Instantaneous electromagnetic interactions
Newton introduced a model of instantaneous direct interactions among massive particles. This model was latter replicated by Coulomb for charged particles and became known as action-at-a-distance; an unfortunate name has generated unending polemics among physicists and philosophers. A better name is direct-particle-interaction. Maxwell electrodynamics and general relativity introduced an alternative model of contact-action, where particles do not interact directly but by means of signals traveling through a mediator. Since the maximum possible speed for any object is the speed of light, interactions are retarded in contact-action models. Althought the Newton and Coulomb models are only valid for low velocities, it is proven that retarded interactions can be derived as approximations to generalized instantaneous interactions beyond Newton and Coulomb.
We will start with the next instantaneous scalar and vector potentials associated to a charge \( e \) placed at \( \boldsymbol{r} \) \[ \phi = \phi(\boldsymbol{x}, t) = \frac{1}{4\pi\epsilon_0} \frac{e}{|\boldsymbol{x} - \boldsymbol{r}(t)|} \] \[ \boldsymbol{A} = \boldsymbol{A}(\boldsymbol{x}, t) = \frac{1}{4\pi\epsilon_0c} \frac{e\boldsymbol{v}(t)}{|\boldsymbol{x} - \boldsymbol{r}(t)|} . \] Expanding both the position and the velocity of the charge around at some early time \( t_0 \) \[ \boldsymbol{r}(t) = \boldsymbol{r}(t_0) + \boldsymbol{v}(t_0) (t-t_0) + \boldsymbol{a}(t_0) (t-t_0)^2 / 2 + \cdots \] \[ \boldsymbol{v}(t) = \boldsymbol{v}(t_0) + \boldsymbol{a}(t_0) (t-t_0) + \cdots \] and assuming that the charge is not accelerating at the initial time \( t_0 = (t - |\boldsymbol{x} - \boldsymbol{r}(t_0)|/c) \) we obtain the retarded potentials \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \frac{e}{|\boldsymbol{x} - \boldsymbol{r}(t_0)| - \boldsymbol{v}(t_0) [ \boldsymbol{x} - \boldsymbol{r}(t_0) ] / c} , \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0c} \frac{e \boldsymbol{v}(t_0)}{|\boldsymbol{x} - \boldsymbol{r}(t_0)| - \boldsymbol{v}(t_0) [ \boldsymbol{x} - \boldsymbol{r}(t_0) ] / c} , \] which are evidently the Lienard & Wiechert potentials found in any textbook.
Note that the Lienard & Wiechert potentials have been derived under the approximation \( \boldsymbol{a}(t_0) = 0 \). This means that Lienard & Wiechert potentials are not complete, and cannot describe the general motion of charged particles. This defficiency of the Lienard & Wiechert potentials explains why physicists have tradittionally added ad hoc reaction-radiation force terms to the field-theoretic equations of motion for curing such issues like net loss of energy on systems of accelerating charged particles. Those ad hoc modifications of the equations follow from a correction to the Lienard & Wiechert potentials \[ \phi^\mathrm{ret} \rightarrow \phi^\mathrm{ret} + \phi^\mathrm{rr} , \] \[ \boldsymbol{A}^\mathrm{ret} \rightarrow \boldsymbol{A}^\mathrm{ret} + \boldsymbol{A}^\mathrm{rr} . \] Not only those correction potentials \( \phi^\mathrm{rr}\) and \( \boldsymbol{A}^\mathrm{rr} \) cannot be derived from the field-theoretic electromagnetic Lagrangian or action —which only gives the Lienard & Wiechert potentials—, but those correction potentials are postulated in a truly inconsistent approach when the advanced solutions to the wave equations are initially rejected —"and this unphysical solution to the wave equation is known as the advanced solution" [1]— only to be used latter as part of the definition of the correction potentials \[ \phi^\mathrm{rr} \stackrel{\mathrm{def}}{=} \frac{\phi^\mathrm{ret} - \phi^\mathrm{adv}}{2} \] \[ \boldsymbol{A}^\mathrm{rr} \stackrel{\mathrm{def}}{=} \frac{\boldsymbol{A}^\mathrm{ret} - \boldsymbol{A}^\mathrm{adv}}{2} . \] If the retarded solutions are the "only physically acceptable solution" [1], how is it possible that they have to be admended by the so-named unphysical solutions? The conventional wisdom does not make any sense. Moreover, even if we ignore those inconsistencies, the resulting equations of motion with reaction-radiation corrections are still subjected to criticism due to odd behaviors and properties. On the other hand the potentials (1) and (2) conserve energy and maintain causality, and do not require ad hoc reaction-radiation corrections.
