publications

2023

1. Article 26

   Dyons with phase δ=nθ

   Ricardo Heras

   The European Physical Journal Plus, 138, 991, 2023

   Abstract HTML PDF

   In a recent paper (Heras in Eur. Phys. J. Plus 138:329, 2023), we have demonstrated that when a dyon encircles an infinitely long solenoid enclosing electric and magnetic fluxes, its wave function accumulates a quantum phase invariant under electromagnetic duality transformations. In this paper, we show that this phase, in conjunction with the Witten effect, gives rise to a topological phase proportional to the vacuum angle \(θ\,\)and thereby connected with CP violation. We show that this phase becomes quantised in a vacuum state \(δ=nθ\,\)and that the most general vacuum state associated with this quantisation identifies with an Abelian form of the \(θ\,\)-vacua. We discuss two hypothetical interference effects in the vacuum where the angle \(θ\,\)could manifest.

2. Article 25

   The quantum phase of a dyon

   Ricardo Heras

   The European Physical Journal Plus, 138, 329, 2023

   Abstract HTML PDF

   When a dyon encircles an infinitely-long solenoid enclosing uniform electric and magnetic fields, its wave function accumulates a duality-invariant quantum phase, which is topological because it depends on a winding number and is nonlocal because the enclosed fields act on the dyon in regions where these fields vanish. Here, we derive this dyon phase and show how its duality symmetry unifies the Aharonov-Bohm phase with its dual phase. We obtain the energy levels, the two-slit interference shift, and the scattering amplitude associated with the duality-invariant quantum phase. Assuming that the dyon has spin 1/2, we show that this spin does not affect the introduced phase. We argue that a spin 1/2 dyon has electric and magnetic moments, the former being greater than the latter because of the Schwinger-Zwanziger quantisation condition.

2022

1. Article 24

   The Aharonov-Bohm effect in a closed flux line

   Ricardo Heras

   The European Physical Journal Plus, 137, 157, 2022

   Abstract HTML PDF

   The Aharonov-Bohm (AB) effect was convincingly demonstrated using a micro-sized toroidal magnet but it is almost always explained using an infinitely-long solenoid or an infinitely-long flux line. The main reason for this is that the formal treatment of the AB effect considering a toroidal configuration turns out to be too cumbersome. But if the micro-sized toroidal magnet is modelled by a closed flux line of arbitrary shape and size then the formal treatment of the AB effect is exact, considerably simplified, and well-justified. Here we present such a treatment that covers in detail the electromagnetic, topological, and quantum-mechanical aspects of this effect. We demonstrate that the AB phase in a closed flux line is determined by a linking number and has the same form as the AB phase in an infinitely-long flux line which is determined by a winding number. We explicitly show that the two-slit interference shift associated with the AB effect in a closed flux line is the same as that associated with an infinitely-long flux line. We emphasise the topological nature of the AB phase in a closed flux line by demonstrating that this phase is invariant under deformations of the charge path, deformations of the closed flux line, simultaneous deformations of the charge path and the closed flux line, and the interchange between the charge path and the closed flux line. We also discuss the local and nonlocal interpretations of the AB effect in a closed flux line and introduce a non-singular gauge in which the vector potential vanishes in all space except on the surface surrounded by the closed flux line, implying that this vector potential is zero along the trajectory of the charged particle except on the crossing point where this trajectory intersects the surface bounded by the closed flux line, a result that questions the alleged physical significance of the vector potential and thereby the local interpretation of the AB effect.

2. Article 23

   Topology, nonlocality and duality in classical electrodynamics

   José A. Heras, and Ricardo Heras

   The European Physical Journal Plus, 137, 641, 2022

   Abstract HTML PDF

   We have recently (Heras et al. in Eur. Phys. J. Plus 136:847, 2021) argued that classical electrodynamics can predict nonlocal effects by showing an example of a topological and nonlocal electromagnetic angular momentum. In this paper, we discuss the dual of this angular momentum which is also topological and nonlocal. We then unify both angular momenta by means of the electromagnetic angular momentum arising in the configuration formed by a dyon encircling an infinitely long dual solenoid enclosing uniform electric and magnetic fluxes and show that this electromagnetic angular momentum is topological because it depends on a winding number, is nonlocal because the electric and magnetic fields of this dual solenoid act on the dyon in regions for which these fields are excluded and is invariant under electromagnetic duality transformations. We explicitly verify that this duality-invariant electromagnetic angular momentum is insensitive to the radiative effects of the Liénard-Wiechert fields of the encircling dyon. We also show how duality symmetry of this angular momentum suggests different physical interpretations for the corresponding angular momenta that it unifies.

