arXiv subjects

H. Arodz

Publications and source records attributed to H. Arodz.

At least 19 recordsLinked to original sources

arXiv2022

Axial momentum and quantization of the Majorana field

New approach to quantization of the relativistic Majorana field is presented. It is based on expansion of the field into eigenfunctions of the axial momentum -- a novel observable introduced recently. Relativistic invariance is used as the main guiding principle instead of canonical formalism. Hidden structure of the quantized Majorana field in the form of real Clifford algebra of Hermitian fermionic operators is unveiled. Generators of the Poincaré transformations in the Fock space are found as solutions of certain operator equations, without invoking the principle of correspondence with classical conserved quantities. Also operators of parity $\hat{\mbox{P}}$ and time reversal $\hat{\mbox{T}}$ are constructed.

hep-th↗

arXiv2020

Relativistic Quantum Mechanics of the Majorana Particle

This article is a pedagogical introduction to relativistic quantum mechanics of the free Majorana particle. This relatively simple theory differs from the well-known quantum mechanics of the Dirac particle in several important aspects. We present its three equivalent formulations. Next, so called axial momentum observable is introduced, and general solution of the Dirac equation is discussed in terms of eigenfunctions of that operator. Pertinent irreducible representations of the Poincaré group are discussed. Finally, we show that in the case of massless Majorana particle the quantum mechanics can be reformulated as a spinorial gauge theory.

quant-ph↗

arXiv2019

Ehrenfest's Theorem Revisited

Historically, Ehrenfest's theorem (1927) is the first one which shows that classical physics can emerge from quantum physics as a kind of approximation. We recall the theorem in its original form. Next, we highlight its generalizations to the relativistic Dirac particle, and to a particle with spin and izospin. We argue that apparent classicality of the macroscopic world can probably be explained within the framework of standard quantum mechanics.

physics.hist-ph↗

arXiv2018

Axial momentum for the relativistic Majorana particle

The Hilbert space of states of the relativistic Majorana particle consists of normalizable bispinors with real components, and the usual momentum operator $- i \nabla$ can not be defined in this space. For this reason, we introduce the axial momentum operator, $ - i γ_5 \nabla$ as a new observable for this particle. In the Heisenberg picture, the axial momentum contains a component which oscillates with the amplitude proportional to $m/E$, where $E$ is the energy and $m$ the mass of the particle. The presence of the oscillations discriminates between the massive and massless Majorana particle. We show how the eigenvectors of the axial momentum, called the axial plane waves, can be used as a basis for obtaining the general solution of the evolution equation, also in the case of free Majorana field. Here a novel feature is a coupling of modes with the opposite momenta, again present only in the case of massive particle or field.

physics.gen-ph↗

arXiv2013

Three-body forces from a classical nonlinear field

Forces in the systems of two opposite sign and three identical charges coupled to the dynamical scalar field of the signum-Gordon model are investigated. Three-body force is present, and the exact formula for it is found. Flipping the sign of one of the two charges changes not only the sign but also the magnitude of the force. Both effects are due to nonlinearity of the field equation.

hep-th↗

arXiv2013

Total screening and finite range forces from ultra-massive scalar fields

Force between static point particles coupled to a classical ultra-massive scalar field is calculated. The field potential is proportional to the modulus of the field. It turns out that the force exactly vanishes when the distance between the particles exceeds certain finite value. Moreover, each isolated particle is surrounded by a compact cloud of the scalar field that completely screens its scalar charge.

hep-th↗

arXiv2012

Note on scale invariance and self-similar evolution in (3+1)-dimensional signum-Gordon model

Several classes of self-similar, spherically symmetric solutions of relativistic wave equation with nonlinear term of the form sign(ϕ) are presented. They are constructed from cubic polynomials in the scale invariant variable t/r. One class of solutions describes a process of wiping out the initial field, another an accumulation of field energy in a finite and growing region of space.

hep-th↗

arXiv2011

Swaying oscillons in the signum-Gordon model

We present a new class of oscillons in the (1+1)-dimensional signum-Gordon model. The oscillons periodically move to and fro in the space. They have finite total energy, finite size, and are strictly periodic in time. The corresponding solutions of the scalar field equation are explicitly constructed from the second order polynomials in the time and position coordinates.

hep-th↗

arXiv2007

Compact oscillons in the signum-Gordon model

We present explicit solutions of the signum-Gordon scalar field equation which have finite energy and are periodic in time. Such oscillons have a strictly finite size. They do not emit radiation.

hep-th↗

arXiv2006

Scaling, self-similar solutions and shock waves for V-shaped field potentials

We investigate a (1+1)-dimensional nonlinear field theoretic model with the field potential $V(ϕ)| = |ϕ|.$ It can be obtained as the universal small amplitude limit in a class of models with potentials which are symmetrically V-shaped at their minima, or as a continuum limit of certain mechanical system with infinite number of degrees of freedom. The model has an interesting scaling symmetry of the 'on shell' type. We find self-similar as well as shock wave solutions of the field equation in that model.