Note that the \( \boldsymbol{a}(t_0) = 0 \) approximation also explains why the Lienard & Wiechert potentials for a moving charge can be obtained from the potentials for a charge at rest, \( \boldsymbol{A}=0 \) and \( \phi \), applying the Lorentz transformations between a frame \( S' \) where the particle is at rest and a frame \( S \) where the particle is moving with velocity \( \boldsymbol{v} \). The Lorentz transformations can be applied because both frames are inertial, that is, the particle is not accelerating. In fact, some derivations of the Lienard & Wiechert potentials explicitly admit that the charge is moving "with uniform velocity \( \boldsymbol{v} \) through a frame \( S \)".
This same approximation is the reason why quantum field theory admits as the only physical admissible states those of free particles, that is, particles are not accelerating. This is picturesquely described in Feynman diagrams
The diagram considers electrons initially in free motion, states 1 and 2, until a photon is emitted and absorbed, states 6 and 5 and, as a consequence, both electrons change their state of motion to new free states 3 and 4. Quantum field theory only can rigorously describe the states before and after the interaction, 1, 2, 3, and 4, respectively; but cannot provide a description of what happens during the interaction, i.e. states 5 and 6.
The instantaneous potentials (1) and (2) can be written in integral form as \[ \phi = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\boldsymbol{y}, t)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A} = \frac{1}{4\pi\epsilon_0 c} \int \frac{\boldsymbol{j}(\boldsymbol{y}, t)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} , \] using the electron density \( \rho(\boldsymbol{y}, t) = e\delta(\boldsymbol{y} - \boldsymbol{r}(t)) \) and current \( \boldsymbol{j}(\boldsymbol{y},t) = \boldsymbol{v}(t) \rho(\boldsymbol{y},t) \). Now we can utilize the equations of motion to relate densities and currents at times \( t \) and \( t_0 \) \[ \phi = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ L(t-t_0) \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A} = \frac{1}{4\pi\epsilon_0 c} \int \exp \Big[ L(t-t_0) \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} . \] \( L \) in the above expression is the Liouvillian. We can now see clearly that a kernel evaluated at present time \( t \) is identical to a modified kernel evaluated at retarded time \( t_0 \) \[ \left\{ \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|}\right\}_t = \left\{ \exp[L(t-t_0)] \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|}\right\}_{t_0} .\] If we approximate the full Liouvillian by its free component \( L^\mathrm{free} = - \boldsymbol{v} \boldsymbol{\nabla_y} \) and integrate the expressions \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0 c} \int \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} , \] we recover the Lienard & Wiechert potentials (5) and (6).
In the former section, we derived the Lienard & Wiechert potentials by neglecting acceleration and higher order terms in a series expansion of positions and velocities around an initial time. We can confirm now that the Lienard & Wiechert potentials are an exact consequence of the free component of the generator of the motion for charged particles, of course, this free component only describes inertial particles. The interaction Liovillian \( L^\mathrm{inter} \) introduces acceleration and other higher-order corrections to the Lienard & Wiechert potentials \[ \phi = \phi^\mathrm{ret} + \mathbb{O}(L^\mathrm{inter}) \] \[ \boldsymbol{A} = \boldsymbol{A}^\mathrm{ret} + \mathbb{O}(L^\mathrm{inter}) \] Those corrections are responsible for preserving causality and conserving energy on (13) and (14).
