Instantaneous electromagnetic interactions

Newtonian gravity introduced a model of instantaneous direct interactions among massive particles. This model was latter replicated by Coulomb for charged particles. Those models have been traditional named the action-at-a-distance model; although, this name is misleading and generated unending polemics among physicists and philosophers about how a particle can act at a distance in a vacuum over distant particles. A better name for this model is direct-particle-interaction. Maxwell electrodynamics and General Relativity introduced an alternative model of contact-action, where particles don't interact directly but by means of a mediator. The former theory uses fields, whereas the latter uses a concept of curved spacetime as mediator. In the contact-action model a particle emits a signal —e.g. a photon— which propagates through the media —e.g. the electromagnetic field— until reaching a far particle, which then feels the force or interaction of the first particle. Since the maximum possible speed is the speed of light, interactions are retarded in this model. The common belief during almost a century has been that the contact-action model is accurate and that instantaneous interactions have been disproved. Nothing more far from reality! In fact, partially due to the defects of the contact-action model, partially due to set of modern experiments, the models of instantaneous interactions are seeing a renaissance in the specialized literature. It is worth to revisit this topic and clarify some misunderstandings in the literature. I will limit to electromagnetic interactions, but the material discussed here can easily ported to gravitation. We will start with the original Coulomb potential at point \( x \) 'generated' by another charge \( e \) placed at \( r \) \[ \phi(x, t) = K \frac{e}{|R(t)|} = K \frac{e}{|x - r(t)|} . \] This potential is instantaneous because depends on the position of the 'source' particle at present time \( t \). Now we will expand the position of the 'source' charge around its position at some early time \( t_0 \) \[ r(t) = r(t_0) + v(t_0) (t-t_0) + a(t_0) (t-t_0)^2 / 2 + \cdots \] and assuming that this charge is non-accelerating at the initial time \( t_0 = (t - |R(t_0)|/c) \) we obtain the next potential \[ \phi(x, t) = K \frac{e}{|R(t_0)| - v(t_0) R(t_0)/c} , \] which is evidently the scalar Lienard & Wiechert potential. The vector Lienard & Wiechert potential can be obtained in the same way if we start from the instantaneous potential \[ A(x, t) = K \frac{ev}{|R(t)|} = K \frac{ev}{|x - r(t)|} . \] Note that the Lienard & Wiechert potentials have been derived under the approximation of particles being in inertial initial states, \( a(t_0) = 0 \). This means that Lienard & Wiechert potentials aren't complete and this explains why they have to be complemented by adding reaction-radiation potentials to the equations of motion for curing such issues like non-conservation of energy on accelerating particles. Not only those additional potentials are obtained ad hoc but the resulting improved equations of motion are still subjected to criticism due to non-physical behaviors. Note that the \( a(t_0) = 0 \) approximation also explains why the Lienard & Wiechert potentials for a moving charge can be obtained from the potentials for a charge at rest \( A=0 \) and \( \phi = \phi_\mathrm{Coulomb} \) applying the Lorentz transformations between the frame where the particle is at rest and the frame where the particle is moving with velocity \( v \). The Lorentz transformations can be applied because the frames are inertial, i.e., the particle is non-accelerating. In fact some derivations explicitly assume that the charge is moving with "with uniform velocity \( v \) through a frame \( S \)". This same approximation is also the reason why the quantum field theory assumes as the only physicall admisible states those of free particles, i.e. non-accelerating. This is picturesquely described in Feynman diagrams
The diagram consider particles in free motion, until a virtual photon is emitted and absorbed and both electrons change their state of motion to a new inertial state. Quantum field theory only can rigorously describe the initial and the final states —before and after the interaction— but not cannot provide an accurate description of what happens during the interaction. The Lienard & Wiechert potentials have been traditionally associated to retarded interactions because positions and velocities of the 'source' particles are evaluated at early time \( t_0 \). However, I have shown how they can be derived from instantaneous potentials that are function of positions and velocities at present time \( t \). This urge to consider what is the origin of the myth of retarded interactions. For such goal I will start again with the Coulomb potential without any lost of generality, because the application to the vector potential \( A \) is straightforward, \[ \phi(x, t) = K \frac{e}{|x - r(t)|} \] which will be rewritten as \[ \phi(x, t) = K \int \frac{\rho(y,t)}{|x - y|} \mathrm{d}y \] using a pure state electron density in position space \( \rho(y,t) = e\delta(y - r(t)) \). Now we can use the equation of motion to relate the present density to a previous density \[ \phi(x, t) = K \int \exp[L(t-t_0)] \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y .