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Michael K. -H. Kiessling

Publications and source records attributed to Michael K. -H. Kiessling.

At least 37 records · Page 2Linked to original sources
arXiv2021

Electrically non-neutral ground states of stars

To approximately compute the non-relativistic ground state of an electrically non-neutral star, an exactly solvable model was recently introduced, and partly solved, by Krivoruchenko, Nadyozhin, and Yudin. The model generalizes the well-known Lane--Emden equation of a polytropic gas ball of index $n=1$ to a two-fluid setting. Here its complete solution is presented in terms of simple elementary functions; it is also generalized to a more-than-two-fluid setting where it remains exactly solvable. It is shown that, given the number of nuclei, a maximal negatively and a maximal positively charged solution exists, plus a continuous family of solutions which interpolates between these extremes. Numerical comparisons show that this exactly solvable model captures the qualitative behavior of the more physical model it is supposed to approximate. Furthermore, it correctly answers the question: how non-neutral can the star be? The answer is independent of the speed of light $c$ and the Planck quantum $h$. It supports Penrose's weak cosmic censorship hypothesis, in the sense that the bounds on the excess charge are compatible with the bound on the charge of a Reissner--Weyl--Nordström black hole.

gr-qc↗
arXiv2020

How much electric surcharge fits on ... a white dwarf star?

The question how much surplus of electric charge ("surcharge") fits on an object is generally very difficult to answer. Here it is shown that it is easy to answer when the object is a \emph{failed white dwarf} star (a brown dwarf in its ground state) made of protons and electrons: Given the number of protons, how many electrons can there be? Surprisingly, the answer (in the form: as few as $\mathfrak{A}$ and as many as $\mathfrak{B}$) is independent of the speed of light $c$ and the Planck quantum $h$, even when the star is stabilized against collapse by relativistic quantum mechanics.

gr-qc↗
arXiv2020

A Lorentz-Covariant Interacting Electron-Photon System in One Space Dimension

A Lorenz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac's formalism of multi-time wave functions, i.e., wave functions $Ψ(\mathbf{x}_{ph},\mathbf{x}_{el})$ where $\mathbf{x}_{el},\mathbf{x}_{ph}$ are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold $\{\mathbf{x}_{el}=\mathbf{x}_{ph}\}$, compatible with particle current conservation. The corresponding initial-boundary-value problem is proved to be well-posed. Electron and photon trajectories are shown to exist globally in a Hypersurface Bohm--Dirac theory, for typical particle initial conditions. Also presented are the results of some numerical experiments which illustrate Compton scattering as well as a new phenomenon: photon capture and release by the electron.

math-ph↗
arXiv2020

On general-relativistic hydrogen and hydrogenic ions

This paper studies how the static non-linear electromagnetic-vacuum spacetime of a point nucleus with negative bare mass affects the self-adjointness of the general-relativistic Dirac Hamiltonian for a test electron, without and with an anomalous magnetic moment. The study interpolates between the previously studied extreme cases of a test electron in (a) the Reissner--Weyl--Nordström spacetime (Maxwell's electromagnetic vacuum), which supports a very strong curvature singularity with negative infinite bare mass, and (b) the Hoffmann spacetime (Born or Born--Infeld's electromagnetic vacuum) with vanishing bare mass, which features the mildest possible curvature singularity. The main conclusion reached is: {on electrostatic spacetimes of a point nucleus with a strictly negative bare mass} (which may be $-\infty$) essential self-adjointness fails unless the radial electric field diverges sufficiently fast at the nucleus and the anomalous magnetic moment of the electron is taken into account. Thus on the Hoffmann spacetime with (strictly) negative bare mass the Dirac Hamiltonian of a test electron, with or without anomalous magnetic moment, is not essentially self-adjoint. All these operators have self-adjoint extensions, though, with the usual essential spectrum $(-\infty,-\mEL c^2]\cup[\mEL c^2,\infty)$ and an infinite discrete spectrum located in the gap $(-\mEL c^2,\mEL c^2)$

