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I just read this: Anapolar Dark Matter

I'm not sure i've heard the term 'anapolar' before, so i google and i found this:

http://en.wikipedia.org/wiki/Toroidal_moment

This confuses me, my understanding is that spherical harmonics spanned 'all possible moments' that you could need to reproduce any kind of field. So what really is this 'unipolar/toroidal moment' and what relationship does it hold with the known and true spherical harmonics?

Is this some kind of 4D (3+1 Minkowski) spherical harmonic moment, that is fundamentally and geometrically inequivalent to the usual 3D spherical harmonics?

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1 Answer 1

Notice that the wikipedia article says that the polar and axial toroidal moments correspond to higher orders in frequency.

In the electrodynamic multipole expansion, all charge and current distributions can be expanded into a complete set of electric and magnetic multipole coefficients. These electrodynamic multipole coefficients can be further expanded in orders of frequency.

That is they appear if you expand the general result with non-static (time varying) distributions.

You'll also note that they fill in the unclaimed spaces in the symmetry space of $\text{Parity} \otimes \text{Time}$.

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but, we already call those moments multipolar, and it is also in that same article, mentioned that the anapolar term is independent from the multipole expansion. WTH? –  lurscher Jun 11 '13 at 3:25
    
No, these are different than the time varying multipoles. They have to be to have different symmetries. The usual proof about multipoles bothers itself only about static configurations. I'm not familiar enough with these to have an example on tap, however. The relationship will be roughly similar to that between vectors and pseudo vectors. You can span a whole Cartesian space with vectors, and you can span it with pseudo-vectors as well, but to get the maximum possible choice of transformation properties you need both. –  dmckee Jun 11 '13 at 3:27
    
if SE would allow a smilie with the dropped jaw, i would be using it now. Where can i read about this unknown side of a subject i thought i understood well? –  lurscher Jun 11 '13 at 3:29
    
isn't a time varying spherical harmonics, just a 4D spherical harmonic? In that case, the family of basis functions change a bit (some Gamma functions might appear) –  lurscher Jun 11 '13 at 3:31
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My weak little experimental brain strains over this stuff. I struggle with axial interaction in the weak force even though I have done experiments to extract the axial form factor of the proton. ::sigh:: –  dmckee Jun 11 '13 at 3:33
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