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Note: I will expand this question with more specific points when I have my own internet connection and more time (we're moving in, so I'm at a friend's house).

This question is broad, involved, and to some degree subjective.

(I started out as a physics-only student, but eventually decided to add a mathematics major. I am greatly interested in mathematics; the typical curriculum required for physics students is not deep or thorough enough; mathematics is more general (that means work!); and it only requires a few more classes. Naturally, I enjoy mathematics immensely.)

This question asks mainly of undergraduate-level study, but feel free to discuss graduate-level study if you like.

Please do not rush your answer or try to be comprehensive. I realize the StackOverflow model rewards quick answers, but I would rather wait for a thoughtful, thorough (on a point) answer than get a fast, cluttered one. (As you probably know, revision produces clear, useful writing; and a properly-done comprehensive answer would take more than a reasonable amount of time and effort.) If you think an overview is necessary, that is fine.

For a question this large, I think the best thing to do is focus on a specific area in each answer.


Update: To Sklivvz, Cedric, Noldorin and everyone else: I had to run off before I could finish, but I wanted to say I knew I would regret this; I was cranky and not thinking clearly, mainly from not eating enough during the day. I am sorry for my sharp responses and for not waiting for my reaction to pass. I apologize.

Re: Curricula:

Please note that I am not asking about choosing your own curriculum in college or university. I did not explicitly say that, but several people believed that was my meaning. I will ask more specific questions later, but the main idea is how a physics student should study mathematics (on his or her own, but also by choosing courses if available) to be a competent mathematician with a view to studying physics.

I merely mentioned adding a mathematics major to illustrate my conclusion that physics student need a deeper mathematical grounding than they typically receive.

And now I have to run off again.

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closed as not constructive by David Z Apr 14 '13 at 3:31

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

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"the typical curriculum required for physics students is not deep or thorough enough; mathematics is more general (that means work!); and it only requires a few more classes. " this seems a bit contradictory to me... –  Cedric H. Nov 4 '10 at 20:48
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Surely I am overreacting, and am lacking in civility and courtesy--please forgive me. However, casting a "close" vote without an explanation @Sklivvz. How would one "make this of international interest"? More rightly asked, what is localized about it? Surely this is one of the broadest questions one can ask! Who is interested in conformational topological field theory? How many professional physicists specialize in low-temperature physics? –  Mark C Nov 4 '10 at 21:17
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@Mark C: the main problem I have with this question is that you are writing constantly things like "This question is broad, involved", "Please do not rush your answer ", "insulting", "promise" ... just ask your question and let people answer if they understand what you want. –  Cedric H. Nov 4 '10 at 21:25
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not all university systems allow one to choose the curricula. not all university systems have undergraduate/graduate separation. i don't even know exactly what "adding a major" means. this said, the question has merit, and can be saved. note that university specific stuff is also sort of off topic. the basic question which has merit is: what approach/topics in maths are useful to study physics (or mathematical physics)? the rest of the question basically confuses me... i don't know how your university works (nor should i care). –  Ebenezer Sklivvze Nov 4 '10 at 21:57
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Also the "Please do not rush your answer or try to be comprehensive." is flamebait, or at least meta- material!? –  Ebenezer Sklivvze Nov 4 '10 at 21:59

8 Answers 8

I feel very strongly about this question. I believe that for an experimentalist, it's fine to not go very deeply into advanced mathematics whatsoever. Mostly experimentalists need to understand one particular experiment at a time extremely well, and there are so many skills an experimentalist needs to focus all of their time/energy on developing as students.

I believe experimentalists should derive their physical intuition from lots of time spent in the lab, whereas theoreticians should develop their physical intuition from a sense of "mathematical beauty" in the spirit of Dirac.

Theoreticians, in my opinion, should study mathematics like math majors, almost forgetting about physics for a time; this is the point I feel so strongly about. The thing is that math is such a big subject, and once you have the road map of what is important for theoretical physics; then it really takes years of study to learn all the mathematics. I think it's so bad how many physics proffessors, who are themselves experimentalists, teach math improperly to young theoretician's. I personally had to unlearn many of the things I thought I knew about math, once I took a course based on Rudin's "Principle's of Analysis".

