In the following we first derive the classical form of the EM interaction from STF Direct Particle Interaction. From that we show the steps to derive the relativistic Newton-Lorentz equation of motion for a charge in a given EM field. And then we derive the Maxwell equations for the EM fields for a given current.
The conditions under which these derivation succeed are a tool for understanding the relationship between Classical Electrodynamics (CED) and Schwarzschild, Tetrode, Fokker Direct Particle Interaction (STF DPI).
As before we use the shorthand
\[a \circ b = {a^\mu }{b_\mu },\quad {\left( a \right)^2} = {a^\mu }{a_\mu }\,.\]
The relationship with Maxwell field theory with self-action excluded can be established as follows. First introduce an explicit common speed parameter (time) for each of the charges. Due to time-reparameterization invariance of the integrals we are free to choose a common laboratory time \(t = q_{\left( i \right)}^0\). Then define the 4-velocities
\[\begin{align} {v_{\left( i \right)}}\left( t \right) & \equiv \left\{ {v_{\left( i \right)}^\mu \left( t \right)} \right\} \\ &=\left( {1,\frac{{dq_{\left( i \right)}^1\left( t \right)}}{{dt}},\frac{{dq_{\left( i \right)}^2\left( t \right)}}{{dt}},\frac{{dq_{\left( i \right)}^3\left( t \right)}}{{dt}}} \right) \\ &= \left( {1,\frac{{d{{\bf{q}}_{\left( i \right)}}\left( t \right)}}{{dt}}} \right) \\ &= \left( {1,{{\bf{v}}_{\left( i \right)}}\left( t \right)} \right)\,. \end{align}\]
Once again due to time re-parameterization invariance it does not matter that these are not true Lorentz 4-vectors. Now explicate the speed parameter into the integrals:
\[{I_{mass}} = - \sum\limits_{i = 1}^N {{m_i}\int {dt} \sqrt {{{\left( {{v_{\left( i \right)}}\left( t \right)} \right)}^2}} } \]
\[{I_{DPI}} = - \frac{{{e^2}}}{{4\pi }}\sum\limits_{i \ne j = 1}^N {{\sigma _i}{\sigma _j}\int {dt} \int {dt'} {v_{\left( i \right)}}\left( t \right) \circ {v_{\left( j \right)}}\left( {t'} \right)\delta \left( {{{\left( {{q_{\left( i \right)}}\left( t \right) - {q_{\left( j \right)}}\left( {t'} \right)} \right)}^2}} \right)} \,.\]
Here and throughout we use the Heaviside-Lorentz system with \(c=1\). Indefinite integrals imply integration over the full range of the integration parameter.
Now use \(\int {{d^3}x} \delta^3 \left( {{\bf{x}} - {{\bf{q}}_{\left( i \right)}}\left( t \right)} \right) = 1\) to write the DPI action as
\[\begin{align}
{I_{DPI}} =& - \frac{1}{{4\pi }}\sum\limits_{i \ne j = 1}^N {{e_i}{e_j}\int {{d^4}x} \int {{d^4}x'} {v_{\left( i \right)}}\left( t \right) \circ {v_{\left( j \right)}}\left( {t'} \right)\delta \left( {{{\left( {x - x'} \right)}^2}} \right)} \\
\times & {\delta ^3}\left( {{\bf{x}} - {{\bf{q}}_{\left( i \right)}}\left( t \right)} \right){\delta ^3}\left( {{\bf{x'}} - {{\bf{q}}_{\left( j \right)}}\left( {t'} \right)} \right)
\,.\end{align}\]
Recalling that the classical 4-currents are
\[{j_{\left( i \right)}}\left( x \right) = {e_i}{v_{\left( i \right)}}\left( t \right){\delta ^3}\left( {{\bf{x}} - {{\bf{q}}_{\left( i \right)}}\left( t \right)} \right)\]
(these are true Lorentz vectors) the DPI action is
\[{I_{DPI}} = - \frac{1}{{4\pi }}\sum\limits_{m \ne n = 1}^N {{e_m}{e_n}\int {{d^4}x} \int {{d^4}x'} {j_{\left( m \right)}}\left( x \right) \circ {j_{\left( n \right)}}\left( {x'} \right)\delta \left( {{{\left( {x - x'} \right)}^2}} \right)}\,. \]
The total current is
\[j\left( x \right) = \sum\limits_{m = 1}^N {{j_{\left( m \right)}}\left( x \right)} \]
in which terms the DPI action is
\[{I_{DPI}} = - \frac{1}{{4\pi }}\int {{d^4}x} \int {{d^4}x'} j\left( x \right) \circ j\left( {x'} \right)\delta \left( {{{\left( {x - x'} \right)}^2}} \right) - {I_{self}}\]
where
\[{I_{self}} = - \frac{1}{{4\pi }}\sum\limits_{m = 1}^N {\int {{d^4}x} \int {{d^4}x'} {j_{\left( m \right)}}\left( x \right) \circ {j_{\left( m \right)}}\left( {x'} \right)\delta \left( {{{\left( {x - x'} \right)}^2}} \right)} \]
is the self action. If we now define
\[A\left( x \right) \equiv \frac{1}{{4\pi }}\int {{d^4}x'} j\left( x \right)\delta \left( {{{\left( {x - x'} \right)}^2}} \right)\label{potential}\]
then the DPI action can be written
\[{I_{DPI}} = - \int {{d^4}x}j\left( x \right) \circ A\left( x \right) - {I_{self}}\label{I_interaction}\,.\]
We recognize \( \left. \delta \left( x^2 \right) \middle/ 4 \pi \right.\) as Green's function for the wave equation
\[\partial ^2 \left. \delta \left( x^2 \right) \middle/ 4 \pi \right. = \delta ^4\left( x \right)\]
and therefore \(A\left( x \right)\) satisfies
\[{\partial ^2}A\left( x \right) = j\left( x \right)\label{ihe}\,.\]
Distinct from Maxwell theory here \eqref{ihe} is a side effect of the definition \eqref{potential}, and by itself is incomplete.
