in the present article is meant to represent a solid background for newcomers in the field, the results reviewed here will soon become obsolete, as this field is undergoing rapid development.