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In Dirac's 'Classical theory of radiating electrons', submitted in 1938, the electromagnetic radiated field is defined as:

$$F_\text{rad}^{\mu\nu}=F_\text{ret}^{\mu\nu}-F_\text{adv}^{\mu\nu}$$

Where $F_\text{ret}^{\mu\nu}$ is the electromagnetic tensor for the retarded field, which can be obtained from the Lienard-Wiechert potentials, and $F_\text{adv}^{\mu\nu}$ is the electromagnetic field for the advanced field, in which the charge effect propagates backwards in time.

I wonder which is the meaning of this definition. If an electron accelerates at a given time, the effect of that acceleration will propagate in two different regions of the space for the retarded and the advanced field (in the forward and backwards light cone). Also, for a given point $P$ we can have $F_\text{rad}^{\mu\nu}\neq 0$ even if the particle has accelerated neither at the retarded time $t_\text{ret}$ nor at the advanced time $t_\text{adv}$ but it has accelerated between $t_\text{ret}$ and $t_\text{adv}$.

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  • $\begingroup$ I've edited your question to use MathJax, it's better than importing images since this way the text remains searchable. $\endgroup$ – Philip Apr 3 at 16:24
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Advanced solution is just a particular solution for time-dependent EM field, there is no reason to call it "propagating backwards in time".

The definition

$$ F_{rad} = F_{ret} - F_{adv} \tag{*} $$ is the simplest combination of two distinct particular solutions (retarded and advanced solution of Maxwells's equations with sources) that results in a homogeneous solution. That is, the combination $F_{ret} - F_{adv}$ solves the Maxwell equations where sources vanish:

$$ \partial_\mu F^{\mu\nu} = 0. $$

This is the homogeneous wave equation that describes radiation in free space free of sources, hence the choice to label resulting $F$ with "rad" in (*).

Dirac is simply seeking a new nice definition of that part of total field of point particle that can be reasonably called "only radiation, without the bound Coulomb-like field" and he was impressed by the difference of retarded and advanced field. There may be other alternative definitions for "radiation field", depending on the context. For example, "retarded field minus the Coulomb-like quasi-static field that depends on position and velocity only". Dirac just chose (*) because it is nice and interesting.

This field (*) has an interesting property when it is due to point particle source; Dirac found its value in the vicinity of the particle is finite and directly related to value of the Lorentz-Abraham force $k\dot{\mathbf a }$.

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  • $\begingroup$ Thank you for your response. I appreciate your explanation that (*) is a homogeneous field that propagates through space without any source. The field caused by acceleration, which only propagates towards the future in the retarded solution, can be seen as a continuation of the field caused by the acceleration in the advanced solution. $\endgroup$ – Sergio Prats Apr 4 at 18:51

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