Classical Field Theory Research Articles

Application of Herglotz’s variational principle to electromagnetic systems with dissipation

This work applies the contact formalism of classical mechanics and classical field theory, introduced by Herglotz and later developed in the context of contact geometry, to describe electromagnetic systems with dissipation. In particular, we study an electron in a non-perfect conductor and a variation of the cyclotron radiation. In order to apply the contact formalism to a system governed by the Lorentz force, it is necessary to generalize the classical electromagnetic gauge and add a new term in the Lagrangian. We also apply the k-contact theory for classical fields to model the behavior of electromagnetic fields themselves under external damping. In particular, we show how the theory describes the evolution of electromagnetic fields in media under some circumstances. The corresponding Poynting theorem is derived. We discuss its applicability to the Lorentz dipole model and to a highly resistive dielectric.

Spectral walls at one loop

A spectral wall is a surface in a moduli space of classically BPS solitons where an internal excitation crosses the continuum mass threshold. It has recently been shown that spectral walls in classical field theory repel solitons whose corresponding internal excitations are excited. We investigate, for the first time, the fate of spectral walls in a quantum theory, calculating the instantaneous acceleration of a wave packet of classically BPS antikinks in the presence of an impurity. We find that, in the quantum theory, the antikinks are repelled even when the bound mode is not excited. Perhaps due to this repulsive force, the dramatic classical effects of the spectral wall are not seen in our quantum calculation.

Classical Kalb-Ramond field theory in curved spacetimes

We propose to develop the Kalb-Ramond theory in four-dimensional spacetime at the level of a classical field theory by following the same formal development steps as in Maxwell theory of standard electrodynamics. Solutions of Kalb-Ramond theory in the presence of static sources in various curved spacetimes are then analyzed. A question that we address here is that of a possible Kalb-Ramond polarization in curved spacetimes, like one can encounter a dielectric polarization in ordinary Maxwell electrodynamics in certain types of spacetimes.

Field-Theoretical Representation of Interactions between Particles: Classical Relativistic Probability-Free Kinetic Theory

It was proven that the class of stable interatomic potentials can be represented exactly as a superposition of Yukawa potentials. In this paper, an auxiliary scalar field was introduced to describe the dynamics of a system of neutral particles (atoms) in the framework of classical field theory. In the case of atoms at rest, this field is equivalent to the interatomic potential, but in the dynamic case, it describes the dynamics of a system of atoms interacting through a relativistic classical field. A relativistic Lagrangian is proposed for a system consisting of atoms and an auxiliary scalar field. A complete system of equations for the relativistic dynamics of a system consisting of atoms and an auxiliary field was obtained. A closed kinetic equation was derived for the probability-free microscopic distribution function of atoms. It was shown that the finite mass of the auxiliary field leads to a significant increase in the effect of interaction retardation in the dynamics of a system of interacting particles.

Application of Hirota operators for controlling soliton interactions for Bose-Einstien condensate and quintic derivative nonlinear Schrödinger equation

With the help of the Hirota bilinear method (HBM), we study soliton interactions of quasi-1D Bose-Einstein Condensate system (BECs) with dipole-dipole attraction and repulsion. BEC is an extended form of nonlinear Schrödinger equation (NLSE) and it consists of quadratic-cubic nonlinearities, linear gain or loss and time modulated dispersion. Due to its spatially varying coefficients property it has significance in the field of fluid dynamics, classical and quantum field theories, nonlinear optics and physics etc. We will also discuss soliton interactions with graphically descriptions for QDNLSE. We obtain some parabolic, anti parabolic, M-shaped, W-shaped, butterflies, bright, anti dark, V-shaped, S-shaped and other solitons for our governing models.

