Finite Self-energy Research Articles

Self-energy problem, vacuum polarization, and dual symmetry in Born–Infeld-type U(1) gauge theories

We extensively explore three different aspects of Born-Infeld (BI) type nonlinear $U(1)$ gauge-invariant modifications of Maxwell's classical electrodynamics (also known as BI-type nonlinear electrodynamics) and bring some new perspectives on these theories. First, within the framework of exponential $U(1)$ gauge theory, it is explicitly proved that although the electric field at the location of the elementary point charges is not finite, but the total electrostatic field energy is finite. Motivated by this observation together with a wealth of evidence, we conjecture that all theories in 4-dimensional spacetime that belong to the BI family result in finite self-energy for elementary charged particles. In higher dimensions, it is found that the weak-field coupling limit of BI-type theories, which is identified as the weak field limit of effective Euler-Heisenberg (EH) theory, does not possess a regularizing ability to make the self-energy of a point charge finite, which implies that the conjecture may not hold for some BI-type theories. However, we explicitly prove that BI, logarithmic and exponential $U(1)$ gauge theories in arbitrary dimensions result in finite self-energy as well. Next, we classically study the problem of vacuum polarization effects and then systematically make a connection between all BI-type theories and QED. It is shown that all the BI-type theories classically predict the vacuum polarization effects, in which the final results are exactly in one-to-one correspondence with QED and the effective EH theory up to the leading order of corrections. Finally, we present a new, simple proof indicating that ...

A non-singular continuum theory of point defects using gradient elasticity of bi-Helmholtz type

ABSTRACTIn this paper, we develop a non-singular continuum theory of point defects based on a second strain gradient elasticity theory, the so-called gradient elasticity of bi-Helmholtz type. Such a generalised continuum theory possesses a weak nonlocal character with two internal material lengths and provides a mechanics of defects without singularities. Gradient elasticity of bi-Helmholtz type gives a natural and physical regularisation of the classical singularities of defects, based on higher order partial differential equations. Point defects embedded in an isotropic solid are considered as eigenstrain problem in gradient elasticity of bi-Helmholtz type. Singularity-free fields of point defects are presented. The displacement field as well as the first, the second and the third gradients of the displacement are derived and it is shown that the classical singularities are regularised in this framework. This model delivers non-singular expressions for the displacement field, the first displacement gradient and the second displacement gradient. Moreover, the plastic distortion (eigendistortion) and the gradient of the plastic distortion of a dilatation centre are also non-singular and are given in terms of a form factor (shape function) of a point defect. Singularity-free expressions for the interaction energy and the interaction force between two dilatation centres and for the interaction energy and the interaction force of a dilatation centre in the stress field of an edge dislocation are given. The results are applied to calculate the finite self-energy of a dilatation centre.

Casimir Energies for Isorefractive or Diaphanous Balls

It is known that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside the spherical boundary is the same as that outside, so the self-energy of a perfectly conducting spherical shell is finite, as is the energy of a dielectric-diamagnetic sphere with ε μ = 1 , a so-called isorefractive or diaphanous ball. Here we re-examine that example and attempt to extend it to an electromagnetic δ -function sphere, where the electric and magnetic couplings are equal and opposite. Unfortunately, although the energy expression is superficially ultraviolet finite, additional divergences appear that render it difficult to extract a meaningful result in general, but some limited results are presented.

Short Range Gravity Potential Self Energy and Nonlinear Potential Dependent Lorentz Transformation

Using Poisson equation beside Maxwell distribution law for very hot star core consisting of elementary particles one can find new Poisson equation. This equation predicts existence of short range gravity force. This short force may have a link with short range nuclear force, thus raises a hope in unifying gravity and nuclear force. This short range field beside long range field secures singular finite self-energy. This central role of potential in unifying self-energy for high relativistic particles at star cores requires seeking Lorentz transformation that accounts for effect of fields. This work derived new Lorentz transformation for accelerated frame. This transformation is potential dependent and reduces to SR and explains time dilation and gravitational red shift.

