Abstract algebra or ODE for physics

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Discussion Overview

The discussion revolves around the choice between taking an abstract algebra course or an upper division ordinary differential equations (ODE) class for a math major interested in double majoring in physics. The conversation explores the relevance and applicability of each subject to physics, particularly in the context of the participant's academic trajectory.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • One participant suggests that differential equations are prevalent in physics and that not taking ODEs would be a disservice.
  • Another participant agrees that a second course on differential equations would likely be more beneficial than abstract algebra, noting limited applications of abstract algebra in physics.
  • Some participants acknowledge that abstract algebra, particularly group theory, has applications in physics, such as symmetry in quantum mechanics.
  • One participant mentions specific mathematical structures like Von Neumann algebras and C* algebras, indicating their relevance to quantum mechanics, though they express uncertainty about the details.
  • Another participant points out that while abstract algebra can be useful, a lower division course may not cover advanced applications that are more relevant to physics.
  • There is a suggestion that if the participant intends to pursue a PhD, taking abstract algebra could be worthwhile.

Areas of Agreement / Disagreement

Participants express differing views on the utility of abstract algebra versus ODEs in physics. While some emphasize the importance of ODEs, others highlight the potential relevance of abstract algebra, particularly in advanced topics. No consensus is reached regarding which course is definitively more beneficial.

Contextual Notes

The discussion reflects varying levels of familiarity with the applications of abstract algebra in physics, and there are indications that the relevance of certain mathematical concepts may depend on the specific area of physics one intends to pursue.

battousai
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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.
 
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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.
 
Last edited:
im fine with proofs
 
battousai said:
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.
 
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.
 
homeomorphic said:
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. :smile:

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

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