\] \( L \) in the above expression is the Liouvillian. We can now see clearly which is the equivalence between instantaneous and retarded potentials. A kernel \( 1/|x - y| \) evaluated at present time \( t \) is identical to a modified kernel evaluated at retarded time \( t_0 \) \[ \left\{ \frac{1}{|x - y|}\right\}_t = \left\{ \exp[L(t-t_0)] \frac{1}{|x - y|}\right\}_{t_0} .\] If we replace the full Liouvillian by its free part \( L^\mathrm{free} = - v \nabla \) we obtain the kernel of the Lienard & Wiechert potentials \[ \exp[L^\mathrm{free}(t-t_0)] \frac{1}{|x - y|} = \frac{1}{\kappa |x - y|} ,\] with \( \kappa = 1 - v(x-y)/|x-y|c \). Note that the denominator of the kernel being linear in space variables implies that the power series expansion of the exponential vanishes identically after the linear term in the Liouvillian. In the introduction we derive the Lienard & Wiechert potentials by neglecting acceleration and higher order terms. We can now confirm that the Lienard & Wiechert potentials are an exact consequence of the free component of the full Liouvillian, this free component of course describe inertial particles. The interaction Liovillian will introduce acceleration and higher-order corrections to the Lienard & Wiechert potentials. Corrections to the Lienard & Wiechert potentials will be explored elsewhere, now we want to identify some flaws have remained unnoticed in the electromagnetic literature during decades. Let us start with the ordinary 'wave' equation for the scalar potential \[ \square \phi = -4\pi K \rho \] Note that the right-hand-side contains the instantaneous charge density, not the density at early times. Now the ordinary literature integrates the equation and obtains the approximated potential \[ \phi = K \int \rho(t',y) \frac{\delta(t-t'-|x-y|/c)}{|x-y|} \mathrm{d}y \, \mathrm{d}t' \] If we integrate first on position and then on time we obtain the Lienard & Wiechert potential. If we integrate on time then we would obtain \[ \phi(x, t) = K \int \exp[L^\mathrm{free}(t-t_0)] \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y .\] However the standard literature gives the wrong result \[ \phi(x, t) = K \int \frac{\rho(y,t_0)}{|x - y|} \mathrm{d}y ,\] with a retarded density \( \rho(y,t_0) \) which is the origin of the myth of retarded interactions. This discrepancy is due to the standard literature performing the integration of the delta function without careful analysis of the functional dependences of the argument of the delta function on the variable of integration. The correct integration is as follows. First we let \( s \equiv t' + |x-y|/c - t \) be the new variable of integration. We have \( \mathrm{d}s / \mathrm{d}t' = 1 + \mathrm{d}|x-y|/c\mathrm{d}t' \) and \[ \phi = K \int \rho(t',y) \frac{\delta(s)}{|x-y| (\mathrm{d}s / \mathrm{d}t')} \mathrm{d}y \, \mathrm{d}s \] Much care has to be taken on evaluating the term \( \mathrm{d}s / \mathrm{d}t' \); on a first attempt we could assume that \( |x-y| \) doesn't depend on time, which implies \( \mathrm{d}s / \mathrm{d}t' = 1 \) and recover the incorrect expression for \( \phi \) with a retarded density. The subtle issue is that \( |x-y| \) doesn't depend on time only outside the path of the 'source' particle, but in this trivial case the potential is identically zero. Within the particle path the term \( |x-y| \) depends on time via the density \( \rho(y,t') = e\delta(y - r(t')) \). Therefore, it is better to leave the term \( \mathrm{d}s / \mathrm{d}t' \) in the integral without evaluating it when performing the integration on \( s \) and use \( y = r(t') \) at then end, when integrating on space coordinates. With this rigorous method we will obtain \[ \phi(x, t) = K \frac{e}{|x - r(t)|} , \] in full agreement with the mechanical result. A pair of final remarks. First, I have focused on retarded potentials but it is possible to obtain the advanced potentials when integrating the equation of motion taking some future time as baseline \( \rho(y,t) = \exp[L(t-t_F)] \rho(y,t_F) \); there is no violation of causality because the equations of motion are deterministic and can be integrated both backward and forward in time. I have also focused on electromagnetic interactions but the same arguments can be applied to gravitation resulting on instantaneous potentials \( h_{\mu\nu} \).

