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For example, I am learning QFT...but as a math/physics double major I kind of cringe at some of the assumptions made in my physics book and since my formal math skills are a bit below my physics skills, I can't always prove why a technique we use works. I decided I would look for a book on functional calculus at the university library to see if I could look over some material, and all I found was the legit stuff like functional analysis and harmonic analysis etc.

I opened up one of the books, (Functional Analysis and Related Fields by T. Kato) and I was just overwhelmed by how much more math I needed to be at that level still. I'm taking Real Analysis, but following a formal math curriculum, Harmonic Analysis or Functional Analysis feels so far away. There are hundreds of books like this one, though there is plenty of overlap, and there are so many subfields and interesting results in each that I don't even know what will be worth learning later. I read my texts in physics and they refer to results in sometimes (to me) obscure math papers or text books and I have to wonder whether the person who the physical theory originated with just played around with the math and later found a rigorous paper to back it, or if they really have such an extensive background in mathematics.

I had, from stumbling around on blogs, heard mention of more interesting algebraic structures such as the Octonions and various other fields, groups, rings etc. For example, I covered some material on Grassman algebra, but only at a surface level and I feel like I need to really know what I'm doing to use something. Lie Algebras are another topic which I can give you a definition of, and examples of how we use them in QFT...but I can't prove any fundamental results with them. I haven't had the time to formally study them yet. I know only enough basic differential geometry and topology to float along in the basic results of GR, but not nearly enough to worry about any sort of advanced material. Even Hilbert spaces are honestly a pretty significant step from a first or second course in Analysis, there's some real math going on in some of our most simple quantum systems.

Am I just impatient? Is there an opportunity for formal study later, when I get to graduate school? Or will I just need to wait until I have the time to sit down and read about all the different mathematical topics I need to know, before I can even worry about reading classic papers (though I do read some currently, it's tough trying to read something like Feynman...I don't get far) let alone reading modern mathematical physics papers....

Anyways, that was kind of rant-ish, but I was just wondering if anyone else ever got the same feeling. I suppose I should be glad I've got time to learn things...