2021

1. Article 22

   Can classical electrodynamics predict nonlocal effects?

   José A. Heras, and Ricardo Heras

   The European Physical Journal Plus, 136, 847, 2021

   Abstract HTML PDF

   Classical electrodynamics is a local theory describing local interactions between charges and electromagnetic fields and therefore one would not expect that this theory could predict nonlocal effects. But this perception implicitly assumes that the electromagnetic configurations lie in simply connected regions. In this paper, we consider an electromagnetic configuration lying in a non-simply connected region, which consists of a charged particle encircling an infinitely long solenoid enclosing a uniform magnetic flux, and show that the electromagnetic angular momentum of this configuration describes a nonlocal interaction between the encircling charge outside the solenoid and the magnetic flux confined inside the solenoid. We argue that the nonlocality of this interaction is of topological nature by showing that the electromagnetic angular momentum of the configuration is proportional to a winding number. The magnitude of this electromagnetic angular momentum may be interpreted as the classical counterpart of the Aharonov-Bohm phase.

2. Article 21

   Reply to comments ‘On Feynman’s handwritten notes on electromagnetism and the idea of introducing potentials before fields’

   José A. Heras, and Ricardo Heras

   European Journal of Physics, 42, 018002, 2021

   Abstract HTML PDF

   We reply to some comments made by Davis (2020 Eur. J. Phys. 40 018001) on our paper (2020 Eur. J. Phys. 41 035202), by arguing that Davis’s assertions are unsupported in some cases and are unsatisfactory in other cases.

2020

1. Article 20

   On Feynman’s handwritten notes on electromagnetism and the idea of introducing potentials before fields

   José A. Heras, and Ricardo Heras

   European Journal of Physics, 41, 035202, 2020

   Abstract HTML PDF

   In his recently discovered handwritten notes on ’An alternate way to handle electrodynamics’ dated on 1963, Richard P Feynman speculated with the idea of getting the inhomogeneous Maxwell’s equations for the electric and magnetic fields from the wave equation for the vector potential. With the aim of implementing this pedagogically interesting idea, we develop in this paper the approach of introducing the scalar and vector potentials before the electric and magnetic fields. We consider the charge conservation expressed through the continuity equation as a basic axiom and make a heuristic handle of this equation to obtain the retarded scalar and vector potentials, whose wave equations yield the homogeneous and inhomogeneous Maxwell’s equations. We also show how this axiomatic-heuristic procedure to obtain Maxwell’s equations can be formulated covariantly in the Minkowski spacetime.

2. Article 19

   Helmholtz’s theorem for two retarded fields and its application to Maxwell’s equations

   José A. Heras, and Ricardo Heras

   European Journal of Physics, 41, 035201, 2020

   Abstract HTML PDF

   An extension of the Helmholtz theorem is proved, which states that two retarded vector fields \(F_1\,\)and \(F_2\,\)satisfying appropriate initial and boundary conditions are uniquely determined by specifying their divergences \(∇⋅F_1 =0\,\,\)and \(\,\,∇⋅F_2 =0\,\,\)and and their coupled curls \(-∇\times F_1-∂F_2/∂t\,\,\)and \(∇\times F_2-(1/c^2)∂F_1∂/ t\), where \(c\,\)is the propagation speed of the fields. When a corollary of this theorem is applied to Maxwell’s equations, the retarded electric and magnetic fields are directly obtained. The proof of the theorem relies on a novel demonstration of the uniqueness of the solutions of the vector wave equation

2019

1. Article 18

   The classical counterpart of the Aharonov-Bohm phase

   Ricardo Heras

   arXiv-eprint, 1902.01694, 2019

   Abstract HTML PDF

   The existence of the Aharonov-Bohm phase shows that the magnetic field may produce nonlocal effects in quantum mechanics. It is generally believed that such a nonlocal behavior of the magnetic field is not possible in classical physics and that this represents a clear difference between classical and quantum mechanics. Contrary to these beliefs, we argue that the classical counterpart of the Aharonov-Bohm phase, which is identified here with the electromagnetic angular momentum of the configuration formed by an electric charge moving around an infinitely-long solenoid enclosing a uniform magnetic flux, shows that the magnetic field may produce nonlocal effects in classical mechanics. We discuss this momentum in detail by putting special emphasis on its topological and nonlocal features. The momentum is topological because it depends on the number of windings the electric charge carries out around the solenoid and is nonlocal because the magnetic flux has no local consequences at any point on the charge trajectory. The topological feature allows us to introduce the concept of accumulated electromagnetic angular momentum and the nonlocal feature allows us to speak of a dynamical nonlocality attributable to the equations of classical electrodynamics.