In this section we will deal only with the scalar potential without any loss of generality because the extension of the arguments and techniques to the vector potential is straightforward. The standard electromagnetic literature proposes the next integral equation as retarded solution to the wave equation \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int\int \rho(\boldsymbol{y}, t') \frac{\delta(t-t'-|\boldsymbol{x}-\boldsymbol{y}|/c)}{|\boldsymbol{x}-\boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \, \mathrm{d}t' . \] If we integrate first on position and then on time we obtain the Lienard & Wiechert potential (5). If instead we integrate first on time then we would obtain (16) \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} .\] However the standard literature gives the wrong result \[ \phi^\mathrm{ret} \stackrel{\mathrm{wrong}}{=} \frac{1}{4\pi\epsilon_0} \int\frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} . \] This discrepancy between (21) and (22) has its origin on subtle functional dependences on the time integration. The correct integration of (20) is as follows. First we let \( s \equiv (t' + |\boldsymbol{x}-\boldsymbol{y}|/c - t) \) be the new variable of integration. We have \( \mathrm{d}s / \mathrm{d}t' = 1 + (\mathrm{d}|\boldsymbol{x}-\boldsymbol{y}|/c\mathrm{d}t') \) and \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int\int \Big( \frac{\mathrm{d}s}{\mathrm{d}t'} \Big) \rho(\boldsymbol{y}, t') \frac{\delta(s)}{|\boldsymbol{x}-\boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \, \mathrm{d}s .\] Much care has to be taken when evaluating the term \( (\mathrm{d}s / \mathrm{d}t') \). The standard literature assumes that \( |\boldsymbol{x}-\boldsymbol{y}| \) does not depend on time \(t'\), set \( (\mathrm{d}s / \mathrm{d}t') = 1 \) and integrates on \(s\) to yield the incorrect expression (22). The subtle issue is that this assumption is only valid for points \( \boldsymbol{y} \) outside the path of the particle, \( \boldsymbol{y} \neq \boldsymbol{r}(t') \), but in this trivial case the retarded potential is identically zero \(\phi^\mathrm{ret} = 0\) by virtue of the delta function on the density \( \rho(y,t') = e\delta(\boldsymbol{y} - \boldsymbol{r}(t')) \). Within the particle path, the term \( |\boldsymbol{x}-\boldsymbol{y}| \) depends implicitly on time \(t'\) via the density because \( \boldsymbol{y} = \boldsymbol{r}(t') \). When this implicit time dependence is considered in \( (\mathrm{d}s / \mathrm{d}t') \), we obtain the correct expression (21), in full agreement with the mechanical result. The same arguments and methods can be used to demonstrate that the correct expression for the vector potential \(\boldsymbol{A}^\mathrm{ret}\) is given by \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0c} \int\exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] and not by the incorrect expression \[ \boldsymbol{A}^\mathrm{ret} \stackrel{\mathrm{wrong}}{=} \frac{1}{4\pi\epsilon_0c} \int\frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} .\]
We take the expressions (16) and (17), that produce the Lienard & Wiechert potentials of classical electrodynamics, and replace classical densities, currents, and Liouvillians by their quantum analogs. Using the quantum identity between Liouvillian and the Hamiltonian \( \exp [ L^\mathrm{free}\tau ] F = \exp [ iH^\mathrm{free}\tau/\hbar ] F \exp [ -iH^\mathrm{free}\tau/\hbar ] \) for arbitrary \(\tau\) and \(F\), we obtain the next quantum potentials \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ iH^\mathrm{free}(t-t_0)/\hbar \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ -iH^\mathrm{free}(t-t_0)/\hbar \Big] \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0 c} \int \exp \Big[ iH^\mathrm{free}(t-t_0)/\hbar \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ -iH^\mathrm{free}(t-t_0)/\hbar \Big] \mathrm{d}\boldsymbol{y} . \] Quantum densities and currents are given by \( \rho = e \sum_i \sum_j c_i^{*} c_j u_i^\dagger u_j \) and \( \boldsymbol{j} = \sum_i \sum_j c_i^{*} c_j u_i^\dagger c\boldsymbol{\alpha} u_j \) respectively. Here \( u_k = u_k(\boldsymbol{y}) \) are solutions to the time-independent Dirac free equation: \( H^\mathrm{free} u_k = E_k u_k \). Further replacing the time delay by its value and with the energy difference expressed as \( E_i - E_j = \hbar w_{ij} \), we obtain \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} e \sum_i \sum_j c_i^{*} c_j \int u_i^\dagger \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] u_j \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} e \sum_i \sum_j c_i^{*} c_j \int u_i^\dagger \frac{ \boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] u_j \mathrm{d}\boldsymbol{y} , \] which is just the quantum electrodynamics result, albeit those expressions are not usually found in the literature. To make contact with the usual expressions in the literature, we need to evaluate the interaction energy \( V_{ee} = \int \rho \phi^\mathrm{ret} - \boldsymbol{j} \boldsymbol{A}^\mathrm{ret} \mathrm{d}\boldsymbol{x} \), \[ V_{ee} = \frac{1}{4\pi\epsilon_0} e^2 \sum_i \sum_j \sum_k \sum_l c_i^{*} c_k^{*} c_j c_l \int\int u_i^\dagger u_k^\dagger \frac{ 1 - \boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] u_j u_l \mathrm{d}\boldsymbol{x}\mathrm{d}\boldsymbol{y} . \] This is the standard result found in quantum electrodynamics literature, with the associated one-photon operator in covariant gauge \[ V^\mathrm{\gamma CG} = \frac{1}{4\pi\epsilon_0} e^2 \frac{ 1 - \boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] . \] The more popular Coulomb & Breit operator used in molecular physics and quantum chemistry can be derived from the above operator by expansion of the exponential and retaining terms up to quadratic order in the speed of light. After some simple but tedious operations we obtain \[ V^\mathrm{CB} = \frac{1}{4\pi\epsilon_0} e^2 \frac{ 1 }{|\boldsymbol{x} - \boldsymbol{y}|} - \frac{1}{4\pi\epsilon_0} e^2 \frac{ \boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} + \frac{1}{4\pi\epsilon_0} \frac{e^2}{2} \Big[ \frac{\boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} - \frac{(\boldsymbol{x} - \boldsymbol{y})\boldsymbol{\alpha}(\boldsymbol{x} - \boldsymbol{y})\boldsymbol{\alpha}}{|\boldsymbol{x} - \boldsymbol{y}|^3} \Big] , \] consisting on the sum of the Coulomb operator, the Gaunt operator, and the Breit retardation operator.
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 \( \rho(\boldsymbol{y},t) = \exp[L(t-t_F)] \rho(\boldsymbol{y},t_F) \); contrary to a common myth, there is no violation of causality because the mechanical 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 \[ h_{\mu\nu} = -\frac{16 \pi G}{c^4} \int \frac{\sigma_{\mu\nu}(\boldsymbol{y}, t)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} .\]
[1] Electromagnetic Theory, Lecture Notes 2005 (Accessed May 2017): Poisson, Eric.
Simple derivation of Lienard & Wiechert potentials
We will start with the next instantaneous scalar and vector potentials associated to a charge \( e \) placed at \( \boldsymbol{r} \) \[ \phi = \phi(\boldsymbol{x}, t) = \frac{1}{4\pi\epsilon_0} \frac{e}{|\boldsymbol{x} - \boldsymbol{r}(t)|} \] \[ \boldsymbol{A} = \boldsymbol{A}(\boldsymbol{x}, t) = \frac{1}{4\pi\epsilon_0c} \frac{e\boldsymbol{v}(t)}{|\boldsymbol{x} - \boldsymbol{r}(t)|} . \] Expanding both the position and the velocity of the charge around at some early time \( t_0 \) \[ \boldsymbol{r}(t) = \boldsymbol{r}(t_0) + \boldsymbol{v}(t_0) (t-t_0) + \boldsymbol{a}(t_0) (t-t_0)^2 / 2 + \cdots \] \[ \boldsymbol{v}(t) = \boldsymbol{v}(t_0) + \boldsymbol{a}(t_0) (t-t_0) + \cdots \] and assuming that the charge is not accelerating at the initial time \( t_0 = (t - |\boldsymbol{x} - \boldsymbol{r}(t_0)|/c) \) we obtain the retarded potentials \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \frac{e}{|\boldsymbol{x} - \boldsymbol{r}(t_0)| - \boldsymbol{v}(t_0) [ \boldsymbol{x} - \boldsymbol{r}(t_0) ] / c} , \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0c} \frac{e \boldsymbol{v}(t_0)}{|\boldsymbol{x} - \boldsymbol{r}(t_0)| - \boldsymbol{v}(t_0) [ \boldsymbol{x} - \boldsymbol{r}(t_0) ] / c} , \] which are evidently the Lienard & Wiechert potentials found in any textbook.
