In 2009 Insane Clown Posse put out that "fuckin' magnets, how do they work?" song and everybody made fun of them but I was just sitting there at the time thinking "I sure do wish someone would explain to me how the fuck force-carrying bosons work"
Like the way magnets work is they peturb a U(1) gauge field (which I think corresponds to the symmetry of the EM phase). I get that part. What I don't get is why, once you have a field, that makes, like… shit move
@mcc If a matter field carries a charge under the gauge transformation generator, then its partial derivative violates the gauge symmetry when it appears in the Lagrangian. Replacing it with a gauge-covariant derivative defined as a sum of the partial derivative and a term coupling the matter field to the gauge field proportional to the charge cancels the extra phase factor and generates the interaction.
@andrea Okay. Okay that's great, thanks. Let me just think about that for three weeks straight
@mcc (The curvature-covariant derivatives in general relativity, incidentally, do exactly the same thing but for diffeomorphisms instead of gauge transformations)
@mcc @mhoye there's category theory in there somewhere if you look hard enough at representation theory and recoupling coefficients (where the pentagon identity on monoidal categories corresponds to tree rotations as a change of basis in a vector space of intertwiners)
You don't really see representations other than the fundamental and adjoint ones in gauge theory though; gravity is more complicated and does make arbitrary-order tensor powers of the fundamental rep of the Poincare group
@andrea @mcc @mhoye If you all want category theory with your physics: https://arxiv.org/abs/1310.7930
(I understand the first two, maybe three words)
@glaebhoerl @mcc @mhoye heh I think I am cursed with being able to understand about two-thirds of a whole sentence