What field of study to pursue in graduate school?

[Ishida52134]

So, I will have to start applying to graduate school in the near future, and I am still stuck between pure mathematics and theoretical physics. Both of these areas of study interest me vastly, and I have had to find a good balance between them, taking an equal number of graduate courses in both, and self studying both areas as well. However, I am stuck on what to pursue in graduate school, since I have to choose a specific field. For people like me, who like pure mathematics like algebra and analysis, as well as theoretical physics like quantum mechanics, general relativity, and elementary particle physics, would pursuing mathematical physics be the most reasonable path?

thanks for the advice.

 

[Hypersphere]

What you need to find out isn't really whether you can balance the number of courses, but whether you will prefer doing research in one field over the other. After all, that is what you'll be doing for most of gradschool (i.e. not taking courses).

That said, mathematical physicists is often taken to mean the people that try to put parts of theoretical physics on a rigorous (i.e. acceptable to mathematicians) foundation. That might be right for you (hard to tell), but certainly doesn't imply taking an equal number of physics and mathematics grad courses.

 

[Ishida52134]

By taking an equal number of courses in both areas, I just mean that I am equally interested in both fields. And also, I have participated in both theoretical physics and pure math research groups before, so I still don't really know yet.

Then again, exactly what areas are considered theoretical physics and pure mathematics? Is nuclear physics theory and condensed matter theory considered theoretical physics, and is research in combinatorics, probability theory, and dynamical systems considered pure math?
I always thought that the main pillars of theoretical physics were quantum mechanics and relativity, and mathematics were algebra, analysis, and geometry. How exactly do those other topics fall in?

 

[SophusLies]

For people like me, who like pure mathematics like algebra and analysis, as well as theoretical physics like quantum mechanics, general relativity, and elementary particle physics, would pursuing mathematical physics be the most reasonable path?

I'm assuming that you're a senior. As a math/physics person, I can that the best thing right now is talk to professors in each field, math and physics, and see the actual research. I spent a quarter as an undergrad working on a math research project, by the end of the quarter I knew that I didn't think like a mathematician enough to produce much. I also did the same, although many more quarters in physics research. Physics seemed much more natural for me to think of ideas and apply principles. For grad school, I applied to both math and physics departments but ended up choosing physics because of my affinity for it.

To be blunt, I've seen a lot of people that claim they can do both math and physics but in almost all cases they have strengths in only one. Experiencing the research of each subject should quickly show your decision.

 

[Hypersphere]

By taking an equal number of courses in both areas, I just mean that I am equally interested in both fields. And also, I have participated in both theoretical physics and pure math research groups before, so I still don't really know yet.

Maybe SophusLies has the right approach then: which field are you better at and can hope to be more successful in?

If there is truly no way for you to decide, you might want to take a chance or use a random decision method. It's usually not entirely impossible to switch programs,and it's quite possible to work with a professor in the other department (at least within reason). Until then, there is certainly time to look more into research.

BThen again, exactly what areas are considered theoretical physics and pure mathematics?

That is a classification issue which is ultimately down to subjective opinion and different choices of definitions. Essentially, some people consider "theoretical" an adjective, in which case theoretical physics is just a way to distinguish it from experimental physics, while others (mainly the public, apparently) seems to consider "Theoretical Physics" a proper noun encompassing the parts of physics cool enough to be the subject of TV documentaries. Actually, few physicists talk about "theoretical physics" if they get a chance to say "subfield X theory".

You might get a better idea by reading the relevant wikipedia pages, but there isn't really a final word on any of these.
Theoretical physics
Mathematical physics
Fields of mathematics
Pure mathematics

In the end, classification issues shouldn't matter that much. Early Jag Panzer might be considered a speed metal band, a (US) power metal band or a heavy metal band but the choice of genre doesn't change the sound of their music. The essence matters, not the label. (Feel free to replace this example by any other cultural entity without a clear genre.)

Is nuclear physics theory and condensed matter theory considered theoretical physics, and is research in combinatorics, probability theory, and dynamical systems considered pure math?

Not always (see above). That is why you should try to find an area that interests you, without paying attention to how it's classified.

I always thought that the main pillars of theoretical physics were quantum mechanics and relativity, and mathematics were algebra, analysis, and geometry. How exactly do those other topics fall in?

If quantum physics is considered a pillar, it would encompass condensed matter - at least the solid variety; as well as all of nuclear physics, particle physics, quantum information and so on. Quantum mechanics (and QFT) is more of a framework than a current research field (neglecting the associated challenges for mathematical physicists, that is). Basically, if you count QM and GR as pillars of theoretical physics, you are only leaving out the fields which use essentially classical physics, i.e. fluid mechanics, plasma, lots of accelerator physics etc. These fields can also be very theoretical, and also give rise to problems for mathematical physics.

As for math, the whole idea of "purity" as an ideal to strive for annoys me a bit. Anyway, combinatorics would usually be called pure I guess. According to the wikipedia link above, dynamical systems is considered part of pure mathematics too. Probability theory is a tricky one, as it is used throughout statistics and analysis (look up stochastic analysis) as well as number theory.