Abstract algebra or ODE for physics

[battousai]

currently i am a math major, still unsure whether pure or applied. i am also looking to double major in physics. which class would be more helpful to me: abstract algebra, or (upper division) ODE class? I have taken the lower division DE class already.

 

[MathWarrior]

Differential Equations are everywhere in physics you would be doing yourself a disservice by not taking ODE's. Plus abstract algebra is basically really a lot of proofs, if you haven't had a course on proof writing you will be at a disadvantage.

 

[battousai]

im fine with proofs

 

[MathWarrior]

im fine with proofs

A second course on differential equations is still likely to be way more useful to you then abstract algebra. Some concepts in abstract algebra were used to derive concepts in quantum mechanics, but that is really the only physics application of it, I could be wrong. But even searching online paints that same picture. This site describes a mathematical physicist and what they might use in their job. I see no mention of abstract algebra.

 

[homeomorphic]

Some concepts in abstract algebra were used to derive concepts in quantum mechanics, but that is really the only physics application of it, I could be wrong.

Yeah, that's pretty much wrong.

Abstract algebra can be useful in physics. I'm not an expert on it. But the thing is, group theory, one branch of abstract algebra is all about symmetry. Symmetry is an important concept in physics. For example, you might have a U(1)-symmetry which keeps track of the phase of an electron. And the standard model uses a U(1) cross SU(2) cross SU(3), (maybe modulo Z_6 or something, I forget, since I don't actually understand this stuff).

I'm sure there are lots of other applications that I'm not aware of. For example, things like Von Neumann algebras and C* algebras. Those are examples of "rings", so ring theory can be relevant there. I don't know what the applications are there, but I know some physicists are interested in them. For starters, QM deals with Hilbert spaces. The bounded operators on a Hilbert space are sort of the canonical example of a C* algebra. In QM, a lot of the operators are unbounded, but there are ways to try to approximate them with bounded operators or some such thing. Again, this isn't my specialty.

Another area is quantum groups--these are not actually groups. Again, they are algebras.

Quantum groups have their origins in physics and may be relevant to the physics of anyonic condensed matter systems, exactly solvable models in statistical mechanics, or maybe quantum gravity.

These are just some examples. Depends on what you want to do exactly.

 

[MathWarrior]

Abstract algebra can be useful in physics.

I'm sure their are more uses beyond what I described, as I am not a physicist, but given that its a lower division abstract algebra course. He will not likely get to many of these things that you mention.

If he intends to go for a full PhD on the other hand by all means take it.