Note that the Lienard & Wiechert potentials have been derived under the approximation \( \boldsymbol{a}(t_0) = 0 \). This means that Lienard & Wiechert potentials are not complete, and cannot describe the general motion of charged particles. This defficiency of the Lienard & Wiechert potentials explains why physicists have tradittionally added ad hoc reaction-radiation force terms to the field-theoretic equations of motion for curing such issues like net loss of energy on systems of accelerating charged particles. Those ad hoc modifications of the equations follow from a correction to the Lienard & Wiechert potentials \[ \phi^\mathrm{ret} \rightarrow \phi^\mathrm{ret} + \phi^\mathrm{rr} , \] \[ \boldsymbol{A}^\mathrm{ret} \rightarrow \boldsymbol{A}^\mathrm{ret} + \boldsymbol{A}^\mathrm{rr} . \] Not only those correction potentials \( \phi^\mathrm{rr}\) and \( \boldsymbol{A}^\mathrm{rr} \) cannot be derived from the field-theoretic electromagnetic Lagrangian or action —which only gives the Lienard & Wiechert potentials—, but those correction potentials are postulated in a truly inconsistent approach when the advanced solutions to the wave equations are initially rejected —"and this unphysical solution to the wave equation is known as the advanced solution" [1]— only to be used latter as part of the definition of the correction potentials \[ \phi^\mathrm{rr} \stackrel{\mathrm{def}}{=} \frac{\phi^\mathrm{ret} - \phi^\mathrm{adv}}{2} \] \[ \boldsymbol{A}^\mathrm{rr} \stackrel{\mathrm{def}}{=} \frac{\boldsymbol{A}^\mathrm{ret} - \boldsymbol{A}^\mathrm{adv}}{2} . \] If the retarded solutions are the "only physically acceptable solution" [1], how is it possible that they have to be admended by the so-named unphysical solutions? The conventional wisdom does not make any sense. Moreover, even if we ignore those inconsistencies, the resulting equations of motion with reaction-radiation corrections are still subjected to criticism due to odd behaviors and properties. On the other hand the potentials (1) and (2) conserve energy and maintain causality, and do not require ad hoc reaction-radiation corrections.
Note that the \( \boldsymbol{a}(t_0) = 0 \) approximation also explains why the Lienard & Wiechert potentials for a moving charge can be obtained from the potentials for a charge at rest, \( \boldsymbol{A}=0 \) and \( \phi \), applying the Lorentz transformations between a frame \( S' \) where the particle is at rest and a frame \( S \) where the particle is moving with velocity \( \boldsymbol{v} \). The Lorentz transformations can be applied because both frames are inertial, that is, the particle is not accelerating. In fact, some derivations of the Lienard & Wiechert potentials explicitly admit that the charge is moving "with uniform velocity \( \boldsymbol{v} \) through a frame \( S \)".
This same approximation is the reason why quantum field theory admits as the only physical admissible states those of free particles, that is, particles are not accelerating. This is picturesquely described in Feynman diagrams
Rigorous relation between instantaneous and retarded interactions
The instantaneous potentials (1) and (2) can be written in integral form as \[ \phi = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\boldsymbol{y}, t)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A} = \frac{1}{4\pi\epsilon_0 c} \int \frac{\boldsymbol{j}(\boldsymbol{y}, t)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} , \] using the electron density \( \rho(\boldsymbol{y}, t) = e\delta(\boldsymbol{y} - \boldsymbol{r}(t)) \) and current \( \boldsymbol{j}(\boldsymbol{y},t) = \boldsymbol{v}(t) \rho(\boldsymbol{y},t) \). Now we can utilize the equations of motion to relate densities and currents at times \( t \) and \( t_0 \) \[ \phi = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ L(t-t_0) \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A} = \frac{1}{4\pi\epsilon_0 c} \int \exp \Big[ L(t-t_0) \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} . \] \( L \) in the above expression is the Liouvillian. We can now see clearly that a kernel evaluated at present time \( t \) is identical to a modified kernel evaluated at retarded time \( t_0 \) \[ \left\{ \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|}\right\}_t = \left\{ \exp[L(t-t_0)] \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|}\right\}_{t_0} .\] If we approximate the full Liouvillian by its free component \( L^\mathrm{free} = - \boldsymbol{v} \boldsymbol{\nabla_y} \) and integrate the expressions \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0 c} \int \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} , \] we recover the Lienard & Wiechert potentials (5) and (6).
