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Main Question or Discussion Point
What background one needs to have to study QFT? I have a good background in calculus, linear algebra, PDEs, and quantum mechanics (at Shankar's level). Are these enough?
As a physics graduate student doing fairly mathematical Condensed Matter Theory, I WHOLLY agree with the above statements.Notice how neither graduate nor undergraduate physics departments require students to study real analysis, differential geometry, abstract algebra, or topology (from what I know). Taking any of these courses is not a horrible idea, but it is important to recognize that the math department has a very different culture and purpose for the study of these subjects than physics, and you can probably get by simply learning what you need on your own rather than going through any of these courses (none of which are even remotely trivial).
Well, I am self-studying the material; I'm not taking any courses...Notice how neither graduate nor undergraduate physics departments require students to study real analysis, differential geometry, abstract algebra, or topology (from what I know). Taking any of these courses is not a horrible idea, but it is important to recognize that the math department has a very different culture and purpose for the study of these subjects than physics, and you can probably get by simply learning what you need on your own rather than going through any of these courses (none of which are even remotely trivial).
The only major missing piece in your first post list was Lagrangian mechanics, but the basic formalism can be understood quickly by a good student.