How do people obtain the math for theoretical physics?

[malignant]

I'm confused about how people learn the required math for theoretical physics. Do they take the math during undergrad or is it taught during grad school or do they just learn it on their own time? The physics undergraduate program here requires calc 1-3, linear algebra, odes, pdes but that's very little compared to what's needed in grad level theoretical physics. I'm probably going to fit complex analysis and abstract algebra in but there's still a lot more to be learned.

My question:
Should I be working through textbooks in my free time or will grad school cover the rest? I'd gladly learn the required math in my free time (I have a year of undergrad left) but if it'll be taken care of in grad school I'd use my free time elsewhere.

Any suggestions?

 

[BiGyElLoWhAt]

Well, let me offer my opinion as an undergrad:
We just organized a (rather small) group of students to begin working through some of the things that we were interested in, and the consensus we came to is this: We're going to SR, (recap) Linear Algebra, then move on to tensor calculus before hitting GR.
We are doing this because we cannot learn all this as undergrads. We essentially have the same math requirements as you, but you don't learn about manifolds in linear algebra, you also don't learn about tensors (well.. kinda) in Linear, so we're taking it upon ourselves to learn this. Me personally, I want to do this because I'm impatient and don't want to wait until grad school.

That being said, I would be very inclined to believe that any math that would be required for your degree will be provided by your graduate theoretical program. Will it cover all of the math that you will need to work as a theoretical physicist? Absolutely not. Will all the prerequisite math be covered in undergrad school? Absolutely not.

My suggestion is this: If you are content with knowing the math, do self study. If you want it to show on paper that you know the math, consider a masters in math, or maybe a PhD, post-doc, or something along those lines.

 

[homeomorphic]

I don't think you necessarily need to study it on your own time. I think people just tend to learn what they need along with the physics. If you really needed to spend extra time, physics departments would require you to take extra classes. Not that it's a bad idea to do extra math, per se. It's just sort of optional. The math that you need varies wildly depending on what you end up doing. If you're really into math, maybe you do a double major. If math isn't your thing or you have other priorities, I think it works out fine to just do whatever the requirements are. My background is in math, but I considered trying for a physics PhD and interacted with a lot of physics students in grad school. I'm not sure that most physics grad students even know what a group is, so taking a whole abstract algebra class might be sort of tangential. Also, I took a graduate level GR class and although it might have been nice to know what a manifold was at some points, the concept was never used explicitly, although if you want to be really advanced in GR, you do need more math than that. So, there's probably going to be less math than you think. I think some physicists should absolutely have more math awareness, but I don't think you have to be the one to play that role, unless you like math. My impression is that physics professors have tend to have a bit higher math level than the grad students. Maybe they have to teach themselves more for research purposes.

 

[BiGyElLoWhAt]

Again, i am an undergrad currently, but if you arent that into math, you shouldn't become a theoretical physicist. If you are good at math, but wouldn't state math as one of your fundamental interests, then you should become a particle/quantum physicist, but not theoretical. As my friends and i have been talking, our currunt mathematical capacity is far too insufficient to be able to reasonably describe certain phenomenon, which means we need an entire new branch of mathematics to be invented. If that concept doesn't appeal to you IMHO you should not become a theoretical physicist.

 

[malignant]

Again, i am an undergrad currently, but if you arent that into math, you shouldn't become a theoretical physicist. If you are good at math, but wouldn't state math as one of your fundamental interests, then you should become a particle/quantum physicist, but not theoretical. As my friends and i have been talking, our currunt mathematical capacity is far too insufficient to be able to reasonably describe certain phenomenon, which means we need an entire new branch of mathematics to be invented. If that concept doesn't appeal to you IMHO you should not become a theoretical physicist.

I started out as a math major so I'd say I'm quite interested in it but I wouldn't be able to fit more than an additional 2 courses.

Any idea which 2 of these 6 would be the best to have upon entering grad school? : abstract algebra, advanced linear algebra, complex analysis, topology, real analysis 1, real analysis 2.

