9
$\begingroup$

I am currently a math phd student specializing in algebraic geometry aspiring to work at the boundaries of the the fields of mathematics and physics and so, was looking into the field of mathematical physics. Unlike many other fields in pure mathematics or theoretical physics, there doesn't seem to be much of a clear path in terms of studying the fundamentals for this field as most of the "mathematical physics" books are simply mathematical methods used in physics. In stark contrast, many fields have more or less clear roadmaps on what books to study.

As such, I do not really understand the basic knowledge that a mathematical physicist should have? Do they specialize in a particular area of mathematics or is it mostly topology and geometry or must they know other applicable areas such as functional analysis as well and to what depth?

So I was wondering, how exactly should one begin to learn the fundamentals for doing research in mathematical physics? In terms of a specific goal, what should one learn to begin to understand things such as Ed Witten's research papers? Should one ideally start through Nakahara's book? And would it help to study topics such as tqft and gauge theory? Even though it's too late now, should one apply to a mathematics or physics graduate program if their interest is in mathematical physics?

$\endgroup$
  • 1
    $\begingroup$ My favourite book so far on gauge theory is "Mathematical Aspects of Quantum Field Theory" by Faria-Melo -- you might find it useful. $\endgroup$ – Eivind Dahl Oct 7 '14 at 21:56
  • $\begingroup$ Thanks for the recommendation! But I am still no quite sure what I should be studying in particular. As for qft specifically, I also did a physics major as an undergraduate and took a qft class that used P&S. Would it be ideal to learn physics subjects from both the "mathematician's" and "physicist's" perspectives? $\endgroup$ – Sky123 Oct 7 '14 at 21:59
6
$\begingroup$

This got too long for a comment but is meant to be an extended one. I'm note quite the guy to say being mostly interested in it from a structural/mathematical perspective. Forgive me if I'm not telling you anything new.

You can definitely do TQFT within the confines of pure math. If what you want is the standard model you'll do well to understand your representation theory, as types of particles correspond to fundamental representations of Lie groups (U(1)xSU(2)xSU(3) in the standard model, times the Poincaré group if you do the analysis.) From there a quantum field is a section of a vector bundle associated to the representation over space-time satisfying a variational principle (an extremal of an action) involving suitably equivariant connections (which are incidentally your bosons). Faria-Melo develops this and in fact exhibits the standard model in this framework.

They leave out a clear analysis of how representations tie in with types of particles, but this is done by Baez and Huerta in this text (http://math.ucr.edu/~huerta/guts/). Basically, elements in your fundamental representations are fermionic particle states, generators of the adjoint representation are bosons that act on your fermions in a way that can be represented by Feynmann diagrams.

Quantization is still fluffy to me, but it appears this is where quantum groups come in: You cannot deform a semi-simple Lie algebra and get a reasonable deformation of its representation theory (it's category of representations). You can however deform its universal enveloping algebra (which is a Hopf algebra, i.e., an object with a favourably interacting product and coproduct). There is a master class on this going on right now which talks about this for the purpose of studying 3-manifold invariants using 3-dimensional field theories. Notes about quantum groups may be found on its web page: http://www.math.ku.dk/english/research/conferences/2014/tqft/ They have incidentally a crash course on operator algebras as well, which is part of the theory that allows you to reasonably deal with infinite dimensional representations of the Poincaré group.

How the functor point of view on field theories relate to the "classical" one developed in Faria-Melo a bit fuzzy to me, but I suspect you may find some answers in Segal's article on conformal field theories (http://www.math.upenn.edu/~blockj/scfts/segal.pdf -- a pretty shitty scan but you'll find it in his 60th birth day thing).

Of course this leaves out nitty-gritty computational aspects of the kind a physicists would be able to tell you about, and I have never gotten close enough to what the physicists do to actually wanting to renormalize anything (something you apparently need to do because of self-interacting particles producing diverging integrals). This is definitely a pretty big part of QFT you'll be missing if you don't study the physicists approach as well.

It appears the big unifying idea in any case is that a physical system should be invariant under choice of presentation (gauge) up to a group or automorphisms (gauge transformation) and that this is true for classical systems (Lorentz or Poincaré invariance of space or space-time) as well as quantum systems (other Lie groups acting on a vector bundle of states) and that all of physics are more or less fall out as properties of stuff with the right symmetries. This appears to be what physicists and mathematicians agree on either way, so you can't go wrong studying representations.

Aside from Faria-Melo here are some notes I like to look at:

These notes are pretty explicit about the kind of mathematics they use math.lsa.umich.edu/~idolga/physicsbook.pdf

These notes on Lie groups and representation theory are very good. staff.science.uu.nl/~ban00101/lie2012/lie2010.pdf They come with video lectures. webmovies.science.uu.nl/WISM414

$\endgroup$
  • $\begingroup$ Thanks for the useful insight! So I guess there is a lot to learn regarding tqft and approaching problems from both a mathematician's and physicist's perspective. On the other hand, I was more or less wondering what I should learn to do mathematical physics in general and not specifically any area such as tqft. (I just mentioned it as an inquiry of it being one of the things I should learn about). But what you say is very interesting and I will definitely look more into it. $\endgroup$ – Sky123 Oct 7 '14 at 23:05
  • $\begingroup$ Yeah well, principal bundles with their connections, vector bundles, lie groups and their representations, operator algebras for infinite dimensional stuff, Lagrangian and Hamiltonian mechanics for the dynamics of states (or the unifying frame work of symplectic geometry), and as always category theory. Cohomology becomes useful for understanding existence of spin structures etc. I'll let someone with more experience write a more comprehensive answer :) $\endgroup$ – Eivind Dahl Oct 7 '14 at 23:12
  • $\begingroup$ So in general, mathematical physicists start by learning those topics? Do they usually emphasize learning a lot in a specific area of math like I had to do with alg geo and reading research papers, or just learning it at the level of a graduate textbook? As for references, do you have any recommendations besides your previous one? The only ones I know of at the moment are nakahara and naber. In addition, I would need to learn them from a more applied perspective and not a more pure mathematics one right? $\endgroup$ – Sky123 Oct 7 '14 at 23:18
  • $\begingroup$ For example, I learned category theory from Maclane and its applications from the topology books I used, but in practice, would I only need it at the level of Geroch's book? $\endgroup$ – Sky123 Oct 7 '14 at 23:21
  • $\begingroup$ I couldn't really say. I mean, if you're interested in 1 dimensional field theory as described in Faria-Melo one could say that you don't really ever need to say "category," it's just a convenient way to think. Though if you're interested in d dimensional field theories, you should probably know about d (or at least d-1) categories. I couldn't tell you what mathematical physicists usually do because I'm not one of them. Maybe I'll be able to send one your way but I can't promise anything, heh. $\endgroup$ – Eivind Dahl Oct 8 '14 at 16:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.