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

[wotanub]

I already know about generators, rotations, angular momentum, etc.

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.

 

[jedishrfu]

This article is a little dense but may help your understanding:

http://mysite.du.edu/~jcalvert/phys/groups.htm

 

[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).

 

[wotanub]

This article is a little dense but may help your understanding:

[/QUOTE] Maybe something with co...en useful, because it doesn't seem necessary.

 

[WannabeNewton]

I don't know either. I learned QM without even studying group theory, so I don't see why it's even useful, because it doesn't seem necessary.

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

 

[dkotschessaa]

Maybe something with concrete examples of why it is useful? It lists a few examples, but those don't seem that complicated to solve with just plain old linear algebra and differential equations.

The article seems awfully mathy without getting to the point. What's the motivation? It seems like its just showing that some things can be thought about with group theory if you are so inclined, but you don't really have to. Why would I want to learn about a whole new branch of math to solve perturbation problems I can already solve?
I don't know either. I learned QM without even studying group theory, so I don't see why it's even useful, because it doesn't seem necessary.

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

 

[robphy]

Possibly useful reading
http://www.physics.oregonstate.edu/~stetza/Lie.pdf
http://arxiv.org/abs/math-ph/0005032 (Hall)
http://arxiv.org/abs/0810.1019 (Neumaier)
(I haven't read through these myself.)

 

[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.

 

[espen180]

Maybe something with concrete examples of why it is useful? It lists a few examples, but those don't seem that complicated to solve with just plain old linear algebra and differential equations.

The article seems awfully mathy without getting to the point. What's the motivation? It seems like its just showing that some things can be thought about with group theory if you are so inclined, but you don't really have to. Why would I want to learn about a whole new branch of math to solve perturbation problems I can already solve?

I don't know either. I learned QM without even studying group theory, so I don't see why it's even useful, because it doesn't seem necessary.

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 [itex]p_x=e^{i\hbar \frac{d}{dx}}[/itex], where [itex]\frac{d}{dx}[/itex] 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.

 

[WannabeNewton]

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.

What paper was this by the way?

 

[espen180]

What paper was this by the way?

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.

 

[WannabeNewton]

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

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.

 

[espen180]

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.

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

 

[wotanub]

Possibly useful reading
(Neumaier)
(I haven't read through these myself.)

Exactly what I wanted; thank you.