# What's the quickest way to understand group theory in physics?

1. Jun 6, 2013

### wotanub

When I see questions about SO(3), SU(3), and lie groups as it pertains to quantum mechanics, I always hold off on getting into the discussion because I think maybe I don't know what that means. It all seems really familiar, but in a language I'm not familiar with.

2. Jun 6, 2013

3. Jun 6, 2013

### dkotschessaa

Do you need to cover the math background more thoroughly before applying it to physics?

(I don't know. I didn't even know group theory was applicable to physics as I just do math these days).

4. Jun 6, 2013

5. Jun 6, 2013

### WannabeNewton

It's extremely useful. You will see applications of Lie group theory in not only quantum mechanics and QFT but also classical field theory (finite group theory shows up as well in certain places of physics and chemistry). Note that studying group theory is different from study group theory for physicists (the former is obviously going to be much more formal and go into topics beyond what you might see in physics). Since you said you wanted a quick introduction, I'm assuming you don't want to learn all the background mathematics for a rigorous account of Lie group theory (which would require first working through a text on $C^{\infty}$ manifolds up to things like universal covering groups along with knowledge of homotopy theory). Here's a recent thread that might be of help: https://www.physicsforums.com/showthread.php?t=694546

Last edited: Jun 6, 2013
6. Jun 6, 2013

### dkotschessaa

Well, since I'm a math guy, my motivation is kind of "because it's cool/beautiful/etc." just from a mathematics perspective." It's very useful within mathematics (see here: http://en.wikipedia.org/wiki/Group_theory#Applications_of_group_theory )

I don't know the physics but I can intuitively see why it makes sense, when for example I'm dealing with symmetries. Perhaps you have another way of solving those problems, but group theory may allow you to generalize to find solutions to other problems you can't do the same way.

Looking forward to other responses.
-Dave K

7. Jun 6, 2013

### robphy

Last edited by a moderator: May 6, 2017
8. Jun 6, 2013

### WannabeNewton

Symmetries are definitely one of the larger motivating factors for the formalism. See, for example, chapter 3 of Ballentine's "Quantum Mechanics-A Modern Development", starting with section 3.2 which starts off with the Galilei group. Isometry groups of space-time solutions to Einstein's equations (which happen to be Lie groups) are also important in general relativity, especially when studying cosmological models.

9. Jun 6, 2013

### espen180

Group theory really shines only when you go to higher physics.
-In QM, the symmetry group of space-time is what gives rise to observable quantities. More precisely, the generators of the associated Lie algebra generate the observables. Take for example $p_x=e^{i\hbar \frac{d}{dx}}$, where $\frac{d}{dx}$ is a generator of the Galilean symmetry group. Same for the Poincaré group in relativistic mechanics.
-The way I understand quantum field theory, it relies on gauge groups to work. You feed it gauge groups and it spits out particles and their reactions.
-In the general theory of relativity, general covariance can be understood as follows: there is no canonical way of passing from a state of your system to a subgroup of the diffeomorphism group of your spacetime manifold, isomorphic to some fixed nontrivial Lie group.

I saw the last assertion in a paper by John Baez, and don't really understand it myself, but I think it's neat anyway.

Last edited: Jun 6, 2013
10. Jun 6, 2013

### WannabeNewton

11. Jun 6, 2013

### espen180

Sorry. It is "Higher Dimensional Algebra and Topological Quantum Field Theory", on page 2.
Arxiv: http://arxiv.org/abs/q-alg/9503002

He doesn't give a reference though.

12. Jun 6, 2013

### WannabeNewton

I don't see why you're apologizing ! Thank you for the link. As an aside (if you're interested), this is essentially what Wald talks about in section 4.1 of his text except he phrases it as General Covariance => no preferred vector fields pertaining only to space-time geometry may appear in the laws of physics. It is quite clear, however, how to relate this to Baez's statement. It is nice how one can easily encode these physical statements using the language of group theory.

13. Jun 6, 2013

### espen180

Thanks for the tip. I haven't gotten around to Wald yet, but it's definitely on my reading list.

14. Jun 11, 2013

### wotanub

Exactly what I wanted; thank you.