 

[Fredrik]

Not sure how to choose two of those, especially when I don't know exactly what's in them. I'd say that abstract algebra is the least relevant topic. You should learn some basic concepts from that course (group, ring, field, homomorphism, isomorphism, etc.) but you can do that quickly on your own. Topology is very useful if you're going to study either functional analysis (to better understand the mathematics of QM) or differential geometry (to understand the mathematics of GR), but it's often studied after a course on real analysis. It's also not really useful on its own. It's just the tool you need to be able to study functional analysis and differential geometry.

Linear algebra is the most useful topic, but since you say "advanced linear algebra", I'm assuming that you have already studied a book on linear algebra. Did it cover things like diagonalization and include a lot of proofs, or was it just about linear systems of equations and matrix multiplication? If it's the former, then maybe you just need to study a good book ("Linear algebra done wrong" by Sergei Treil) on your own to refresh your memory and fill in some gaps. If it's the latter, you definitely need to take this course.

Complex analysis is pretty useful. You will learn about analytic functions and contour integrals, and you will finally get to see a proof of the fundamental theorem of algebra, which says that every polynomial has a root. And the subject is easier and more fun than real analysis.

 

[Arsenic&Lace]

The physics undergraduate program here requires calc 1-3, linear algebra, odes, pdes but that's very little compared to what's needed in grad level theoretical physics.
Any suggestions?

The overwhelming majority of work in theoretical physics begins and ends with linear algebra, 18th century calculus, and its applications (e.g. ODE's/PDE's), so what you're saying here is simply false, with the exception of a course in applied complex analysis. Sometimes mathematicians will try to do physics and you get things like topological order or string theory, and in that case you might need to go on a sojourn to the pure math department, but you need to ask yourself whether or not that's a good idea.

 

[QuantumCurt]

In many cases, the math used in physics just represents a small piece of a whole field of math. Differential geometry is very useful for General Relativity, but there is also a lot of material covered in differential geometry that really has nothing at all to do with General Relativity. Statistics and probability is very useful in physics, but there's also a lot of material covered in these courses that has next to nothing to do with physics. This suggests that one can learn the math that they need as they learn the physics.

I plan to go to grad school for physics in some area of particle physics, which is very mathematically intensive. The mathematical intensity of this area is one of the reasons that I'm double majoring in physics and math. However, I also genuinely love mathematics purely for its own merits as well. I considered simply doing a minor in math, but after looking at what would be involved in it, I realized that it simply didn't cover as much math as I want to take.

 

[JorisL]

It really depends on what your focus will be during grad school.
Currently we are studying phase transitions in a rigorous way, meaning we look at it from a measure theory point of view.
It is organized in the same form as for example a real analysis text, i.e. state a theorem and prove it in an exact way.

Another example is when you are studying Lie groups and algebras. Some basic notions from topology like compactness etc. are really helpful.
In Belgium we covered that in an introductory course on real analysis. Knowing about Lie groups helped me with for example QFT and the follow up introducing supergravity.

In general I do agree with Fredrik, use his fleshed out advise.
Also if you have the basic notions of all this, you can study/read up on extra maths when needed. It's not really necessary to do this exactly, an understanding usually suffices!

My main point is mostly this;
I believe a course in real analysis is essential. It should cover sequences and their convergence, the convergence of series and preferably some basic introduction to various types of spaces e.g. metric spaces, compact spaces etc.

 

[Arsenic&Lace]

My main point is mostly this;
I believe a course in real analysis is essential. It should cover sequences and their convergence, the convergence of series and preferably some basic introduction to various types of spaces e.g. metric spaces, compact spaces etc.

Can you post a paper where your rigorous look at phase transitions produced interesting new experimental predictions? I'm generally skeptical of this sort of science. Even the standard, "non-rigorous" look at a phase transitions is often tragically useless, if elegant (applications to lipid bilayers or biopolymers being particularly egregious examples which come to mind).