2. Article 17

   Comment on “Monopole production via photon fusion and Drell-Yan processes: MadGraph implementation and perturbativity via velocity-dependent coupling and magnetic moment as novel features,”

   Ricardo Heras

   arXiv-eprint, 1902.11138, 2019

   Abstract HTML PDF

   In a recent study on monopole production [Eur. Phys. J. C (2018) 78: 966], Baines et al added the potential of a magnetic dipole to the Wu-Yang potentials for the Dirac monopole and claimed that this modified Wu-Yang configuration does not affect the Dirac quantisation condition. In this comment, we argue that their claim is incorrect by showing that their modified Wu-Yang configuration leads to an infinite number of quantisation conditions. In their study, they also incorrectly identified the magnetic field of the monopole with the magnetic field of the Dirac string and its attached magnetic monopole.

2018

1. Article 16

   Dirac quantisation condition: a comprehensive review

   Ricardo Heras

   Contemporary Physics, 59, 331, 2018

   Abstract HTML PDF

   In most introductory courses on electrodynamics, one is taught the electric charge is quantised but no theoretical explanation related to this law of nature is offered. Such an explanation is postponed to graduate courses on electrodynamics, quantum mechanics and quantum field theory, where the famous Dirac quantisation condition is introduced, which states that a single magnetic monopole in the Universe would explain the electric charge quantisation. Even when this condition assumes the existence of a not-yet-detected magnetic monopole, it provides the most accepted explanation for the observed quantisation of the electric charge. However, the usual derivation of the Dirac quantisation condition involves the subtle concept of an ’unobservable’ semi-infinite magnetised line, the so-called ’Dirac string,’ which may be difficult to grasp in a first view of the subject. The purpose of this review is to survey the concepts underlying the Dirac quantisation condition, in a way that may be accessible to advanced undergraduate and graduate students. Some of the discussed concepts are gauge invariance, singular potentials, single-valuedness of the wave function, undetectability of the Dirac string and quantisation of the electromagnetic angular momentum. Five quantum-mechanical and three semi-classical derivations of the Dirac quantisation condition are reviewed. In addition, a simple derivation of this condition involving heuristic and formal arguments is presented.

2017

1. Article 15

   Dirac Quantization Condition Holds with Nonzero Photon Mass

   Alfred S. Goldhaber, and Ricardo Heras

   arXiv-eprint, 1710.03321, 2017

   Abstract HTML PDF

   Dirac in 1931 gave a beautiful argument for the quantization of electric charge, which required only the existence in the universe of one magnetic monopole, because gauge invariance of the interaction between the pole and any charge could hold only if the product of the charge and the pole strength were quantized in half-integer multiples of the reduced Planck constant. However, if the photon had a nonzero mass, implying exponential decrease of the flux out of an electric charge, then Dirac’s argument might seem to fail. We demonstrate that the result still should hold. The key point is that magnetic charge, unlike electric charge, cannot be screened, so that on any surface enclosing the pole Dirac’s string, or equally the Wu-Yang gauge shift, must be present, and to make either of these invisible to charged particles the quantization condition is required.

2. Article 14

   Readers weigh in on how to teach physics

   Ricardo Heras

   Physics Today, 70, 11, 11, 2017

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3. Article 13

   Alternative routes to the retarded potentials

   Ricardo Heras

   European Journal of Physics, 38, 055203, 2017

   Abstract HTML Journal Highlights PDF

   Two procedures to introduce the familiar retarded potentials of Maxwell’s equations are reviewed. The first well-known procedure makes use of the Lorenz-gauge potentials of Maxwell’s equations. The second less-known procedure applies the retarded Helmholtz theorem to Maxwell’s equations. Both procedures are compared in the context of an undergraduate presentation of electrodynamics. The covariant form of both procedures is discussed for completeness. As a related discussion, two procedures to introduce the unfamiliar instantaneous potentials of Maxwell’s equations are also reviewed. The first procedure applies the standard Helmholtz theorem to Maxwell’s equations and the second one uses the Coulomb-gauge potentials of Maxwell’s equations. The retarded and instantaneous forms of the potentials of Maxwell’s equations are briefly commented upon. The retarded Helmholtz theorem is used to introduce the retarded potentials of Maxwell’s equations with magnetic monopoles.