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Being a Theoretical Physics Grad student, I absolutely could not agree with the third paragraph more. To reiterate, it of course depends greatly on what kind of 'physics' (VERY broad term!) student you are. Your question simply asks '...a physics student...'. I absolutely regret so much every single physics lecture I ever sat in during my undergraduate time. When there is so so so SOO much Maths to learn, I feel that every single one of those hours was wasted. I got better understanding from the Maths-Physics courses that I took in the maths department... –  Flint72 Apr 12 at 13:13
    
(comment character limit is annoying!)... usually the year after doing a similar in physics, which became superfolus. I am now at the stage where I am a full year (or more!) behind my peers in the department who did their undergraduate study only in maths. It's sad, but theres nothing that I can do about it not but try to work hard, read and study as much advanced maths as I can, and try to catch up! The point of this little rant, if you want to be a theorist, don't make the mistake that I made, and listen to the advice of Matt above! –  Flint72 Apr 12 at 13:14

It is important when studying mathematics to do so with the following perspective

Mathematicians Allow Useless Non-computable Fantasy Objects

Mathematicians often choose to live in a world where the axiom of choice is true for sets of size the continuum. This is idiotic for many reasons, even for them, but it is especially idiotic for physics. There are easy intuitive arguments that establish that every set has a volume, or Lebesgue measure, and they go like this:

Given any set S in a big box B, choose points randomly and consider when they land in S. In the limit of many throws, define the measure of S to be the volume of B times the fraction of points which land in S. When this works, and it always works, every set is measurable.

This definition is not allowed in mathematics, because the concept of randomly choosing a point requires taking a limit of the random process of choosing the digits at random. The limiting random process must be defined separately from the approximation processes within usual mathematics, even when the approximations almost always converge to a unique answer! The only reason for this is that there are axiom of choice constructions of non-measurable sets, so that the argument above cannot be allowed to go through. This leads to many cumbersome conventions which inhibit understanding.

If you read mathematics, keep in the back of your head that every set of real numbers is really measurable, that every ordinal is really countable (even the ones that pretend to be uncountable collapse to countable ones in actual models of set theory), and that all the fantasy results of mathematics come from mapping the real numbers to an ordinal. When you map the real numbers to an ordinal, you are pretending that some set theory model, which is secretly countable by the Skolem theorem, contains all the real numbers. This causes the set of real numbers to be secretly countable. This doesn't lead to a paradox if you don't allow yourself to choose real numbers at random, because all the real numbers you can make symbols for are countable, because there are only countable many symbols. But, if you reveal this countability by admitting a symbol which represents a one-to-one map between some ordinal and the real numbers, you get Vitali theorems about non-measurable sets. These theorems can never impact physics, because these "theorems" are false in every real intepretation, even within mathematics.

Because of this, you can basically ignore the following:

  • Advanced point set topology--- the nontrivial results of point-set topology are useless, because they are often analyzing the choice structure of the continuum. The trivial results are just restating elementary continuity properties in set theoretic language. The whole field is bankrupt. The only useful thing in it is the study of topologies on discrete sets.
  • Elementary measure theory: while advanced measure theory (probability) is very important, the elementary treatments of measure theory are basically concerning themselves with the fantasy that there are non-measurable sets. You should never prove a set is measurable, because all sets are measurable. Ignore this part of the book, and skip directly to the advanced parts.

Discrete mathematics is important

This is a little difficult for physicists to understand at first, because they imagine that continuous mathematics is all that is required for physics. That's a bunch of nonsense. The real work in mathematics is in the discrete results, the continuous results are often just pale shadows of much deeper combinatorial relations.

The reason is that the continuum is defined by a limiting process, where you take some sort of discrete structure and complete it. You can take a lattice, and make it finer, or you can take the rationals and consider Dedekind cuts, or you can take decimal expansions, or Cauchy sequences, or whatever. It's always through a discrete structure which is completed.

This means that every relation on real numbers is really a relation on discrete structures which is true in the limit. For example, the solution to a differential equation

$${d^2x\over dt^2} = - x^2$$

Is really an asymptotic relation for the solutions of the following discrete approximations

$$ \Delta^2 X_n = -\epsilon x_n^2$$

The point is, of course, that many different discrete approximations give the same exact continuum object. This is called "existence of a continuum limit" in mathematics, but in statistical physics, it's called "universality".

