Math/Physics Double Major Benefits for Math/Physics Grad School

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Well, mathematical physics is fine, but I think it's better to either start as physics or math and then move to it. To do research in mathematical physics means you should be able to do research level work in both physics and math. You'd be better off just doing research in whatever subject you master first. Trying to master both will obviously take longer, and your goal is getting to research asap. Having said that, I have no idea what the competition is like for this field.

By mathematical physics I mean rigorous stuff like axiomatic quantum field theory.
 
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People change fields all the time. You're certainly not bound to what you do research in as an undergraduate for the rest of your life. The idea is that you need to convince people you can do research. That is, people (ie grad school admissions) should see that you're capable of taking something ridiculously over your head and making some sense out of it, and if you can find something new about it and share it with the scientific community, all the better. The main skills are resourcefulness and problem-solving.
 
Well yeah, but it should be related to the field you're applying to. I'm not sure how much doing math research will help with your physics grad school application (given probably none of the adcoms will understand what you did - I'm concerned people in het won't understand what I did in my specialized het area).

For best results, you need to set your goals straight as early as possible.