4. Article 12

   Commentary: How to teach me physics: Tradition is not always a virtue,

   Ricardo Heras

   Physics Today, 70, 3, 10, 2017

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5. Article 11

   Four easy routes to the Lorentz transformations: addendum to ’Lorentz transformations and the wave equation’

   Ricardo Heras

   European Journal of Physics, 38, 019401, 2017

   Abstract HTML PDF

   In this paper I briefly discuss and compare four easy derivations of the Lorentz transformations. Two of these derivations assume the invariance of the Minkowski spacetime interval in inertial frames and the other two assume the invariance of the d’Alembert operator in these frames. These derivations are suitable for a first view of special relativity. Finally, I discuss the comment made by Di Rocco on my original paper, ’Lorentz transformations and the wave equation’ (2016 Eur. J. Phys. 37 025603).

2016

1. Article 10

   The Helmholtz theorem and retarded fields

   Ricardo Heras

   European Journal of Physics, 37, 065204, 2016

   Abstract HTML PDF

   Textbooks frequently use the Helmholtz theorem to derive expressions for electrostatic and magnetostatic fields but they do not usually apply this theorem to derive expressions for time-dependent electric and magnetic fields, even when there is no formal objection to doing so because the proof of the theorem does not involve time derivatives but only spatial derivatives. Here we address the question as to whether the Helmholtz theorem is useful in deriving expressions for the fields of Maxwell’s equations. We show that when this theorem is applied to Maxwell’s equations we obtain instantaneous expressions of the electric and magnetic fields, which are formally correct but of little practical usefulness. We then discuss two generalizations of the theorem which are shown to be useful in deriving the retarded fields.

2. Article 9

   Lorentz transformations and the wave equation

   Ricardo Heras

   European Journal of Physics, 37, 025603, 2016

   Abstract HTML PDF

   In this note we explicitly show how the Lorentz transformations can be derived by demanding form invariance of the d’Alembert operator in inertial reference frames.

3. Article 8

   Magnetorotational instabilities and pulsar kick velocities

   Ricardo Heras

   New Astronomy, 43, 6, 2016

   Abstract HTML PDF

   At the end of their birth process, neutron stars can be subject to a magnetorotational instability in which a conversion of kinetic energy of differential rotation into radiation and kinetic energies is expected to occur at the Alfvén timescale of a few ms. This birth energy conversion predicts the observed large velocity of neutron stars if during the evolving of this instability the periods are of a few ms and the magnetic fields reach values of \(10^16\)G.

2014

1. Article 7

   A review of Voigt’s transformations in the framework of special relativity

   Ricardo Heras

   arXiv-eprint, 1411.2559, 2014

   Abstract HTML PDF

   In 1887 Woldemar Voigt published the paper "On Doppler’s Principle," in which he demanded covariance to the homogeneous wave equation in inertial reference frames, assumed the invariance of the speed of light in these frames, and obtained a set of spacetime transformations different from the Lorentz transformations. Without explicitly mentioning so, Voigt applied the postulates of special relativity to the wave equation. Here, we review the original derivation of Voigt’s transformations and comment on their conceptual and historical importance in the context of special relativity. We discuss the relation between the Voigt and Lorentz transformations and derive the former from the conformal covariance of the wave equation.

2. Article 6

   The wave equation in the birth of spacetime symmetries

   Ricardo Heras

   arXiv-eprint, 1407.3425, 2014

   Abstract HTML PDF

   In 1887 Voigt published a paper dedicated to the Doppler effect in which he demanded form invariance to the wave equation in inertial frames and obtained a set of spacetime transformations now known as the Voigt transformations. In 1905 Poincaré showed that the wave equation was also invariant under the Lorentz transformations. Voigt and Lorentz transformations are then closely related, but this relation is not widely known in the standard literature. In this paper we derive the Lorentz transformations from the invariance of the D’Alembert operator (\(□^2=□’^2\)) and the Voigt transformations from the conformal invariance of the D’Alembert operator (\(□^2=(1/γ^2)□’^2\), where \(γ=1/\sqrt1-v^2/c^2\)). The homogeneous scalar wave equation is then invariant under the Lorentz transformations and conformally invariant under the Voigt transformations. We suggest a presentation of special relativity in which the Voigt transformations are commented after discussing the Galilean transformations but before presenting the Lorentz transformations.