math-ph↗
arXiv2020

The force on a point charge source of the classical electromagnetic field

It is shown that a well-defined expression for the total electromagnetic force $f^{em}$ on a point charge source of the classical electromagnetic field can be extracted from the postulate of total momentum conservation whenever the classical electromagnetic field theory satisfies a handful of regularity conditions. Amongst these is the generic local integrability of the field momentum density over a neighborhood of the point charge. This disqualifies the textbook Maxwell-Lorentz field equations, while the Maxwell-Bopp-Lande-Thomas-Podolsky field equations qualify, and presumably so do the Maxwell-Born-Infeld field equations. Most importantly, when the usual relativistic relation between the velocity and the momentum of a point charge with bare rest mass $m_b \neq 0$ is postulated, Newton's law $\dot{p} = f$ with $f = f^{em}$ becomes an integral equation for the point particle's acceleration; the infamous third-order time derivative of the position which plagues the Abraham-Lorentz-Dirac equation of motion does not show up. No infinite bare mass renormalization is invoked, and no ad hoc averaging of fields over a neighborhood of the point charge. The approach lays the rigorous microscopic foundations of classical electrodynamics.

physics.class-ph↗
arXiv2020

Microscopic foundations of kinetic plasma theory: The relativistic Vlasov--Maxwell equations and their radiation-reaction-corrected generalization

It is argued that the relativistic Vlasov--Maxwell equations of the kinetic theory of plasma approximately describe a relativistic system of $N$ charged point particles interacting with the electromagnetic Maxwell fields in a Bopp--Landé--Thomas--Podolsky (BLTP) vacuum, provided the microscopic dynamics lasts long enough.The purpose of this work is not to supply an entirely rigorous vindication, but to lay down a conceptual road map for the microscopic foundations of the kinetic theory of special-relativistic plasma, and to emphasize that a rigorous derivation seems feasible. Rather than working with a BBGKY-type hierarchy of $n$-point marginal probability measures, the approach proposed in this paper works with the distributional PDE of the actual empirical 1-point measure, which involves the actual empirical 2-point measure in a convolution term.The approximation of the empirical 1-point measure by a continuum density, and of the empirical 2-point measure by a (tensor) product of this continuum density with itself, yields a finite-$N$ Vlasov-like set of kinetic equations which includes radiation-reaction and nontrivial finite-$N$ corrections to the Vlasov--Maxwell-BLTP model. The finite-$N$ corrections formally vanish in a mathematical scaling limit $N\to\infty$ in which charges $\propto 1/\surd{N}$. The radiation-reaction term vanishes in this limit, too. The subsequent formal limit sending Bopp's parameter $\varkappa\to\infty$ yields the Vlasov--Maxwell model.

physics.plasm-ph↗
arXiv2019

On the Schrödinger spectrum of a hydrogen atom with electrostatic Bopp-Landé-Thomas-Podolsky interaction between electron and proton

The Schrödinger spectrum of a hydrogen atom, modelled as a two-body system consisting of a point electron and a point proton, interacting with a modification of Coulomb's law proposed, in the 1940s, by Bopp, Landé--Thomas, and Podolsky (BLTP). The BLTP theory hypothesizes the existence of an electromagnetic length scale of nature --- the Bopp length $\varkappa ^{-1}$ ---, to the effect that the electrostatic pair interaction deviates significantly from Coulomb's law only for distances much shorter than $\varkappa^{-1}$. Rigorous lower and upper bounds are constructed for the Schrödinger energy levels of the hydrogen atom, $E_{\ell,n}(\varkappa)$, for all $\ell\in\{0,1,2,...\}$ and $n >\ell$. The energy levels $E_{0,1}(\varkappa)$, $E_{0,2}(\varkappa)$, and $E_{1,2}(\varkappa)$ are also computed numerically and plotted versus $\varkappa^{-1}$. It is found that the BLTP theory predicts a non-relativistic correction to the splitting of the Lyman-$α$ line in addition to its well-known relativistic fine-structure splitting. Under the assumption, that this splitting doesn't go away in a relativistic calculation, it is argued that present-day precision measurements of the Lyman-$α$ line suggest that $\varkappa^{-1}$ must be smaller than $\approx 10^{-18}\,\mathrm{m}$. Finite proton size effects are found not to modify this conclusion. As a consequence, the electrostatic field energy of an elementary point charge, although finite in BLTP electrodynamics, is much larger than the empirical rest energy of an electron. If, as assumed in all `renormalized theories' of the electron, the empirical rest mass of a physical electron is the sum of its bare rest mass plus its electrostatic field energy ($/c^2$), then in BLTP electrodynamics the electron has to be assigned a negative bare rest mass.