Relativistic correlation effects on the x-ray spectra of Li-like ions

The wavelengths and rates of electric dipole transitions between states with $n=2$ and $n=1$ of doubly excited Li-like ions have been studied for some selected ions in the range $13\ensuremath{\le}Z\ensuremath{\le}54$ using fully relativistic multiconfiguration Dirac-Fock wave functions in the active space approximation with the inclusion of finite nuclear size, Breit interaction, self-energy, and vacuum polarization. A detailed discussion on the effects of intercomplex correlation around $Z=26$ and intracomplex correlation around $Z=37$ leading to irregularities and sharp discontinuities in the x-ray rates noticed for a few transitions has been provided. An unusually large contribution of Breit interaction has been found for intercomplex correlation in certain cases. The present results are compared with other available experimental and theoretical data. The errors associated with the transitions are highlighted for some experimentally available lines taking into account the uncertainties on the fine-structure energy levels and also on the line strengths.

Brane world regularization of point particle classical self-energy

Physical effects in brane worlds models emerge by the incorporation of field modes coming from extra dimensions with the usual four dimensional ones. Such effects can be tested with well established experiments to set bounds on the parameters of the brane models. In this work we extend a previous result which gave finite electromagnetic potentials and self energies for a source looking pointlike to an observer sitting in a 4D Minkowski subspace of a single brane of a Randall-Sundrum spacetime including compact dimensions, and along which the source stretches uniformly. We show that a scalar particle produces a nonsingular static potential, possess a finite self-energy and that technically its analysis is very similar to the electrostatic case. As for the latter, we use the deviations from the Coulomb potential to set bounds on the anti de Sitter radius of the brane model on the basis of two experiments, namely, one of the Cavendish type and other being the scattering of electrons by Helium atoms. We found these are less stringent than others previously obtained using the Lamb shift in Hydrogen.

Low-energy effects in brane worlds: Liennard–Wiechert potentials and Hydrogen Lamb shift

Testing extra dimensions at low-energies may lead to interesting effects. In this work a test point charge is taken to move uniformly in the 3D subspace of a (3 + n)-brane embedded in a (3 + n + 1)-space with n compact and one warped infinite spatial extra dimensions. We found that the electromagnetic potentials of the point charge match standard Liennard–Wiechert’s at large distances but differ from them close to it. These are finite at the position of the charge and produce finite self-energies. We also studied a localized Hydrogen atom and take the deviation from the standard Coulomb potential as a perturbation. This produces a Lamb shift that is compared with known experimental data to set bounds for the parameter of the model. This work provides details and extends results reported in a previous Letter.

Finite self-energy of pointlike sources for the Reissner-Nordström metric

It is shown that charged fluid distributions, matched to the Reissner-Nordstrom metric, may posses finite self-energy in the point particle limit if negative energy densities are allowed and the appropriate definition of bare mass is used. Several models are analyzed to illustrate the point.

Radiation reaction and the electromagnetic energy-momentum of moving relativistic charged membranes

The charged membrane of Dirac provides a stable electron model with finite self-energy. Its total mass m has been previously calculated from the Hamiltonian of the membrane. To complete the picture we evaluate it here on the basis of the energy-momentum of its self-field (radiation reaction) and obtain the same result showing the consistency of the model. We show explicitly that the old 43-problem does not arise. The electron's stability (the vanishing of total Tμv,v) is du to surface tension κ of the membrane, but the model is as simple as the point particle, with two parameters κ and e; the surface tension parameter κ can be expressed in terms of the mass m.

Scalar-vector topological soliton

We present classical scalar-vector equations which admit soliton solutions in three space dimensions. Exact spherical solutions are obtained which are everywhere regular and resemble charged particles of finite self-energy. The corresponding 4-current is identically conserved and leads to quantized charges. The scale of the soliton is unique and determined by boundary conditions, which also ensure its topological stability.

Dirac's shell model of the electron and the general theory of moving relativistic charged membrabes

A covariant theory of a moving charged membrane in an arbitrary dimension coupled to the electromagnetic field is presented. In particular we obtain a new formulation of Dirac's model of the electron as a charged spherical shell with internal oscillations and finite self-energy, and its generalization to a moving shell.

Evaluation of finite lowest-order self-energies

We show that the various lowest-order self-energies can all be made finite on account of the proposal by Danos and Rafelski that the space-time integrations in the covariant perturbation expansion of theS-matrix are to be open instead of closed. We also discuss how we can evaluate these finite lowest-order self-energies.