To the OP, always try to see with utmost clarity where various core subjects tie back to experiment, which is the heart and soul of physics. Maybe you can find papers where some technique from real analysis proved essential to predicting some experimental observation down the line. My own personal opinion, which is of course just an opinion, is that this sort of thing is extremely rare. The point is that to validate opinions you hear in this thread, be sure to try and obtain examples where there is a clear paper trail from the mathematical discipline which has been suggested back to experimental results.

 

[homeomorphic]

I'm not sure the influence of math is that direct that you can follow the paper trail down. I don't think functional analysis would have arisen without real analysis preceding it, and there's a lot of influence from there on quantum mechanics.

 

[Arsenic&Lace]

If you can't draw the dots between real analysis and experimental discoveries, why would anybody consider a suggestion that real analysis is useful for physicists as anything other than a hunch? It may be a lot of connections and dots, but there still should be a path between that subject and solid experimental predictions, which are the objective of physics.

The currency of theoretical physics is experimental predictions, so suggestions in this thread should involve some degree of showing the OP the money.

 

[homeomorphic]

Well, all I'm saying is that if you are going to say functional analysis is useful, which, to some degree, the physics community accepts on some level, then real analysis will help you because it will help you understand what functional analysis is about. You can't even understand what a Hilbert space actually is unless you know the completeness property, which is a concept from real analysis. So, I think there is a path.

I think understanding is important, too, as well as predictions. And it doesn't necessarily have to result in experimental predictions. Suppose you enjoy a subject more because you understand it better, even if it offers no other advantage. It's extremely useful to enjoy your work.

 

[Arsenic&Lace]

Considering that very few physics textbooks spare much thought to the notion of Cauchy completeness in Hilbert spaces, and considering that the intuition for Hilbert spaces can be developed entirely by analogy to finite vector spaces, I fail to see how that justifies functional analysis for a physicist.

What does your understanding even mean if it cannot be related to your objectives? If I "understand" what a Hilbert space is from the point of view of functional analysis, and this understanding does not help me to achieve my goals, how can I validate it as understanding? To me understanding can be measured by how many meaningful questions you can answer about it correctly, and the correctness of the answer and meaning of the question is dictated by the goal you are trying to achieve. A meaningful question about quantum mechanics relates to your ultimate goal; for instance, what does a wavefunction tell me about the system's physical state? How do I calculate the wavefunction of a hydrogen atom in a uniform electric field? If I am given an initial wavefunction, how can I determine how it will evolve in time? What is density of states of a Fermi gas? Personally, I cannot think of a physical goal, with it's associated questions, which requires reference to say, functional analysis, in order to be answered.

Now you may say that it is the mathematical questions that are interesting, but again the mathematical questions addressed by pure mathematics seem to me generally irrelevant to the mathematical questions posed by a physicist. A mathematician might want to know if, given a differential equation and any initial condition, solutions exist, are unique, are smooth, globally defined etc. A physicist just wants to know how to numerically approximate a solution or analytically obtain one for his/her specific equation. I cannot personally think of a mathematical question posed by a topic such as real or functional analysis which coincides with the rather more mundane sort of mathematical question posed by a physicist (how do I integrate this high-dimensional integral? how do I calculate the mean first passage time on a Markov chain with these properties? what are the solutions to this differential equation etc).

 

[homeomorphic]

intuition for Hilbert spaces can be developed entirely by analogy to finite vector spaces, I fail to see how that justifies functional analysis for a physicist.

