Lie Groups, Lie Algebras, Spinors, Belt Tricks, SU(2)<=>SO(3), Quantum Mechanics, and the meaning of life

Lie Groups, Lie Algebras, Spinors, Belt Tricks, SU(2)SO(3), Quantum Mechanics, and the meaning of life.

Other urls found in this thread:

youtube.com/watch?v=kpeP3ioiHcw
en.wikipedia.org/wiki/Lie_algebra
en.wikipedia.org/wiki/Group_action
en.wikipedia.org/wiki/Lie_bracket_of_vector_fields#Properties
twitter.com/NSFWRedditGif

>That feel when you will never understand this shit

Hey Veeky Forums

So I just got done with a Quantum Mechanics class that touched on symmetries and group theory applications in QM at the end. Unfortunately it was poorly presented, but fortunately it turned me on to group representations and Lie Groups.

So I really want to understand QM spin. I've tried several times over the years, and I am ready to admit that it's a very difficult concept. But I think it's because I've never seen it well presented. I was looking around on youtube the other day and stumbled across this guys lectures from CO College of Mines on Lie Groups:

youtube.com/watch?v=kpeP3ioiHcw

While the presentation wasn't super high level, the dude describes the exponential map in an intuitive way and motivates discovering deeper mathematics by looking for deeper, often times algebraic, connections.

So, to understand spin, I really need to understand Lie Groups and Lie Algebras. Right now I am scratching my head over the motivation for a Lie Algebra. I understand that the generators of the algebra form a basis of the tangent space of the group manifold, but don't understand the motivation for the definition of Lie Bracket, which is often the commutator of group elements. It smells of determinant to me, but I don't see the connect, and the wiki article doesn't motivate it's definitions.

en.wikipedia.org/wiki/Lie_algebra

Anyway, I was wondering if anyone could shed some light on these fascinating topics. Ultimately I want a solid understanding of spin, but I need to get there step by step. This thread is for people who want the same, or who have the insight to help the more naive of us out a little.

>but don't understand the motivation for the definition of Lie Bracket

Consider the lie group [math]G[/math]. Lie groups are smooth manifolds so you can define a diffeomorphism which is a left group action, [math{L_g}:G \to G[/math].

A vector field on G, [math]v \in \Gamma \left( {TG} \right)[/math], is called left invariant if [math]\left( {\operatorname{d} {L_g}} \right)\left( v \right) = v[/math].

The lie bracket is a product on the tangent bundle of a manifold. [math] \left[ { \cdot , \cdot } \right]:\Gamma \left( {TG} \right) \times \Gamma \left( {TG} \right) \to \Gamma \left( {TG} \right) [/math].

Consider the set of left invariant vector fields on G. [math] \mathfrak{g} = \left\{ {v \in \Gamma \left( {TG} \right)|\left( {\operatorname{d} {L_g}} \right)\left( v \right) = v} \right\} [/math]

To make this set a vector space, we need to equip the product [math]\left[ { \cdot , \cdot } \right][/math].

The vector space [math]\left( {\mathfrak{g},\left[ { \cdot , \cdot } \right]} \right)[/math] is the lie algebra of G.

This space is then isomorphic to the tangent space of G at the identity.

All you need to do is go through Stewart's Galois theory then through Jacobson's Lie algebras and you that will give you a good introduction.

...

>diffeomorphism
Had to look up this, but looks like a bijective smooth map between manifolds, check.

>group action
Also had to look up this. The wiki is ok:
en.wikipedia.org/wiki/Group_action
But talks about things in the most abstract way possible, which isn't bad per se, but not great for beginners. Anyway I made the connection between the example of matrices of a group rep and the vector spaces they act on, and saw buried in the example precisely this. So I get this concept. check.

[math]{L_g}:G \to G[/math] (Just to correct your post)

What do you mean by [math]TG[/math]? What's T? Is [math]\Gamma[/math] the set of all vector fields on [math]G[/math]?

I just spent 30 minutes staring at that gif. Do i have a problem?

Perhaps illustrating this post with SO(2) would be more illuminating? I'm starting to think TG means tangent space, but don't really understand.

[math] {L_g}:G \to G [/math] is simply the map [math] x \mapsto g \cdot x [/math] for all [math]x \in G[/math] and fixed [math]g \in G[/math].

[math] TG = \coprod\limits_{x \in G} {{T_x}} G[/math] is the tangent bundle of G, which is the disjoint union of the tangent spaces.

For any vector bundle [math]E[/math], [math]\Gamma \left( E \right)[/math] denotes the set of sections of E.

By definition, a vector field on a manifold M is a section of the tangent bundle of M. So [math]\Gamma \left( {TM} \right)[/math] denotes the set of vector fields on M.