physics.atom-ph↗
arXiv2019

The Einstein-Infeld-Hoffmann legacy in mathematical relativity. Part I: The classical motion of charged point particles

Einstein, Infeld, and Hoffmann (EIH) claimed that the field equations of general relativity theory alone imply the equations of motion of neutral matter particles, viewed as point singularities in space-like slices of spacetime; they also claimed that they had generalized their results to charged point singularities. While their analysis falls apart upon closer scrutiny, the key idea merits our attention. This rapport identifies necessary conditions for a well-defined general-relativistic joint initial value problem of $N$ classical point charges and their electromagnetic and gravitational fields. Among them, in particular, is the requirement that the electromagnetic vacuum law guarantees a finite field energy-momentum of a point charge. This disqualifies the Maxwell(--Lorentz) law used by EIH. On the positive side, if the electromagnetic vacuum law of Bopp, Landé--Thomas, and Podolsky (BLTP) is used, and the singularities equipped with a non-zero bare rest mass, then a joint initial value problem can be formulated in the spirit of the EIH proposal, and shown to be locally well-posed --- \emph{in the special-relativistic zero-$G$ limit}. With gravitational coupling (i.e. $G>0$), though, changing Maxwell's into the BLTP law and assigning a bare rest mass to the singularities is by itself not sufficient to obtain even a merely well-defined joint initial value problem: the gravitational coupling also needs to be changed, conceivably in the manner of Jordan and Brans--Dicke.

gr-qc↗
arXiv2019

The influence of gravity on the Boltzmann entropy of a closed universe

This contribution inquires into Clausius' proposal that "the entropy of the world tends to a maximum.'" The question is raised whether the entropy of "the world" actually does have a maximum; and if the answer is "Yes!," what such states of maximum entropy look like, and if the answer is "No!," what this could entail for the fate of the universe. Following R. Penrose, "the world" is modelled by a closed Friedman--Lemaitre type universe, in which a three-dimensional spherical "space" is filled with "matter" consisting of $N$ point particles, their large-scale distribution being influenced by their own gravity. As "entropy of matter" the Boltzmann entropy for a (semi-)classical macrostate, and Boltzmann's ergodic ensemble formulation of it for an isolated thermal equilibrium state, are studied. Since the notion of a Boltzmann entropy is not restricted to classical non-relativistic physics, the inquiry will take into account quantum theory as well as relativity theory; we also consider black hole entropy. Model universes having a maximum entropy state and those which don't will be encountered. It is found that the answer to our maximum entropy question is not at all straightforward at the general-relativistic level. In particular, it is shown that the increase in Bekenstein--Hawking entropy of general-relativistic black holes does not always compensate for the Boltzmann entropy of a piece of matter swallowed by a black hole.

physics.hist-ph↗
arXiv2019

Hilbert's "monkey saddle" and other curiosities in the equilibrium problem of three point particles on a circle for repulsive power law forces

This article determines all possible equilibrium arrangements (proper as well as pseudo) under repulsive power law forces of three point particles on the unit circle. These are the critical points of the sum over the three (standardized) Riesz pair interaction terms, each given by $V_s(r)= s^{-1}\left(r^{-s}-1 \right)$ when the real parameter $s \neq 0$, and by $V_0(r) := \lim_{s\to0}V_s(r) = -\ln r$; here, $r$ is the chordal distance between the particles in the pair. The bifurcation diagram which exhibits all these equilibrium arrangements together as functions of $s$ features three obvious "universal" equilibria, which do not depend on $s$, and two not-so-obvious continuous families of $s$-dependent non-universal isosceles triangular equilibria. The two continuous families of non-universal equilibria are disconnected, yet they bifurcate off of a common universal limiting equilibrium (the equilateral triangular configuration), at $s=-4$, where the graph of the total Riesz energy of the 3-particle configurations has the shape of a "monkey saddle." In addition, one of the families of non-universal equilibria also bifurcates off of another universal equilibrium (the antipodal arrangement), at $s=-2$. While the bifurcation at $s=-4$ is analytical, the one at $s=-2$ is not. The bifurcation analysis presented here is intended to serve as template for the treatment of similar N-point equilibrium problems on the d-dimensional sphere for small N.