The definition \eqref{potential} demands, in addition to \eqref{ihe}, that the potential satisfy the Lorenz gauge \(\partial \circ A = 0\). The absence of gauge freedom in the potential is a consequence of the particular form of the STF action. Other forms of the direct particle interaction exist in which the derived potential possess gauge freedom. But gauge freedom is irrelevant here because the classical theory can be expressed entirely in terms of the electric and magnetic fields - without reference to a potential.
Notice also that despite appearances there is no freedom to choose a boundary conditions on \(A\left( x \right)\) through a complimentary function solution to \eqref{ihe}. This is because in DPI the value of \(A\left( x \right)\) is already exactly determined by the position and motion of charges (the currents). From this point of view any field theory that depends on \(A\left( x \right)\) or its derivatives (as in the classical Maxwell theory) must be an approximation. The fact that it generally seems one can set a boundary condition on \(A\left( x \right)\) has to do the with great number of very distant currents that contribute to \(A\left( x \right)\) in \eqref{potential}.
The fact that one can set a Cauchy boundary condition in the past or present time (quite often this is the Sommerfeld boundary condition wherein \(A\left( x \right)\) and its derivatives are all zero at some `initial` time) relies also an electromagnetic arrow of time. In DPI it implies the emergence of a collective macroscopic arrow of time from pair-wise interactions each of which is reciprocal and intrinsically time symmetric. The observational fact of an electromagnetic arrow of time requires an explanation from DPI - seemingly more urgently than that required of field theory.
To obtain the equation of motion of a single charge label \(m=1\) say in the presence of given fields we could write the STF DPI action as
\[\begin{align}{I_{DPI}} &= - \int {{d^4}x} {j_{\left( 1 \right)}}\left( x \right) \circ {A_{in}}\left( x \right) + {I_{other}} \label{NL_action1}\\ &= - e_1\int {dt} {v_{\left( 1 \right)}}\left( t \right) \circ {A_{in}}\left( {{q_{\left( 1 \right)}}\left( t \right)} \right) + {I_{other}}\label{NL_action2}\end{align}\]
where
\[{A_{in}}\left( x \right) = \frac{1}{{4\pi }}\sum\limits_{m = 2}^N {\int {{d^4}x'} {j_{\left( m \right)}}\left( {x'} \right)\delta \left( {{{\left( {x - x'} \right)}^2}} \right)} \label{Ain}\]
and \({I_{other}}\) is the interaction between all other charges (except the charge with label 1). The total action involving the charge with label 1 is
\[\begin{align}{I_1} = & - {m_1}\int\limits_0^\infty dt \sqrt {{{\left( {{v_{\left( 1 \right)}}\left( t \right)} \right)}^2}} \\ & - e_1 \int {dt} {v_{\left( 1 \right)}}\left( t \right) \circ {A_{in}}\left( {{q_{\left( 1 \right)}}\left( t \right)} \right)\,.\label{I1}\end{align}\]
Extremization of this action by variation of \({{q_{\left( 1 \right)}}\left( t \right)}\) and suppressing the label 1 gives the Newton-Lorentz equation of motion
\[m\frac{d}{{dt}}\frac{{{v^\mu }\left( t \right)}}{{\sqrt {{{\left( {v\left( t \right)} \right)}^2}} }} = eF_{in}^{\mu \nu }\left( {q\left( t \right)} \right){v_\nu }\left( t \right)\label{NLE}\]
where
\[{F^{\mu \nu }} = {\partial ^\mu }{A^\nu } - {\partial ^\nu }{A^\mu } \equiv \left[ {\begin{array}{*{20}{c}}
0&{ - {E^x}}&{ - {E^y}}&{ - {E^z}}\\
{{E^x}}&0&{ - {B^z}}&{{B^y}}\\
{{E^y}}&{{B^z}}&0&{ - {B^x}}\\
{{E^z}}&{ - {B^y}}&{{B^x}}&0
\end{array}} \right]\label{Faraday}\]
is the Faraday tensor.