General equation for non-equilibrium reversible-irreversible coupling (kinetic formulation)

Equations used by scientists and engineers can be classified into two larger groups: the first group includes deterministic and time-reversible equations, whereas the second includes irreversible and stochastic equations. Chemical kinetics equations belong to the second group, whereas the Hamilton equations of classical mechanics belong to the first group. Usually scientist work with equations of only one group, but sometimes it is needed to study situations that require elements of both classes at once; in such situations we need a new kind of equation that combines determinism, stochasticity, time-symmetry, and irreversibility all at once. Take the phenomenological equation for Brownian motion for the average momentum of a particle contained in a fluid, as illustration \[ \frac{\mathrm{d} p}{\mathrm{d}t} = F_\mathrm{ext} - \frac{\zeta}{m} p . \] The term \( F_\mathrm{ext} \) denotes the external deterministic and time-symmetric force, whereas the second term in the right-hand-side describes the irreversible force associated to friction with the fluid. This friction term has the characteristic form of kinetic equations. Since we are looking for a unified description of both terms, a first approach could be to write down the expression for \( F_\mathrm{ext} \) using the Hamiltonian formalism and then to obtain a similar expression for the irreversible term. Using the thermodynamics of moving bodies we find \[ \frac{\partial S}{\partial p} = - \frac{v}{T} , \] which helps us to rewrite the phenomenological equation of motion in a more unified fashion \[ \frac{\mathrm{d} p}{\mathrm{d}t} = - \frac{\partial E}{\partial x} + T\zeta \frac{\partial S}{\partial p} . \] This is the approach taken by the people behind the theory named GENERIC —acronym for General Equation for Non-Equilibrium Reversible-Irreversible Coupling—, which they write for generic state variables \( n \) as \[ \frac{\mathrm{d} n}{\mathrm{d}t} = L \frac{\partial E}{\partial n} + M \frac{\partial S}{\partial n} . \] Here \( L \) and \( M \) are two matrices associated to Hamiltonian flow and friction respectively, in fact \( M \) is named the friction matrix. The idea of extending the Hamiltonian formalism to irreversible phenomena is very interesting, but unfortunately most irreversible process are not so simple as the Brownian motion and the linear relationship between the generator of time-translations for dissipative processes and the gradient of entropy is lost. For instance for chemical reactions the generator is an exponential with the linear form only valid close to chemical equilibrium. For this reason expressions involving a dissipative potential \( \Xi \) have been proposed lately for GENERIC \[ \frac{\mathrm{d} n}{\mathrm{d}t} = L \frac{\partial E}{\partial n} + \frac{\partial \Xi}{\partial (\partial S/\partial n)} . \] Since chemical kinetic equations are known to describe well complex nonlinear phenomena, we will consider an alternative approach. Instead taking the Hamiltonian formalism as base to rewrite the friction term \( (\zeta/m) p \), we will take the kinetic formalism as base to rewrite the deterministic term \(F_\mathrm{ext}\), obtaining the next reformulation of the Brownian motion equation \[ \frac{\mathrm{d} p}{\mathrm{d}t} = \left( K - \frac{\zeta}{m} \right) p . \] Instead writing the concrete expression for \( K \), we will write directly the general equation for abstract state variables \( n \) \[ \frac{\mathrm{d} n}{\mathrm{d}t} = \Omega n \] and the relationship of the generator of time translations \( \Omega\) with the traditional formulation of GENERIC \[ \Omega_{\alpha\beta} = L_{\alpha\beta} \frac{\partial E}{\partial n_\beta} \frac{\partial}{\partial n_\beta} + M_{\alpha\beta} \frac{\partial S}{\partial n_{\beta}} \frac{\partial}{\partial n_\beta} . \] This is not the only possible way to write it. Another possibility that resembles a dissipative generalization of the Poisson brackets of classical mechanics is \[ \Omega_{\alpha\beta} = \frac{\partial E}{\partial n_\alpha} L_{\alpha\beta} \frac{\partial}{\partial n_\beta} + \frac{\partial S}{\partial n_\alpha} M_{\alpha\beta} \frac{\partial}{\partial n_\beta} . \] But all the equations written up to now are deterministic. There are different ways to introduce fluctuations. A heuristic approach using a Fokker & Planck equation was used in the original paper of GENERIC, whereas a systematic approach based in projection operator techniques was used latter. I am exploring a novel approach to fluctuations. The starting point is the assumption that at some initial time \( t^0 \) the state is fully known \( n = n^0 = \overline{n} \), but fluctuations transform the state into a stochastic variable \( n = \overline{n} + \delta n \) at posterior times
We have to combine the pair of equations for the stochastic trajectories and the conditional average \[ \frac{\mathrm{d} \overline{n}}{\mathrm{d}t} = \overline{\Omega} \overline{n} \] \[ \frac{\mathrm{d} n}{\mathrm{d}t} = \Omega n . \] The result I got after some algebra is \[ \frac{\mathrm{d} n}{\mathrm{d}t} = \Omega n - \epsilon (n - \overline{n}), \] where \( \epsilon \) is a positive infinitesimal. Formally, this expression resembles the Zubarev equation used as foundation for an extension of statistical mechanics to irreversible processes, the physical interpretation is however different.

