I am interested in understanding how and whether the transformation properties of a (classical or quantum) field under rotations or boosts relate in a simple way to the directional dependence of the radiation from an oscillating dipole.

For example, the EM field from an oscillating electric dipole $\mathbf{d}(t) = q x(t) \hat{\mathbf{x}}$ pointing along the $x$-axis vanishes in the x-direction (in the far-field). On the other hand, the sound radiating from a similar acoustic dipole vanishes in the $y-z$ plane (in the far-field). This result makes total sense classically because EM radiation consists of transverse waves while acoustic radiation is carried by longitudinal waves (try drawing a picture if this is not immediately obvious). The same holds true even if the fields and dipoles under consideration are treated quantum mechanically.

Now, the acoustic (displacement) field is represented by a scalar field, while the EM field is a (Lorentz) vector field. This leads me to wonder if one can draw some more general conclusions about multipole radiation, based on the transformation properties of the field in question.

Given a tensorial or spinorial field of rank k, and a multipole source of order l, what is the asymptotic angular dependence of the resulting radiation?

In particular, how does the radiation from a "Dirac" dipole look, assuming such a thing can even make sense mathematically?

By the latter I mean writing the classical Dirac equation

$$ (\mathrm{i}\gamma^{\mu}\partial_{\mu} - m)\psi(x) = J(x), $$

for some spinor-valued source field $J(x)$ corresponding to something like an oscillating dipole. (I understand that in general this term violates gauge invariance and so is unphysical, but I am hoping this fact doesn't completely invalidate the mathematical problem.)

Apologies to the expert field theorists if this is all nonsense :)


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