Hodge theory used to produce finite self-energy models of the electron

The relationship between electromagnetic fields and topology is discussed. Hodge theory is generalized to classify static fields in static wormhole models. This generalization is used to show models of the electron exist which have finite self-energy. These models always have 1/R2 electric fields and self-energies of 4πq2/R0, whereR0 is the inner radius of the wormhole.

Fundamental coupling for weak interactions and masses

Weak interactions and the masses of mesons and of the neutron are described by fundamental couplings between muonic neutrinos and electronic leptons or nucleons by means of heavy bosons with spin 0. These couplings yield the charged- and neutral-current interactions, the masses of the muon and the pion as composite systems of leptons and the finite electromagnetic self-energy of the charged fermion. The parity violation in atomic transitions and the anomalous magnetic moment of the muon are the same as in gauge theory results. The theoretical cross-sections of muonic-neutrino scatterings by the electron agree with experimental data. The mass difference of proton and neutron is evaluated from a composite picture of the neutron.

Particle solutions in a unified field theory

A numerical solution is obtained for the electric spherically symmetric field in Bonnor's unified field theory. This solution can be interpreted as a distributed particle with a finite self-energy. Important constants of the theory $k$ and $p$ are evaluated by assuming the particle carries the electron's charge and mass.

Consequences of nonlinearity in the generalized theory of gravitation

This paper contains, for the time-independent spherically symmetric fields, various regular solutions of the field equations. The spectrum of magnetic charges ${\mathcal{I}}_{n}$, $n=0, 1, 2, \dots{}$, with alternating signs has been obtained as a function of mass where ${\mathcal{I}}_{n}\ensuremath{\rightarrow}0$ for $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and where ${\ensuremath{\Sigma}}_{n=0}^{\ensuremath{\infty}}{\mathcal{I}}_{n}=0$. The neutral magnetic core of an elementary particle consists of an infinite number of stratified layers containing magnetic charges of amounts ${\mathcal{I}}_{n}$. The screening caused by the stratified distribution of magnetic charges generates short-range forces. The strength of the coupling between the field and particle is described by ${e}^{2}+{{\mathcal{I}}_{n}}^{2}$, where $n=\ensuremath{\infty}$ corresponds to the distances beyond which there are no short-range forces. The observed mass $M$ of a particle or antiparticle is obtained, as a consequence of the equations of motion, in the form $M{c}^{2}=\frac{1}{2}m{c}^{2}+2{E}_{s}$, where ${E}_{s}$ is the finite self-energy of the particle (or antiparticle) and where $m$ and ${E}_{s}$ have opposite signs. The "bare gravitational mass" $m$, obtained as a constant of integration, is estimated to be of the order of ${10}^{21}$ MeV. The spectrum of fundamental lengths ${r}_{0n}(=[\frac{{(2{G}_{0})}^{\frac{1}{2}}}{{c}^{2}}]{({e}^{2}+{{\mathcal{I}}_{n}}^{2})}^{\frac{1}{2}}\ensuremath{\sim}{10}^{\ensuremath{-}33} \mathrm{cm})$ measures the deviation of the theory from general relativity. The self-energy ${E}_{s}$ in the limit ${r}_{0n}=0$ tends to infinity and the solutions reduce to the corresponding spherically symmetric solutions in general relativity. The spin \textonehalf{} of an elementary particle is found to be the result of the assumption ${{\mathcal{I}}_{n}}^{2}={\ensuremath{\gamma}}_{n}\ensuremath{\hbar}c$ with ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=0$ and its neutral magnetic structure, where the latter exists only for nonvanishing ${\ensuremath{\Gamma}}_{[\ensuremath{\mu}\ensuremath{\nu}]}^{\ensuremath{\rho}}$, the antisymmetric part of the affine connection. The two states of spin correlate with the two possible sequences of signs of ${\mathcal{I}}_{n}$, i.e., ${\mathcal{I}}_{n}$ and $\ensuremath{-}{\mathcal{I}}_{n}$, $n=0, 1, 2, \dots{}$. For the solutions where $e=0$ the symmetries of charge conjugation and parity are not conserved. The latter lead to the assumption of small masses for the two neutrinos ${\ensuremath{\nu}}_{e}$, ${\ensuremath{\nu}}_{\ensuremath{\mu}}$. Conservation of the electric charge multiplicity, i.e., the existence of -1, +1, 0 units of electric charge, is found to be the basis for the existence of four massive fundamental particles $p$, $e$, ${\ensuremath{\nu}}_{e}$, ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ and the corresponding antiparticles $\overline{p}$, ${e}^{+}$, ${\overline{\ensuremath{\nu}}}_{e}$, ${\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$. Based on a new concept of "vacuum" and new magnetic charge predicted by the theory it may be possible to construct all other elementary particles as bound or resonance states of the "fundamental quartet" $p$, $e$, ${\ensuremath{\nu}}_{e}$, ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ and the "antiquartet" $\overline{p}$, ${e}^{+}$, ${\overline{\ensuremath{\nu}}}_{e}$, ${\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$.