In the former section, we derived the Lienard & Wiechert potentials by neglecting acceleration and higher order terms in a series expansion of positions and velocities around an initial time. We can confirm now that the Lienard & Wiechert potentials are an exact consequence of the free component of the generator of the motion for charged particles, of course, this free component only describes inertial particles. The interaction Liovillian \( L^\mathrm{inter} \) introduces acceleration and other higher-order corrections to the Lienard & Wiechert potentials \[ \phi = \phi^\mathrm{ret} + \mathbb{O}(L^\mathrm{inter}) \] \[ \boldsymbol{A} = \boldsymbol{A}^\mathrm{ret} + \mathbb{O}(L^\mathrm{inter}) \] Those corrections are responsible for preserving causality and conserving energy on (13) and (14).
A subtle mistake has remained unnoticed in the literature
In this section we will deal only with the scalar potential without any loss of generality because the extension of the arguments and techniques to the vector potential is straightforward. The standard electromagnetic literature proposes the next integral equation as retarded solution to the wave equation \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int\int \rho(\boldsymbol{y}, t') \frac{\delta(t-t'-|\boldsymbol{x}-\boldsymbol{y}|/c)}{|\boldsymbol{x}-\boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \, \mathrm{d}t' . \] If we integrate first on position and then on time we obtain the Lienard & Wiechert potential (5). If instead we integrate first on time then we would obtain (16) \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} .\] However the standard literature gives the wrong result \[ \phi^\mathrm{ret} \stackrel{\mathrm{wrong}}{=} \frac{1}{4\pi\epsilon_0} \int\frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} . \] This discrepancy between (21) and (22) has its origin on subtle functional dependences on the time integration. The correct integration of (20) is as follows. First we let \( s \equiv (t' + |\boldsymbol{x}-\boldsymbol{y}|/c - t) \) be the new variable of integration. We have \( \mathrm{d}s / \mathrm{d}t' = 1 + (\mathrm{d}|\boldsymbol{x}-\boldsymbol{y}|/c\mathrm{d}t') \) and \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int\int \Big( \frac{\mathrm{d}s}{\mathrm{d}t'} \Big) \rho(\boldsymbol{y}, t') \frac{\delta(s)}{|\boldsymbol{x}-\boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \, \mathrm{d}s .\] Much care has to be taken when evaluating the term \( (\mathrm{d}s / \mathrm{d}t') \). The standard literature assumes that \( |\boldsymbol{x}-\boldsymbol{y}| \) does not depend on time \(t'\), set \( (\mathrm{d}s / \mathrm{d}t') = 1 \) and integrates on \(s\) to yield the incorrect expression (22). The subtle issue is that this assumption is only valid for points \( \boldsymbol{y} \) outside the path of the particle, \( \boldsymbol{y} \neq \boldsymbol{r}(t') \), but in this trivial case the retarded potential is identically zero \(\phi^\mathrm{ret} = 0\) by virtue of the delta function on the density \( \rho(y,t') = e\delta(\boldsymbol{y} - \boldsymbol{r}(t')) \). Within the particle path, the term \( |\boldsymbol{x}-\boldsymbol{y}| \) depends implicitly on time \(t'\) via the density because \( \boldsymbol{y} = \boldsymbol{r}(t') \). When this implicit time dependence is considered in \( (\mathrm{d}s / \mathrm{d}t') \), we obtain the correct expression (21), in full agreement with the mechanical result. The same arguments and methods can be used to demonstrate that the correct expression for the vector potential \(\boldsymbol{A}^\mathrm{ret}\) is given by \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0c} \int\exp \Big[ L^\mathrm{free}(t-t_0) \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} \] and not by the incorrect expression \[ \boldsymbol{A}^\mathrm{ret} \stackrel{\mathrm{wrong}}{=} \frac{1}{4\pi\epsilon_0c} \int\frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} .\]
Quantum electrodynamics potentials