2013

1. Article 5

   Individualism: The legacy of great physicists

   Ricardo Heras

   Physics Today, online, 2013

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2. Article 4

   Birth accelerations of neutron stars

   Ricardo Heras

   Proceedings of the International Astronomical Union, s291, 399, 2013

   Abstract HTML PDF

   We suggest that neutron stars experienced at birth three related physical changes, which may originate in magneto-rotational instabilities: (i) an increase in period from the initial value \(P_0\)to the current value \(Ps\), implying a change of rotational energy \(∆E_rot\); (ii) an exponential decay of its magnetic field from the initial value B0 to the current surface value Bs, implying a change of radiative energy \(∆E_rad\); and (iii) an increase of space velocity from the initial value v0 to the current value v, implying a change of kinetic energy \(∆E_kin\). These changes are assumed to be connected by \(∆E_rad + ∆E_kin=∆E_rot\). This means that the radiation loss and increase of kinetic energy are both at the expense of a rotational energy loss. It is shown that this energy conversion occurs during times of order of \(10^4\)s if the neutron stars are born with magnetic fields in the range of \(10^15-10^16\)G and initial periods in range 1-20 ms. It is shown that the birth accelerations of neutron stars are of the order of \(10^8\)g.

2012

1. Article 3

   Pulsars are Born as Magnetars

   Ricardo Heras

   Astronomical Society of the Pacific Conference Series, 466, 253, 2012

   Abstract HTML PDF

   This paper suggests the idea that all neutron stars experienced at birth an ultrafast decay of their magnetic fields from their initial values to their current surface values. If the electromagnetic energy radiated during this field decay is converted into kinetic energy of the neutron star via the radiation reaction mechanism then the decay time is of the order of 10-4s provided that the initial magnetic fields lie in the range of \(10^14-10^16G\). This means that all neutron stars are born with magnetic fields typical of magnetars. According to this model the neutron stars acquire their observed high space velocities during the birth ultrafast decay of their magnetic fields. The origin of this field decay points to magnetic instabilities occurring at the end of the birth process.

2011

1. Article 2

   The magnetar origin of pulsars

   Ricardo Heras

   arXiv-eprint, 1104.5060, 2011

   Abstract HTML PDF

   This paper suggests the idea that all neutron stars experienced at birth an ultrafast decay of their magnetic fields from their initial values to their current surface values. If the electromagnetic energy radiated during this field decay is converted into kinetic and rotational energies of the neutron star then the decay time is of the order of \(10^-4\)s provided that the initial magnetic fields lie in the range of \(10^15-10^16\)G and the initial periods in the range 1-20 ms. This means that all neutron stars are born with magnetic fields typical of magnetars and periods typical of millisecond pulsars. More explicitly, energy considerations and a birth-ultrafast-magnetic-field decay imply a model that consistently relates birth properties of neutron stars such as the initial period and the initial magnetic field with current properties such as the observed period, the surface magnetic field and the transverse velocity. This model provides a solution to the long-standing problem of finding a physical explanation for the observed high velocities of neutron stars. These stars acquire their space velocities during the birth ultrafast decay of their magnetic fields. The origin of this field decay points to magnetic instabilities, which are inevitable if neutron stars are born as magnetars.

2. Article 1

   Initial accelerations of pulsars caused by external kicks

   Ricardo Heras

   arXiv-eprint, 1103.1929, 2011

   Abstract HTML PDF

   Using energy conservation we show that if a sudden external force of unknown nature causes a newborn pulsar of mass \(M=1.4M_⊙\)and radius \(R=10\)km to have the acceleration a0 then we can derive the relation \(a_0≃k^-1v B^-2/3 \)s, where \(v\)is the velocity of the pulsar, \(B_0\)its initial magnetic field and \(k≈10^-17G^-2/3\)s. This relation predicts that a newborn pulsar with \(v=450\)km experienced the acceleration \(a_0≃10^13\)g when \(B_0≃10^13\)G. An external force producing such an acceleration seems not to be physically feasible. This pulsar could have experienced the more realistic acceleration \(a_0\simeq10^9\)g if \(B_0≃10^18\)G. But this huge magnetic field seems to be unrealistic.