When studying differential equations, the discrete structures are too elementary for people to remember them. But in quantum field theory, there is no continuum definition right now. We must define the quantum field theory by some sort of lattice model explicitly (this will always be true, but in the future, people will disguise the underlying discrete structure to emphasize the universal asymptotic relations, as they do for differential equations). So keep in mind the translation between continuous and asymptotic discrete results, and that the discrete results are really the more fundamental ones.

So do study, as much as possible:

  • Graph theory: especially results associated with the Erdos school
  • Discrete group theory: this is important too, although the advanced parts never come up.
  • Combinatorics: the asymptotic results are essential.
  • Probability: This is the hardest to recommend because the literature is so obfuscatory. But what can you do? You need it.

Don't study mathematics versions of things that were first developed in physics

The mathematicians did not do a good job of translating mathematics developed in physics into mathematics. So the following fields of mathematics can be ignored:

  • General relativity: Read the physicists, ignore the mathematicians. They have nothing to say.
  • Stochastic processes: Read the physicists, ignore the mathematicians. They don't really understand path integrals, so they have nothing to say. The usefulness of this to finance has had a deleterious effect, where the books have become purposefully obfuscated in order to disguise elementary results. All the results are in the physics literature somewhere in most useful form.
  • Quantum fields: Read the physicists, especially Wilson, Polyakov, Parisi, and that generation. they really solved the problem. The mathematicians are useless. Connes-Kreimer are an exception to this rule, as is but they are bringing back to life results of Zimmermann which I don't think anybody except Zimmermann ever understood. Atiyah/Segal on topological fields is also important, and Kac might as well be a physicist.

Physics is the science of things that are dead. No logic.

There are many results in mathematics analyzing the general nature of a computation. These computations are alive, they can be as complex as you like. But physics is interested in the dead world, things that have a simple description in terms of a small computation. Things like the solar system, or a salt-crystal.

So there is no point to studying logic/computation/set-theory in physics, you won't even use it. But I think that this is short sighted, because logic is one of the most important fields of mathematics, and it is important for it's own sake. Unfortunately, the logic literature is more opaque than any other, although Wikipedia and math-overflow do help.

  • Logic/computation/set-theory: You will never use it, but study it anyway.
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-1: pointlessly dogmatic, and mostly wrong. You've missed the main point of continuous methods, which is to make things easier, not harder. Your quest for discretized solutions to everything seems to have caused the omission of Lie groups, which play a central role in understanding symmetry, but are continuous objects with very few finite subgroups. Also, you are misinterpreting the Löwenheim-Skolem theorem. –  Scott Carnahan Oct 21 '11 at 6:07
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@Scott: I have no "quest" for discretized solutions--- you are misinterpreting. What I said is that you have to understand continuous results as limits of discrete ones, and be conscious of the limiting process. I agree that continuous methods make things easier, in those cases where you already know the continuum structure, but people tend to believe they have exhausted the continuum, and they haven't. The renormalization process gives new continuum structures which haven't been given a continuum description yet, but their discrete description exists, and the limit is hard. –  Ron Maimon Oct 21 '11 at 18:12
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@Scott: I did not forget about Lie groups, its just that everyone already knows these. I tried to focus only on things that not everyone knows already. I understand the Lowenheim Skolem theorem like the back of my hand, I am not misinterpreting it. It proves that any axiomatic system has a countable model. This countable model is the real thing one studies, sorry to disagree with 90% of working mathematicians (not 90% of logicians, however). The fact that mathematicians get this wrong all the time means that it needs to be said by me. –  Ron Maimon Oct 21 '11 at 18:15
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focussing on the Axiom of Choice, especially by putting it in the first place, the chief place, of the answer, is crack-pottery. Focussing on computability is done by some people, but is dogmatic and off-topic. I had to downvote this provocative, detailed, but skewed and useless answer for those two reasons. –  joseph f. johnson Jan 14 '12 at 19:01
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@Joseph f. Johnson: Thank goodness its finally getting downvoted, I was worriedvthat I was preaching to the choir. –  Ron Maimon Jan 15 '12 at 0:00

Obviously this is not a comprehensive list, and my aim is simply to give you a pointer to the basic material you need to cover early on. As you progress you may become more specialised and your field may have particular mathematical techniques and formalisms which are particular to it.