We take the expressions (16) and (17), that produce the Lienard & Wiechert potentials of classical electrodynamics, and replace classical densities, currents, and Liouvillians by their quantum analogs. Using the quantum identity between Liouvillian and the Hamiltonian \( \exp [ L^\mathrm{free}\tau ] F = \exp [ iH^\mathrm{free}\tau/\hbar ] F \exp [ -iH^\mathrm{free}\tau/\hbar ] \) for arbitrary \(\tau\) and \(F\), we obtain the next quantum potentials \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} \int \exp \Big[ iH^\mathrm{free}(t-t_0)/\hbar \Big] \frac{\rho(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ -iH^\mathrm{free}(t-t_0)/\hbar \Big] \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0 c} \int \exp \Big[ iH^\mathrm{free}(t-t_0)/\hbar \Big] \frac{\boldsymbol{j}(\boldsymbol{y}, t_0)}{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ -iH^\mathrm{free}(t-t_0)/\hbar \Big] \mathrm{d}\boldsymbol{y} . \] Quantum densities and currents are given by \( \rho = e \sum_i \sum_j c_i^{*} c_j u_i^\dagger u_j \) and \( \boldsymbol{j} = \sum_i \sum_j c_i^{*} c_j u_i^\dagger c\boldsymbol{\alpha} u_j \) respectively. Here \( u_k = u_k(\boldsymbol{y}) \) are solutions to the time-independent Dirac free equation: \( H^\mathrm{free} u_k = E_k u_k \). Further replacing the time delay by its value and with the energy difference expressed as \( E_i - E_j = \hbar w_{ij} \), we obtain \[ \phi^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} e \sum_i \sum_j c_i^{*} c_j \int u_i^\dagger \frac{1}{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] u_j \mathrm{d}\boldsymbol{y} \] \[ \boldsymbol{A}^\mathrm{ret} = \frac{1}{4\pi\epsilon_0} e \sum_i \sum_j c_i^{*} c_j \int u_i^\dagger \frac{ \boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] u_j \mathrm{d}\boldsymbol{y} , \] which is just the quantum electrodynamics result, albeit those expressions are not usually found in the literature. To make contact with the usual expressions in the literature, we need to evaluate the interaction energy \( V_{ee} = \int \rho \phi^\mathrm{ret} - \boldsymbol{j} \boldsymbol{A}^\mathrm{ret} \mathrm{d}\boldsymbol{x} \), \[ V_{ee} = \frac{1}{4\pi\epsilon_0} e^2 \sum_i \sum_j \sum_k \sum_l c_i^{*} c_k^{*} c_j c_l \int\int u_i^\dagger u_k^\dagger \frac{ 1 - \boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] u_j u_l \mathrm{d}\boldsymbol{x}\mathrm{d}\boldsymbol{y} . \] This is the standard result found in quantum electrodynamics literature, with the associated one-photon operator in covariant gauge \[ V^\mathrm{\gamma CG} = \frac{1}{4\pi\epsilon_0} e^2 \frac{ 1 - \boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} \exp \Big[ iw_{ij}|\boldsymbol{x} - \boldsymbol{y}|/ c \Big] . \] The more popular Coulomb & Breit operator used in molecular physics and quantum chemistry can be derived from the above operator by expansion of the exponential and retaining terms up to quadratic order in the speed of light. After some simple but tedious operations we obtain \[ V^\mathrm{CB} = \frac{1}{4\pi\epsilon_0} e^2 \frac{ 1 }{|\boldsymbol{x} - \boldsymbol{y}|} - \frac{1}{4\pi\epsilon_0} e^2 \frac{ \boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} + \frac{1}{4\pi\epsilon_0} \frac{e^2}{2} \Big[ \frac{\boldsymbol{\alpha}\boldsymbol{\alpha} }{|\boldsymbol{x} - \boldsymbol{y}|} - \frac{(\boldsymbol{x} - \boldsymbol{y})\boldsymbol{\alpha}(\boldsymbol{x} - \boldsymbol{y})\boldsymbol{\alpha}}{|\boldsymbol{x} - \boldsymbol{y}|^3} \Big] , \] consisting on the sum of the Coulomb operator, the Gaunt operator, and the Breit retardation operator.
Final remarks
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 \( \rho(\boldsymbol{y},t) = \exp[L(t-t_F)] \rho(\boldsymbol{y},t_F) \); contrary to a common myth, there is no violation of causality because the mechanical 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 \[ h_{\mu\nu} = -\frac{16 \pi G}{c^4} \int \frac{\sigma_{\mu\nu}(\boldsymbol{y}, t)}{|\boldsymbol{x} - \boldsymbol{y}|} \mathrm{d}\boldsymbol{y} .\]
References
[1] Electromagnetic Theory, Lecture Notes 2005 (Accessed May 2017): Poisson, Eric.