/sci/ - Science & Math

Formerly >>14825203

Talk maths.

>>11514131
It's the symmetry group of the Higgs Mexican hat potential [math]V(\phi)[/math]. What this means is that, if the Higgs potential [math]V[/math] achieves a minimum [math]V(\phi^*) =V^*[/math] somewhere at [math]\phi^*\in \mathbb{C}[/math], it must achieve [math]V^*[/math] at every point in the orbit [math]\operatorname{Orb}_{U(1)}(\phi^*)\cong S^1 \subset\mathbb{C}[/math] of [math]\phi^*[/math] under [math]U(1)[/math]. A specific choice of [math]\phi^*[/math] in this orbit you take as the vev [math]\langle \phi\rangle = \phi^*[/math] in the quantum theory spontaneously breaks this [math]U(1)[/math] symmetry.

>>11199695
A bit later meant 10 minutes until I got home to my computer.
But sure, go ahead.

>>11059104
First of all, all spins labeled by [math]j\in \frac{1}{2}\mathbb{Z}[/math] are irreps of [math]SU(2)[/math]. You're not representing anything else for higher spin particles.
Second of all, the very fact that [math]j[/math] can take half-integer values is due to the rep theory on [math]SU(2)[/math]. As a manifold, [math]SU(2)[/math] double covers [math]SO(3)[/math] so all reps of [math]SO(3)[/math] factor through the central extension [math]1 \rightarrow \mathbb{Z}_2 \rightarrow SU(2)\rightarrow SO(3)\rightarrow 1[/math]. You can check that irreps of [math]SO(3)[/math] are labeled by integers by, for instance, computing eigenvalues of the Casimir [math]L^2[/math] and using Schur's lemma. This is why the [math]j[/math]'s for the spherical harmonics are integers.