oyolasigmaz

Hi everyone--

I'm a math major from Turkey, a sophomore, and double majoring with Physics. So far, I plan to finish college in 3,5 years, as I will be done with the courseload of 2 years after this, my third, semester. I will be done with calculus, multivariable calculus, differentials, abstract algebra 1, linear algebra, statistics and probability theory (introductory though) by the end of this semester. I am planning to pursue a career in high energy (particle) physics, and I have a general sense of what I would need in my graduate studies (such as Lie groups and algebra for representations of subatomic particles, differential geometry for relativity theory, and maybe some functional analysis for its connection with quantum mechanics) besides my normal math major curriculum, including the courses abstract algebra 2, applied math, numerical methods, real analysis 1-2, complex analysis, topology, and 2 more area electives. As such courses are not opened for every semester, or for some of them, never, I have to plan them as independent studies, or I should take them from other universities. Also, what sort of a mathematical background I should have in order to learn quantum field theory, if anyone knows? What do you think about my intention of finishing the college early? What else, you think, I should take as math courses? What would you suggest, about anything, in this concept? Thanks a lot in advance for any answer.

Ozan--

mattpi

You mostly seem to have the right idea of the directions to go in. I'd certainly try to cover some material on Lie groups & algebras, and some functional analysis will be essential.

You might like to look at a course in PDEs, including some stuff about distributions, since this may well come in handy when you are doing QFT.

As far as the differential geometry is concerned, you don't need to do too much to understand GR: you would probably be better served looking at a GR textbook (e.g. Wald) when you need it, than doing loads of differential geometry that you won't need.

Depending on how mathematically rigorous you want to be, you might also like to try and find some material on Operator Algebras.

oyolasigmaz

Thanks for your answer.

I guess we're settled down about Lie groups and algebras, and functional analysis to some extent.

The reason I came up with differential geometry was a professor of mine, who said that it makes it a lot easier to learn about relativity, but what you said is also rational. If I simply jump to GR, it may take much more time to learn it independently, but I have no clue if I am going to need DG later. Maybe I might combine them both in an indepent study of one semester? Would that be enough?

I guess I wouldn't be able to start learning about quantum field theory in undergraduate, because as far as I have heard cumulatively from a few people, I need to have strong background of classical mechanics, electrodynamics, quantum mechanics, statistical physics. You also add PDEs on top of them that I have forgotten. PDEs are a must for sure, so I should also add it to my list.

As far as I know, operator algebras have some solid connections with DG and QFT, and I guess it is a branch of functional analysis, so that I will need to learn about them later. But what about OA make them mathematically rigorous? How would they serve me, is also another question I have because I don't really understand it from the terms yet.

Thanks, again, for the answer. It is not easy, at least for me, to ask such questions and get satisfactory answers.

mattpi

The "usual" way of doing QFT is to start off with a Lagrangian (i.e. an expression that - loosely speaking - describes the relevant physical properties of the field you are looking at) and use it to produce the equations you need to understand the underlying quantum theory. Tthis is an entirely physics-based process, and it's generally responsible for the majority of our predictive ability when it comes to HEP-physics. However, this physical approach does not involve much mathematical rigour (i.e. you can't really go around proving theorems about these fields, and suchlike).

Therefore, people like Wightman and Haag & Kastler, and others, have attempted to define axioms that should apply to any quantum field theory. In particular, the Haag-Kastler axioms are concerned with the assignment of an algebra of observables to each region of Minkowski space, and the properties that the assignment must obey - hence the operator algebras. These approaches come under the umbrella term of "Axiomatic QFT", but the various approaches have other names such as "Algebraic QFT" or "Topological QFT" that you might have come across. Given this axiomatic framework it is possible to start proving theorems about quantum field theories in general, and also produce some numerical results that haven't been achieved through the purely physical approach.

My work at the moment is actually related to what is essentially an extension of the Haag-Kastler axioms to curved spacetimes (called Locally Covariant QFT).