Researchgate: Are you kidding?

I lack a Researchgate account, but I noticed that Researchgate has created a fake profile about me where they are archiving works from mine whereas miss-attributing one of them to inexistent coworkers. My paper published on the International Journal of Thermodynamics is miss-attributed to two inexistent coworkers Juan Ramon and Callen Casas-Vazquez, when I am the only author.

I tried to join up to correct this blatant error, but due to lacking an institutional email, my request wasn't processed automatically but followed a manual verification procedure. I provided links to my published works and links to the works already archived by Researchgate. Moreover, during the process, the software automatically found some other works from mine.

Today I received a rejection letter:
Dear Juan Ramón González Álvarez,

Thank you for your interest in ResearchGate. Unfortunately we were unable to approve your account request.

I accepted the rejection, because it is their site and their policies, but I mentioned to them it makes little sense to negate me an account whereas archiving my works on a fake profile with inexistent co-workers. I requested Researchgate to delete my profile and the works from their archive. I just received the next funny reply:
Thanks for getting in touch. When browsing ResearchGate you might come across a profile or publications in your name. This is most likely an author profile.

Author profiles contain bibliographic data of published and publicly available information. They exist to make claiming and adding publications to your profile easier.

If you don't have a ResearchGate profile yet, click on the ?Are you Gonzlez lvarez?? button on the top right-hand side of the page to be guided through the sign-up process. Once you've created an account, you'll be able to manage and edit all of the publications on your profile.

Kind regards,

Ben
RG Community Support

Therefore, Researchgate archives my works and automatically generates a profile about me without my permission, miss-attributes one of my works to inexistent co-workers, rejects my request to join, doesn't solve the miss-attribution issue and finally suggests me to join to edit by myself the profile.

Are those guys kidding or it is just plain incompetence?

Update

Finally Researchgate has deleted the fake accounts of the inexistent coworkers Juan Ramon and Callen Casas-Vazquez, changed the profile about me to one new profile with my full name Juan Ramón González Álvarez, cleaned it, and offered me to join them:
Obviously you have proved your credentials as a researcher now, I would be happy to activate your account and assign these publications to your account, should you choose that option.

Kind regards,

Thomas
RG Community Support

I didn't reply...

Second Update

Just for curiosity, I checked the status of the fake profile that they still maintain about me. They have changed things again. Apart from listing a set of incorrect disciplines with little to no relation to my work, now they only attribute to me a single publication, whereas my work about heat doesn't appear in my profile but appears standalone.