The Different Faces Of The Special Unitary Group ##SU(2)##

[fresh_42]

Differentiation, Spheres, and Fiber Bundles​

The special unitary groups play a significant role in the standard model in physics. Why? An elaborate answer would likely involve a lot of technical terms as Lie groups, Riemannian manifolds or Hilbert spaces, wave functions, generators, Casimir elements, or irreps. This already reveals that entire books could be written about them, and to be honest, they have been written about them. The many aspects are unfortunately found in quite a lot of different books, lectures, or articles. There is furthermore a gap between the language physicists use and the language mathematicians use. The former is often an abbreviation for entire contexts, and the latter is often hidden when used in physics. I will try to shed some light on the mathematical side of the coin, of course, without claiming completeness. It is all about symmetries and derivatives at its heart. Emmy Noether's famous theorem $[1]$ is pretty fundamental - one could easily develop entire physical and mathematical theories just as applications of Noether's theorem. The environment to prove it takes, unfortunately, a while itself to be developed.

The following text is meant to make its readers curious about a group that plays a big role in physics and shed some light on its many facets. I also hope it can be used as a quick reference guide to look up certain terms, definitions, and relations, or at least as an invitation to read more, e.g. in the sources listed at the end.


Continue reading ...

 

[Greg Bernhardt]

Part 2 has arrived!
https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

 

[Zafa Pi]

fresh_42 submitted a new PF Insights post

A Journey to The Manifold - Part I
View attachment 213148Continue reading the Original PF Insights Post.

I'm having trouble right from the get go. If someone says they have a group on a set U, that means if x and y ∈ U then x⋅y ∈ U. But with your example that is not true.

 

[fresh_42]

I'm having trouble right from the get go. If someone says they have a group on a set U, that means if x and y ∈ U then x⋅y ∈ U. But with your example that is not true.

This was a sloppy abbreviation. The group is defined on the open unit disc and the inversion on the open half of it. I combined both, because I didn't want to go through the entire definition and verification of a local Lie group, because this was not the point there. I primarily wanted to give an example which is not a global matrix group and which has a somehow unusual multiplication. I therefore quoted the source of the example for details. But as you ask, here is the actual definition of a local Lie group.

An $n-$parameter local Lie group consists of connected open subsets $\{0\} \in U_0 \subseteq U \subseteq \mathbb{R}^n$, a smooth group multiplication $U \times U \longrightarrow \mathbb{R}^n$ and a smooth inversion $U_0 \longrightarrow U$ with $0$ as identity element and the usual group axioms. The locality is given by the fact that the group operations only need to apply on a local area around the identity element. The same holds for the group axioms: they only have to hold where they are defined. This makes it different from a global Lie group, where those operations need to be defined everywhere.

But you're right, this has been a bit sloppy, since I left the details of the definition to the reader. (On my list of changes for an update. I have to see first where it can be done without taking too much space.)