Superheavy elements and nonlinear electrodynamics

Maxwell’s equations, which underlie electrodynamics, are linear equations. Nonlinear effects, such as photon-photon scattering, are known to arise in quantum electrodynamics, and one might expect them to become important in the case of strong external fields. We investigate the consequences of a class of nonlinear Lagrangians, which includes that of Born and Infeld and whose common property is that they lead to upper limits for the electric-field strength (somewhat analogous to the upper limit for the velocity of a particle in special relativity). These nonlinear Lagrangians also lead to a finite electromagnetic selfenergy for the electron, unlike the case of Maxwellian electrodynamics. The importance of nonlinear effects of course depends upon the size of the upper limit to the electric-field strength. If this upper limit is determined by requiring that the mass of the electron is of an entirely electromagnetic origin, nonlinear effects become very important in determining the eigenvalues of electrons bound to superheavy nuclei. For example, theIs energy eigenvalue is raised by 270 keV forZ= 164. The Lagrangians considered here do not lead to an absolute gap between bound states and the states of the lower continuum; the Iδ energy eigenvalue becomes equal to —0.511 MeV, where the lower continuum begins, forZ=215. An analogy between nonlinear electrodynamics and higherorder vacuum polarization corrections is studied.

Theory of Weak Interactions without Divergences

A theory of weak interactions is suggested which is free of the divergences associated with the fourfermion theory and the usual intermediate-vector-boson theory, without the use of cutoffs. The interaction Lagrangian is a universal, local coupling of the usual weak currents to a vector field operator which is a nonentire function of a minimal set of vector-meson fields of large mass (\ensuremath{\gtrsim}5 GeV). A summation procedure related to that of Efimov and Fradkin is developed for the analysis of such Lagrangians, and $S$-matrix elements are computed thereby. It is shown that the (two) parameters of the Lagrangian may be adjusted to produce finite fermion self-energies and finite second- and fourth-order graphs. A power-counting argument is given that this finiteness then persists to all orders without the use of cutoffs. It is further shown that the parameters may be chosen to produce agreement with the predictions of the Fermi theory at low-energies and momentum transfers. The high-energy deviations from Fermi theory predicted by this theory are analyzed. The problem of ambiguities associated with nonpolynomial Lagrangians is discussed. A prescription for the resolution of the ambiguity is given which is expected to ensure the unitarity of the theory in all orders.

Finite Self-Energy of Classical Point Particles

The infinite mass self-energy difficulties of quantum field theory already occur, as is well-known, in the corresponding classical theories. Although cutoffs may be introduced to effect renormalization in both the classical and quantum cases, such procedures are physically unsatisfactory. We wish to point out in this note that at least for the static (Coulomb-type) contribution, one obtains finite results for the classical self-energies if the gravitational contribution to the total energy is included. Furthermore, it will be shown rigorously (in the static case) that the natural cutoff furnished by general relativity implies that all the mass of a point charge arises from its total self-field and that a neutral particle has no mass.

Gravitation and Electrodynamics

In this paper Einstein's unified field theory is modified, and some of the physical implications of the new theory are examined. It entails: (1) a restriction of 4-current distribution, (2) an electromagnetic field consisting of short- and long-range parts, (3) a finite self-energy for the electron, (4) a classical description of pair production and annihilation as discussed by Feynman in his electrodynamics, (5) the Lorentz-force law for a charged particle moving in an external electromagnetic field, (6) the bending of light grazing the surface of the sun---the same as given by the general theory of relativity.