math-ph↗
arXiv2018

On the Quantum-Mechanics of a Single Photon

It is shown that a Dirac(-type) equation for a rank-two bi-spinor field on Minkowski (configuration) spacetime furnishes a Lorentz-covariant quantum-mechanical wave equation in position-space representation for a single free photon. This equation does not encounter any of the roadblocks that have obstructed previous attempts (by various authors) to formulate a {quantum-mechanical} photon wave equation. In particular, it implies that the photon wave function yields conserved non-negative Born-rule-type quantum probabilities, and that its probability current density four-vector transforms properly under Lorentz transformations. Moreover, the eigenvalues of the pertinent photon Dirac Hamiltonian and the vector eigenvalues of the photon momentum operator yield the familiar Einstein relations $E=\hbarω$ and ${\bf p}=\hbar{\bf k}$, respectively. Furthermore, these spin-1 wave modes are automatically transversal without the need of an additional constraint on the initial data. Some comments on other proposals to set up a photon wave equation are supplied as well.

math-ph↗
arXiv2018

Do particles and anti-particles really annihilate each other?

Supported by results obtained with semi-classical quantization techniques, and with a quantum mechanical "square-root Klein-Gordon" operator, it is argued that Positronium (Ps) may exhibit a proper quantum-mechanical ground state whose energy level lies $\approx 2m c^2$ below its "hydrogenic (pseudo-) ground state" energy, where $m$ is the empirical rest mass of the electron. While the familiar hydrogenic pseudo-ground state of Ps is caused by the Coulomb attraction of electron and anti-electron, modified by small spin-spin and radiative QED corrections, the proper ground state is caused by the magnetic attraction between electron and anti-electron, which dominates over the electric one at short distances. This finding suggests that the familiar "annihilation" of electron and anti-electron is, in reality, simply yet another transition between two atomic energy levels, with the energy difference radiated off in form of photons --- except that the energy difference is huge: about 1 MeV instead of the few eV in a hydrogenic transition. In their proper ground state configuration the two particles would be so close that they would electromagnetically neutralize each other for most practical purposes, thus giving the appearance of an annihilation. Once in such a tightly bound state such pairs would hardly interact with normal matter and not be noticeable --- except through their gravitational effects in bulk! If the existence of such a low-energy ground state is confirmed it would imply that a significant part of the mysterious "dark matter" in the universe may consist of such matter-antimatter bound states.

quant-ph↗
arXiv2018

On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4π\sum_k a_k δ_{s_k}$ in Physics and Geometry

It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2})=4π\sum_{n=1}^N a_n δ_{s_n}$. Moreover, $u$ is real analytic away from the $s_n$. The result can be interpreted in at least two ways: (a) for any number of point charges of arbitrary magnitude and sign at prescribed locations $s_n$ in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as $|s|\to\infty$; (b) for any number of integral mean curvatures assigned to locations $s_n$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime, having lightcone singularities over the $s_n$ but being smooth otherwise, and whose height function vanishes as $|s|\to\infty$. No struts between the point singularities ever occur.

math-ph↗
arXiv2017

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-Variate Normal Random Polynomials

We study expected values of the polynomials $P_N^{}(z)=\prod_{1\leq n\leq N}(X_n^2+z^2)$ whose $2N$ zeros $\{\pm i X_k\}^{}_{k=1,...,N}$ are generated by $N$ identically distributed multi-variate mean-zero normal random variables $\{X_k\}^{N}_{k=1}$ with co-variance ${\rm{Cov}}_N^{}(X_k,X_l)=(1+\frac{σ^2-1}{N})δ_{k,l}+\frac{σ^2-1}{N}(1-δ_{k,l})$. In principle these can be evaluated in closed form for arbitrary $N$, yet commonly available computer algebra handles only $N$ up to a dozen (due to memory constraints). A list of the first three expected polynomials shows that the expressions become unwieldy already for moderate $N$. On the other hand, asymptotic evaluations of the large-$N$ regime for complex $z$ have traditionally been limited to analytic expansion techniques, several rigorous results are proved about this regime for complex $z$. Yet if $z$ is real one can also compute the large-$N$ asymptotics in the "infinite-degree" limit with the help of the familiar relative entropy principle for probability measures, a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to *{signed and complex measures}* governs the $N\to\infty$ asymptotics of the regime of imaginary $z$. Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

