Group theory and Physics

Most of us have group theory in our degree course. Some time ago, I was pretty critical about pure math and its abstract nature ( having nothing to do with reality). I treated them as just mental fantasies, some of which tends to have a structure that agrees with the way nature nature worked.
I went around telling people about some paradoxes in set theory and how Godel's theorem implies that any system based on precise axioms and set of rules is fundamentally flawed. One famous paradox is Russel's paradox.

I now realize that how I felt back then was wrong. I felt that way partly because I am not good at Math and partly because I my thinking was very naive. Which is not really a bad thing, I always learn from my naiveties.
I am now doing a project in Lie Groups under a professor in my college. And I visit a person named Dr. Aravinda from TIFR. He's helping me out with Topology and Differential Geometry. This is amazing because two years ago I didn't think I would find anyone working on these topics ( and also willing to guide me).

Anyway, I thought I'll put up something here that I found worth reading. This is the preface to one of the books that he suggested:

" As a graduate student in experimental physics, I found the study of group theory considered to be a useless "high-brow" luxury. Furthermore all attempts to follow a lecture course resulted in a losing battle with cosets, classes, invariant subgroups, normal divisors and assorted lemmas. It was impossible to learn all the definitions of new terms defined in one lecture and remember them until the next lecture. The result was complete chaos.


It was a great surprise to find later on that (1) Techniques based on group theory can be useful; (2) They can be learned and used without memorizing the large number of definitions and lemmas which frighten the uninitiated. 

Angular momentum is presented in elementary quantum mechanics courses without a detailed analysis of the lie group of continuous rotations in three dimensions. The student learns about angular momentum multiplets and coupling angular momenta without realizing that these are irreducible representations of the rotation group.He also does not realize that the algebraic properties of other lie groups can be applied to physical problems in the same way as he used angular momentum algebra, with no need for classes, cosets etc. . . . . . . "
He then goes on to talk about further applications in creation and annihilation operators, and also talks about quasi spin etc..
The book is Lie groups for Pedestrians by Harry Lipkin.

Group theory is actually fun and beautiful ( a word I rarely use) , but again college takes the life out of it. I am actually lucky that this course is handled in my college by an expert in this topic, but even he is constrained because the college requires you to rush through all the theorems within one semester and one is not given enough time to understand the subtleties. But it is always best continue to reflect on it even after the course is done.





P.S: The weekend sessions have been going on. Everybody has just been too busy to put up the summary. We have had three sessions within a span of four days. Karthik made quite a lot of connections in classical mechanics during his class on friday. I conducted a test on saturday on whatever has been done, to judge the understanding so far. Today was Shruthi's session on QM, which was again taken up mostly by Karthik to make known previously unspoken things.