Nuclear/Particle Physics & Group Theory: Understanding the Benefits

[Spinalcold]

I'm pursuing a degree in nuclear physics. However, I have a huge interest in particle physics (i know they are closely related). I am wondering how much a math course in group theory will help me understand particle physics. I want to minor in math, so I'm going to take some extra math courses above normal physics (my main interest is complex and chaos theory) but I'm wondering if this class would help and by how much.

 

[Sleepy 104]

I'm still an undergrad myself but I can offer some advice. I took a basic Modern Algebra class in the math department two years ago and recently started an independent study on representation theory in physics.

Some good and inexpensive books are: "A Book of Abstract Algebra" by Pinter, "Group Theory and Quantum Mechanics" by Tinkham, and "Group Theory and its Application to Physical Problems" by Hamermesh.

I was recommended to read "Spinors in Physics" by Hladik and "Lie Algebras in Particle Physics" by Georgi next. While I can't personally recommend these last two since I haven't worked on them, I'm sure they're worth looking into.

Hope this helps!

 

[Sleepy 104]

Also, most math classes on group theory and representation theory don't focus on applications to physics. When I was taking my Modern Algebra class I was struggling to see how it could be useful in physics because we would often prove VERY abstract propositions. But once you start reading or talking about group theory and representation theory with someone who has used it in physics you immediately understand.

The key to these subjects is that they are tools which utilize symmetry. A lot of interesting subjects in the Tinkham book are treated with group theoretical methods only, but if you were to learn the same material in a standard undergraduate quantum class you would use a completely different method such as perturbation theory. It's really cool!

 

[Spinalcold]

thanks!

I definitely understand the huge dependence on symmetry in physics, which is why I was thinking about group theory. I keep hearing about SU2 and SU3 and was wondering how much a person can understand without a heavy class such as group theory (i don't know if our prof is really hard but I keep hearing really smart physicists in our department talking about how its incredibly hard, mainly due to having to produce an elegant solution, not just a correct answer.)

I love math, but not sure what path is best with a physics (maybe mathematical physics) degree. I also don't want to take too many uneeded classes either as I'm on student loans.

 

[Sleepy 104]

Honestly, you could probably do fine without taking a formal class on group theory if all you're interested in is the applications to physics. If you need a math class to fulfill your minor than go for it, but otherwise I think you could pick it up by reading books on your own.

Yeah, from what I've picked up in books about quantum field theory and particle physics it seems like the special unitary (SU) groups are the main ones used.

For perspective, the unitary and orthogonal groups weren't mentioned at all in my modern algebra class but I'm currently taking a graduate linear algebra class and they were shortly touched on in regard to isometries in bilinear forms. So to get started just pick up a cheap text on modern algebra and one on group theory in physics and take a look at them.

Best of luck!

 

[homeomorphic]

In the standard model of particle physics, the gauge group is SU(3) cross SU(2) cross U(1) and the Poincare group is also relevant. These are all Lie groups, the general theory of which, you wouldn't see in an introductory class. The Poincare group is all the symmetries of Minkowski space-time, and the gauge group is internal symmetries of the fields.

I would almost think of guys like SU(2) as being more linear-algebraic things that just happen to be groups, which I think explains why a lot of graduate physics students that I have come across get by to some extent without knowing what a group is (not sure what these people were studying, exactly).

So-called "representations" of these groups can be important in particle physics. And when you study representation theory, the "group" properties of guys like SU(2) really comes into play. Basic representation theory is simpler in the case of finite groups. With Lie groups, like those unitary ones, you need big heavy machinery, like the Haar measure, whereas with finite groups, a lot of things you would use the Haar measure for become a simple matter of taking an average over the group. So, the point is, the stuff you learn about in a basic abstract algebra class provides good background knowledge, even if you don't use it directly.

I'm not a physicist, but it's my hunch that knowing more about group theory would be helpful if you want to have a deeper understanding and go beyond blind calculations.