Well, when you are doing that, you are basically doing functional analysis, you are just not doing it rigorously. I have two problems here, one of which is that I don't know my QM that well, and the second of which is that I haven't bothered to try to sort out what you actually "need" or don't need to do everything. But maybe that's part of my point. I don't know that it's good to spend too much time worrying about whether everything is absolutely necessary or whether it just makes things more rigorous or just makes it more elegant. It's good not to waste too much time in mathematical la la land, but on the flip side, I don't see the point of being afraid to even lay a foot in it, if you can't prove that it's absolutely necessary for some experimental purpose. If you are more questioning, maybe you might not believe all the results, unless you are more rigorous, and then you need more of the mathematician's treatment. Much as with calculus, the theorems about Hilbert spaces don't really work without the completeness property. And unlike with calculus, it's not obvious that things actually work out. So, you can always raise the question of whether you are getting the right answer by wrong reasoning. But mathematicians took care of that.

What might be a better example is this:

http://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems

Penrose got a PhD in algebraic geometry and his whole mathematical education, including real analysis, is part of the reason why he was the one to be able to make these contributions. Specifically, he applied differential topology, a branch of topology which depends crucially on concepts from real analysis.

I cannot personally think of a mathematical question posed by a topic such as real or functional analysis which coincides with the rather more mundane sort of mathematical question posed by a physicist (how do I integrate this high-dimensional integral? how do I calculate the mean first passage time on a Markov chain with these properties? what are the solutions to this differential equation etc).

The first thing I would think of is Fourier series. There are physical experiments I can think of where this comes up, although it might be somewhat limited--but you can just throw in all sorts of Sturm-Liouville type problems and so on, if you want a wider range of applicability, since the orthonormal functions that arise in that case are analogous to Fourier series. There are physical predictions lurking there because if the only way you know how to solve your problem is using that Fourier series or those Bessel functions or Hermite polynomials or whatever it is, which is predicting that experimental result, the way you are going to be able to get your error bounds on your answer is using techniques from numerical analysis and its close relative, real analysis.

 

[homeomorphic]

There are a couple other things I should add.

I can't vouch for this personally, but one of the reasons I made the mistake of getting a math PhD was that a topology prof of mine talked about how he had been to conferences where physicists and mathematicians interacted with great benefit to both sides. So, if that is true, there should be some benefit to being able to communicate with mathematicians and learn their language and culture, and to becoming a bit of a mathematician yourself, even. That's debatable, of course--maybe those guys were just string theorists or something, I don't know. As I said, this point of view is what lead me to the mistaken belief that I could still be in touch with physics if I did a math PhD. At least for me, that wasn't really the case--learning physics just ended up slowing me down and hampering my progress in math (of course, physics was an end in itself, so it was not hampering my own goals, but was quite at odds with what my adviser wanted from me). Of course, it could be that if I had stayed the course, eventually, it would have paid off, and I would be able to act as a mathematical consultant to physicists, as my adviser was, ironically.

Another point I can add is, I think, an unimpeachable argument in favor of taking more math classes, at least under certain circumstances. That is the story of how I learned linear algebra. I took linear algebra in the summer after my first year in college. I wasn't very good at math back then. I didn't understand it deeply, and I didn't retain a whole lot of it. Guess where I actually started getting it? That's right, my real analysis class. To some degree, the same was true of calculus, especially sequences and series. I didn't really get it when I took calculus (I was fine at doing integrals and derivatives and so on, but I didn't really understand why Taylor series are the way they are, why any of those silly convergence tests and error bounds worked, etc.), and then when I did analysis, I got it. And those things are useful for physics. You can be as dismissive as you want about being rigorous and all that, but the fact of the matter is that I didn't really get it, rigorously or non-rigorously before I took that class. So, the point is that when you learn a subject, you're not just learning that subject. It can also be an excellent way to review its prerequisite subjects while learning something new. And I think part of it was the insight of the instructor, which I may not have encountered elsewhere. I think it also had to do, not so much with the rigor, but more the stage before the rigor, where I would have to come up with all the intuition to inspire my proofs. Now, of course, some people may have learned linear algebra very successfully elsewhere and so on, and in that case, they would be fine not taking that real analysis class. But if you happened to be like me at that stage, it would have been very beneficial, and in general, I think it can give you a boost to your problem-solving skills. I think it's just too simplistic to boil everything down to direct applicability.