math-ph↗
arXiv2017

Order and Chaos in some deterministic infinite trigonometric products

This paper discusses some infinite trigonometric products which are characteristic functions of simple random walks on the real line; in fact, these define "random Riemann-$ζ$ functions," a notion which is explained. The concept of typicality for random Riemann $ζ$ functions is explained and connected to the Riemann hypothesis. Then it is shown that the distribution functions of these random walks are Schwarz functions. It is also shown that their characteristic function factors into the characteristic function of a Levy stable random variable and a subdominant fluctuating factor. As a corollary it also follows that amateur mathematician Benoit Cloitre's infinite trigonometric product $\prod_{n=1}^\infty \left[\frac23+\frac13\cos\left(\frac{x}{n^{2}}\right)\right] = e^{- C \,\sqrt{|x|} +\varepsilon(|x|)},$ with $|\varepsilon(|x|)| \leq K |x|^{1/3}$ for some $K>0$, and with $ C= \int\frac{\sinξ^2}{2+\cosξ^2}{\rm{d}}ξ;$ numerically, $C = 0.319905585... \sqrtπ$. This confirms a surmise of Benoit Cloitre. The $O\big(|x|^{1/3}\big)$ error bound is empirically found to be accurate for moderately sized $|x|$ but not for larger $|x|$. This difference $\varepsilon(|x|)$ between Cloitre's $\log \prod_{n\geq1}\left[\frac23 +\frac13\cos\left(\frac{x}{n^{2}}\right)\right]$ and its regular trend $-C\sqrt{|x|}$, although deterministic, appears to be an "empirically unpredictable" function. Our probabilistic investigation of this phenomenon connects the fluctuations to the "random Riemann-$ζ$ function with argument 2."

math.CA↗
arXiv2015

Dirac's point electron in the zero-gravity Kerr--Newman world

The results of a study of Dirac's Hamiltonian for a point electron in the zero-gravity Kerr--Newman spacetime are reported; here, "zero-gravity" means G to 0, where G is Newton's constant of universal gravitation, and the limit is effected in the Boyer--Lindquist coordinate chart of the maximal analytically extended, topologically nontrivial, Kerr--Newman spacetime. In a nutshell, the results are: the essential self-adjointness of the Dirac Hamiltonian; the reflection symmetry about zero of its spectrum; the location of the essential spectrum, exhibiting a gap about zero; and (under two smallness assumptions on some parameters) the existence of a point spectrum in this gap, corresponding to bound states of Dirac's point electron in the electromagnetic field of the zero-G Kerr--Newman ring singularity. The symmetry result of the spectrum extends to Dirac's Hamiltonian for a point electron in a generalization of the zero-G Kerr--Newman spacetime with different ratio of electric-monopole to magnetic-dipole moment. The results are discussed in the context of the general-relativistic Hydrogen problem. Also, some interesting projects for further inquiry are listed in the last section.

math-ph↗
arXiv2014

"Magic" numbers in Smale's 7th problem

Smale's 7-th problem concerns N-point configurations on the 2-dim sphere which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a pair. More generally, V_0(r) may be replaced by the standardized Riesz pair-energy V_s(r)= (r^{-s} -1)/s, which becomes - ln r in the limit s to 0, and the sphere may be replaced by other compact manifolds. This paper inquires into the concavity of the map from the integers N>1 into the minimal average standardized Riesz pair-energies v_s(N) of the N-point configurations on the 2-sphere for various real s. It is known that v_s(N) is strictly increasing for each real s, and for s<2 also bounded above, hence "overall concave." It is (easily) proved that v_{-2}(N) is even locally strictly concave, and that so is v_s(2n) for s<-2. By analyzing computer-experimental data of putatively minimal average Riesz pair-energies v_s^x(N) for s in {-1,0,1,2,3} and N in {2,...,200}, it is found that {v}_{-1}^x(N) is locally strictly concave, while v_s^x(N) is not always locally strictly concave for s in {0,1,2,3}: concavity defects occur whenever N in C^{x}_+(s) (an s-specific empirical set of integers). It is found that the empirical map C^{x}_+(s), with s in {-2,-1,0,1,2,3}, is set-theoretically increasing; moreover, the percentage of odd numbers in C^{x}_+(s), s in {0,1,2,3}, is found to increase with s. The integers in C^{x}_+(0) are few and far between, forming a curious sequence of numbers, reminiscent of the "magic numbers" in nuclear physics. It is conjectured that the "magic numbers" in Smale's 7-th problem are associated with optimally symmetric optimal-energy configurations.

math-ph↗