Field focusing on carbon nanohorns

Abstract Carbon nanohorns are a unique sharp conical nanostructure, with various potential applications including field emitters. Inspired by the recently measured negative real part of the dielectric constant of carbon nanotubes in the mid- and far-infrared regions, we numerically investigate the field focusing effect around carbon nanohorns. Our present study calculates the field enhancement factors at the tips of carbon nanohorns, quantitatively representing the degree of field focusing. Our calculations are based on the classical electromagnetic field theory in the quasistatic limit, for a model system consisting of a subwavelength-scale prolate hemispheroidal body protruding from a conducting plane. Field enhancement factors of about 300 and 100 in the air and H2O, respectively, are observed around 0.1 eV (∼10 μm, ∼20 THz) for the incident electric field parallel to the carbon nanohorn. For the far-infrared region, the field enhancement factor ascends to 6 × 106 at 3 meV (∼400 μm, ∼0.7 THz) and 7 × 108 at 2 meV (∼600 μm, ∼0.5 THz) for the nanohorn aspect ratio of 100 and 1000, respectively. Our findings could lead to further improvement of carbon-based electrical and optical devices.

Hasse invariant for the tame Brauer group of a higher local field

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras, which are given a priori as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or period-index bound) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.

On somep‐adic Galois representations and form class groups

Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, and let $H_K$ be its Hilbert class field. For each prime $p$ we examine two $p$-adic abelian objects in view of class field theory. First, we investigate the image of a $p$-adic Galois representation of $\mathrm{Gal}(\overline{\mathbb{Q}}/H_K)$ attached to an elliptic curve with complex multiplication. Second, we construct a $p$-adic form class group that is isomorphic to a quotient group of $\mathrm{Gal}(K_{(p^\infty)}/H_K)$, where $K_{(p^\infty)}$ is the maximal abelian extension of $K$ unramified outside prime ideals lying above $p$.

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.

Faster-than-light solitons in [formula omitted] dimensions

The existence of a faster-than-light particle is in direct opposition to Einstein’s relativity and the principle of causality. However, we show that the theory of classical relativistic fields is not inherently inconsistent with the existence of the faster-than-light particle-like soliton solutions in 1+1 dimensions. We introduce two extended KG models (k-fields) that lead to a zero-energy and a nonzero-energy stable particle-like solution with faster-than light speeds, respectively.

On Lorentz Invariant Complex Scalar Fields

We obtain a Lorentz covariant wave equation whose complex wave function transforms under a Lorentz boost according to the following rule, Ψ x ⟶ e i / ℏ f x Ψ x . We show that the spacetime-dependent phase f x is the most natural relativistic extension of the phase associated with the transformation rule for the nonrelativistic Schrödinger wave function when it is subjected to a Galilean transformation. We then generalize the previous analysis by postulating that Ψ x transforms according to the above rule under proper Lorentz transformations (boosts or spatial rotations). This is the most general transformation rule compatible with a Lorentz invariant physical theory whose observables are bilinear functions of the field Ψ x . We use the previous wave equations to describe several physical systems. In particular, we solve the bound state and scattering problems of two particles which interact both electromagnetically and gravitationally (static electromagnetic and gravitational fields). The former interaction is modeled via the minimal coupling prescription while the latter enters via an external potential. We also formulate logically consistent classical and quantum field theories associated with these Lorentz covariant wave equations. We show that it is possible to make those theories equivalent to the Klein-Gordon theory whenever we have self-interacting terms that do not break their Lorentz invariance or if we introduce electromagnetic interactions via the minimal coupling prescription. For interactions that break Lorentz invariance, we show that the present theories imply that particles and antiparticles behave differently at decaying processes, with the latter being more unstable. This suggests a possible connection between Lorentz invariance-breaking interactions and the matter-antimatter asymmetry problem.

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.

ZERO-CYCLES ON NORMAL PROJECTIVE VARIETIES

AbstractWe prove an extension of the Kato–Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s formula for the Chow groups of $0$ -cycles on such schemes. We identify the Chow group of $0$ -cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasiprojective schemes over algebraically closed fields. This completes the missing p-part in the torsion theorem of Spieß and Szamuely.