hep-th↗

arXiv2005

Field-theoretic Models with V-shaped Potentials

In this lecture we outline the main results of our investigations of certain field-theoretic systems which have V-shaped field potential. After presenting physical examples of such systems, we show that in static problems the exact ground state value of the field is achieved on a finite distance - there are no exponential tails. This applies in particular to soliton-like object called the topological compacton. Next, we discuss scaling invariance which appears when the fields are restricted to small amplitude perturbations of the ground state. Evolution of such perturbations is governed by nonlinear equation with a non-smooth term which can not be linearized even in the limit of very small amplitudes. Finally, we briefly describe self-similar and shock wave solutions of that equation.

hep-th↗

arXiv2005

Chain of impacting pendulums as non-analytically perturbed sine-Gordon system

We investigate a mechanical system consisting of infinite number of harmonically coupled pendulums which can impact on two rigid rods. Because of gravitational force the system has two degenerate ground states. The related topological kink - likely the simplest one presented in literature so far - is a compacton, that is it has strictly finite extension. In the present paper we elucidate the relation of such system with sine-Gordon model. Also, solutions describing waves with large amplitude, and an asymptotic formula for the width of the kink are obtained.

cond-mat.other↗

arXiv2003

Symmetry breaking transition and appearance of compactons in a mechanical system

Recently we have described a mechanical system which exhibits spontaneous breaking of Z_2 symmetry and related topological kinks called compactons. The corresponding field potential is not differentiable at its global minima. Therefore, standard derivation of dispersion relation $ω(k)$ for small perturbations around the ground state can not be applied. In the present paper we obtain the dispersion relation. It turns out that evolution equation remains nonlinear even for arbitrarily small perturbations. The shape of the resulting running wave is piecewise combined from cosh functions. We also analyse dynamics of the symmetry breaking transition. It turns out that the number of produced compacton-anticompacton pairs strongly depends on the form of initial perturbation of the unstable former ground state.

hep-th↗

arXiv2003

Perturbative expansion for the half-integer rectilinear disclination line in the Landau-de Gennes theory

The structure of the half-integer rectilinear disclination line within the framework of the Landau-de Gennes effective theory of nematic liquid crystals is investigated. The consistent perturbative expansion is constructed for the case of $L_2\neq 0$. It turns out that such expansion can be performed around only a discrete subset of an infinite set of the degenerate zeroth order solutions. These solutions correspond to the positive and negative wedge disclination lines and to four configurations of the twist disclination line. The first order corrections to both the order parameter field as well as the free energy of the disclination lines have been found. The results for the free energy are compared with the ones obtained in the Frank-Oseen-Zocher director description.

cond-mat.soft↗

arXiv2002

Topological Compactons

One dimensional topological kink which has strictly finite size without any exponential or power-like tail is presented. It can be observed in a simple mechanical system akin to the one used in order to demonstrate sinus-Gordon solitons.

nlin.PS↗

arXiv2001

Electrostatics of Pagels-Tomboulis effective model

Long time ago Pagels and Tomboulis have proposed a model for the nonperturbative gluodynamics which in the Abelian sector can be reduced to a strongly nonlinear electrodynamics. In the present paper we investigate Abelian, static solutions with external charges in that model. Nonzero total charge implies that the corresponding field has infinite energy due to slow fall off at large distances. For a pair of opposite charges the energy is finite -- it grows like $R^α$, $0<α<1$ with the distance $R$ between the charges.

hep-ph↗

arXiv2001

Perturbations of planar interfaces in Ginzburg-Landau models

Certain dissipative Ginzburg-Landau models predict existence of planar interfaces moving with constant velocity. In most cases the interface solutions are hard to obtain because pertinent evolution equations are nonlinear. We present a systematic perturbative expansion which allows us to compute effects of small terms added to the free energy functional of a soluble model. As an example, we take the exactly soluble model with single order parameter $ϕ$ and the potential $V_0(ϕ) = Aϕ^2 + B ϕ^3 + ϕ^4$, and we perturb it by adding $V_1(ϕ) = {1/2} ε_1 ϕ^2 \partial_i ϕ\partial_i ϕ+ 1/5 ε_2 ϕ^5 + 1/6 ε_3 ϕ^6. $ We discuss the corresponding changes of the velocity of the planar interface.

cond-mat.soft↗

arXiv2000

Evolution of interfaces and expansion in width

Interfaces in a model with a single, real nonconserved order parameter and purely dissipative evolution equation are considered. We show that a systematic perturbative approach, called the expansion in width and developed for curved domain walls, can be generalized to the interfaces. Procedure for calculating curvature corrections is described. We also derive formulas for local velocity and local surface tension of the interface. As an example, evolution of spherical interfaces is discussed, including an estimate of critical size of small droplets.

cond-mat.soft↗