Much of the mathematics used in physics is continuous. This ranges from the elementary calculus used to solve simple Newtonian systems to the differential geometry used in general relativity. With this in mind, it is generally necessary to cover calculus in depth, real and complex analysis, fourier analysis, etc.

Additionally, many physical transformation have very nice group structures, and so covering basic group theory is a very good idea.

Lastly, strong linear algebra is a prerequisite for many of the techniques used in the other areas I have mentioned above, and is also extremely important in the matrix formulation of quantum mechanics. Finding ground states of discrete systems (for example spin networks) means finding the minimum eigenvalue and corresponding eigenvector of the Hamiltonian.

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The question is way too broad. Different areas of physics requires different level (and area) of maths.

One general list is here: Gerard ’t Hooft, Theoretical Physics as a Challenge.

Also one approach is to learn maths when you encounter it in physics (*), making sure each time you learn a bit more than solely to understand (*).

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Yes, sorry, I think today I will have time to put in some more specific questions. –  Mark C Nov 16 '10 at 14:21

The same reason a literature student needs to learn English. You can't express yourself otherwise. Long have the times past when you could have described physical phenomena using words; Faraday did so. At that time unexplained physical phenomena were at human scale and human language was enough. Today, frontiers of physics are far beyond meters, kilograms, amps and few eV. By a chance, we have discovered that Universe is much weirder than we could have ever imagined and so we resort to only expression of absolute meaning - mathematics. I do have urge to elaborate why mathematics is so efficient painting reality, but I usually get accused of Platonic extremism and, having so little time, I'll refrain.

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Read The Road to Reality: a complete guide to the laws of the universe by Roger Penrose. It provides a handy companion for undergraduate / first-year graduate students of physics. | The first sixteen chapters provide - (in outline form) - all the mathematical material needed for an undergraduate major in (specifically theoretical-) physics - written by a foremost theoretical physicist, (i.e. it provides the "depth" that you wouldn't otherwise find in textbooks or other standardized reading materials).

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In brief, she should study it casually for pleasure and intensly as needed.

I find that many of the people that I know in scientific and technological fields who try to study everything get sidetracked and end up studying only obscure and relatively less useful topics. While it might provide for some interesting correlations, I prefer the physician's approach: "When you hear hoofbeats, look for horses, not zebras." The fact is that the bread and butter calculus, algebra, trigonometry, and geometry will get the average scientist a very long way. If you are going on to more advanced fields, differential and linear equations are also very helpful. Learn these fields well enough to use them on a regular basis and learn just enough about other branches of math to be able to spot their utility should the need arise.

PS - If you are looking for recommendations on books, my favorite is Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence.

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I will give a very general and brief answer to the question, How to study maths etc.

Skip the proofs but study the definitions carefully.

Now I will add a very general remark made to me by a wise woman once: she never learned anything by reading (a paper or a book) except when she was reading it in order to solve a problem she had. But I wouldn't want you to think this implies that you should never read something except when you have a problem in mind....

Relevant to the OP is, ¿what is the difference between studying maths the way a mathematician would and the way a physicist would? I will just give two classic quotations. Nicolas Bourbaki (and André Weil) repeated the often said proverb:

« Depuis les Grecs, qui dit mathématiques dit démonstration »

But Dirac told Harish-Chandra

« I am not interested in proofs but only in what Nature does

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Reading theorems and definitions without proofs is like reading the plaque without looking at the sculpture. –  Ron Maimon Jan 15 '12 at 0:04
    
Well, that is Bourbaki's point of view too. The art thief, though, should concentrate on the plaques. I si agree to the very small extent that I would say that us lesser mortals should not imitate Dirac: what worked for the man who was able to re-invent spinors and distributions on his own would not work for me. But he did not even read mathematics papers as far as I know...I suggest physicists at least read them or take the course or something. –  joseph f. johnson Jan 15 '12 at 2:52
    
you should either immitate Dirac, or not do physics. –  Ron Maimon Jan 15 '12 at 4:00

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