On Modified Scattering for 1D Quadratic Klein–Gordon Equations With Non-Generic Potentials

AbstractWe consider the asymptotic behavior of small global-in-time solutions to a 1D Klein–Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to continue the investigation of the occurrence of a novel modified scattering behavior of the solutions that involves a logarithmic slow-down of the decay rate along certain rays. This phenomenon is ultimately caused by the threshold resonance of the linear Klein–Gordon operator. It was previously uncovered for the special case of the zero potential in [51]. The Klein–Gordon model considered in this paper is motivated by the asymptotic stability problem for kink solutions arising in classical scalar field theories on the real line.

Fractional formulation of Podolsky Lagrangian density

Lagrangians which depend on higher-order derivatives appear frequently in many areas of physics. In this paper, we reformulate Podolsky's Lagrangian in fractional form using left-right Riemann-Liouville fractional derivatives. The equations of motion are obtained using the fractional Euler Lagrange equation. In addition, the energy stress tensor and the Hamiltonian are obtained in fractional form from the Lagrangian density. The resulting equations are very similar to those found in classical field theory.

Conformal elasticity of mechanism-based metamaterials

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. However, flexible metamaterials often contain an additional approximate symmetry due to the presence of a designer soft strain pathway. Here we show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the well-studied—conformal—maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a unique method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

The Dynamics of Human Society Evolution: An Energetics Approach

Human society is an open system that evolves by coupling with various known and unknown (energy) fluxes. How do these dynamics unfold precisely? Energetics may provide further insights. We expand on Navier-Stokes’ approach to study non-equilibrium dynamics in a field that evolves over time. Based on social fi eld th eory, an induction of th e cl assical fie ld theories, we define social force, social energy, and the Hamiltonian of an individual in society. The equations for the evolution of an individual and society are sketched out based on the time-dependent Hamiltonian that includes power dynamics. We demonstrate in this paper that Lotka-Volterra-type equations can be derived from the Hamiltonian equation in the social field.

The vanishing of Iwasawa's $\mu$-invariant implies the weak Leopoldt conjecture

Let K denote a number field containing a primitive p -th root of unity; if p=2 , then we assume K to be totally imaginary. If K_\infty/K is a \mathbb{Z}_p -extension such that no prime above p splits completely in K_\infty/K , then the vanishing of Iwasawa's invariant \mu(K_\infty/K) implies that the weak Leopoldt Conjecture holds for K_\infty/K . This is actually known due to a result of Ueda, which appears to have been forgotten. We present an elementary proof which is based on a reflection formula from class field theory. In the second part of the article, we prove a generalisation in the context of non-commutative Iwasawa theory: we consider admissible p -adic Lie extensions of number fields, and we derive a variant for fine Selmer groups of Galois representations over admissible p -adic Lie extensions.

$\ell$-class groups of fields in Kummer towers

Let $\ell$ and $p$ be prime numbers and $K_{n,m}=\mathbb{Q}(p^{\frac{1}{\ell^n}},\zeta_{2\ell^{m}})$. We study the $\ell$-class group of $K_{n,m}$ in this paper. When $\ell=2$, we determine the structure of the $2$-class group of $K_{n,m}$ for all $(n,m)\in \mathbb{Z}_{\geq 0}^2$ in the case $p=2$ or $p\equiv 3, 5\bmod{8}$, and for $(n,m)=(n,0)$, $(n,1)$ or $(1,m)$ in the case $p\equiv 7\bmod{16}$, eneralizing the results of Parry about the $2$-divisibility of the class number of $K_{2,0}$. We also obtain results about the $\ell$-class group of $K_{n,m}$ when $\ell$ is odd and in particular $\ell=3$. The main tools we use are class field theory, including Chevalley's ambiguous class number formula and its generalization by Gras, and a stationary result about the $\ell$-class groups in the $2$-dimensional Kummer tower $\{K_{n,m}\}$.

Electric Field Theory Based on Weber's Electrodynamics

Weber's electrodynamics, as an alternative to classical electromagnetic field theory, can successfully explain many electromagnetic phenomena. However, the Weber's electrodynamics involves integrating interactions among electric particles for each application, which can be sometimes very tedious. In this study, we try to develop an electric field theory, which sums the contribution of each particle as a field, and then apply this field to the particle of interest. This way, the application of Weber's electrodynamics is greatly simplified.