Notice the result \eqref{NLE} and definitions\eqref{NL_action2}, \eqref{I1} and \eqref{Ain} are consistent with the exclusion of self action.
The Maxwell Equations give the Faraday tensor (the electric and magnetic fields) in terms of nominally local currents that are presumed given. Returning to \eqref{ihe}, \eqref{Faraday} with \(\partial \circ A = 0\) give
\[{\partial _\mu }{F^{\mu \nu }} = {j^\nu }\,.\]
Using \(j = \left( {\rho ,{\bf{j}}} \right)\) these are the two Maxwell Equations
\[\begin{align}
\nabla {\bf{.E}} = & \rho \label{MX1}\\
\nabla \times {\bf{B}} - \frac{{\partial {\bf{E}}}}{{\partial t}} = & {\bf{j}} \label{MX2}
\end{align}\,.\]
Using that \(A = \left( {\phi ,{\bf{A}}} \right)\) Eq. \eqref{Faraday} implies
\[\begin{align}
{\bf{E}} &= - \nabla \phi - \frac{{\partial {\bf{A}}}}{{\partial t}} \label{EB1}\\
{\bf{B}} &= \nabla \times {\bf{A}}\label{EB2}
\end{align}\]
Using that
\[\partial \circ A = \frac{{\partial \phi }}{{\partial t}} + \nabla {\bf{.A}} = 0\]
Eqs. \eqref{EB1} and \eqref{EB2} gives
\[\begin{align}
\nabla {\bf{.B}} &= 0 \label{MX3}\\
\nabla \times {\bf{E}} + \frac{{\partial {\bf{B}}}}{{\partial t}} &= {\bf{0}} \label{MX4}
\end{align}\]
Eqs. \eqref{MX1}, \eqref{MX2}, \eqref{MX3} and \eqref{MX4} are the Maxwell equations that give the fields \({\bf{E}}\) and \({\bf{B}}\) in terms of a presumed given current \(j = \left( {\rho ,{\bf{j}}} \right)\).
Unlike the derivation of the Newton-Lorentz equation, the Maxwell equations as stated are consistent with DPI only if self-action is retained. This is because in using \eqref{ihe} information necessary to distinguish the currents is lost in the definitions of both \(j\left(x\right)\) and \(A\left(x\right)\). As a consequence, if \(A\left(x\right)\) is computed from \eqref{ihe} (or more precisely from \eqref{potential}) and \({\bf{E}}\left(x\right)\) and \({\bf{B}}\left(x\right)\) and computed from that \(A\left(x\right)\), and then these \({\bf{E}}\left(x\right)\) and \({\bf{B}}\left(x\right)\) are used in \eqref{NLE} to compute the force on some charge, in general that force will include the force generated by the charge on itself.
However, despite the formal inclusion in the Maxwell equations of all currents in the generation of the potential, and therefore the electric and magnetic fields, in practice one can split the total current into a part that is the local generator of fields (of interest), and the remainder, any of which are subject to respond to those fields.
Another difference between the Maxwell theory and DPI is that the Maxwell theory grants a physical status to the fields in empty space, whereas in DPI they have no physical status except at the location of charges. In Maxwell theory the fields \({\bf{E}}\left(x\right)\) and \({\bf{B}}\left(x\right)\) contribute to a stress-energy tensor at arbitrary \(x\), not on the world-lines of the charges. For example the theory grants an energy density equal to \(\left( {{\bf{E}}\left( x \right) + {\bf{B}}\left( x \right)} \right)/2\) in empty space, and momentum density equal to \({\bf{E}}\left( x \right) \times {\bf{B}}\left( x \right)\). In principle the effects of this stress-energy tensor on the space-time metric (as predicted by general relativity) can be observed off the world-line of charges.
In the Maxwell theory the fields are presumed to carry energy and momentum away from an oscillating dipole into empty space as radiation. If that radiation is not subsequently absorbed by other dipoles then the fields must be granted physical status of their own as host to that energy and momentum. That is, they must be regarded as having their own degrees of freedom, in addition to the 4-positions of the charges. But the fields have no independent degrees of freedom in DPI. They are defined instead entirely in terms of the charges and currents (see \eqref{potential}), and therefore cannot be host to radiation that is never subsequently re-absorbed. Consequently the viability of STF DPI depends on the eventual re-absorption of all EM radiation.
Accordingly, in addition to the interaction \eqref{I_interaction}, the action for the classical Maxwell theory includes a term involving just the fields. Extremization of that action by variation of \(A\left( x \right)\) followed by the exercise of gauge freedom to chose the Lorenz gauge leads to \eqref{ihe}. But that is all. Unlike DPI the Maxwell theory permits any complimentary function solution to that equation, and therefore the imposition of, for example, Cauchy boundary conditions.