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For a theoretical physicist (not a mathematical physicist), is there a need to learn pure mathematics ?

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if you have an absolutely strong abstract imagination, like as they say Stephen Hawking have, then you probably don't need much higher math!! –  Vineet Menon Nov 9 '11 at 4:58
    
Yes - because pure math - is also interesting. It is - really! –  Adobe Nov 9 '11 at 12:07
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7 Answers 7

If you never learn anything besides what other physicists do, the only advantage you will have over them is being smarter or luckier, which means that you will have to be really smart or lucky to get a job/get tenure at a good university/win a Nobel prize.

However, if you learn some pure math that most physicists don't know, you might be able to apply it to physics somehow. This could help you get good results, which could help your career. If you enjoy learning pure mathematics, then by all means learn some. If you don't, then you probably don't need to, but you might want to consider studying a broader range of topics in physics.

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I will tell a story about the Nobel prize winning physicist, Murray Gell-Mann. He tells it himself, I forget where. As a grad student he took, purely out of intellectual curiosity, a course in the maths dept. where he was, a course in pure maths, representations of Lie groups, and in particular he learned pretty well SU(3) (since, after all, it is one of the ones easy to visualise, the Cartan weight diagram is two-dimensional). Later in life when graphing the properties of some of the elementary particles known at that time he saw these weight diagrams in them, except for one, where one weight was missing. So he hypothesised the existence of a new particle to fill in the missing weight...the omega minus particle, and called this arrangement "the eightfold way". If you never want to make a discovery like that, then go ahead, never learn any pure maths simply out of intellectural curiosity...

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Found this related BBC Horizon 1964 video: youtube.com/watch?v=BGeW6Nc6IMQ –  Qmechanic Jan 9 '12 at 20:38
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I will also give my own story: in 1959 I took a math course on sets and group theory, against the advice of my physics major adviser, because I found it interesting. Even though an experimental physicist, it sure came handy when the eightfold way came my way :).

My advise is to take maths courses that intrigue you, even though they are not a prerequisite for a current physics course.

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Yes. If you attempt to learn quantum mechanics, understanding of the Heisenberg picture requires a reasonable grasp of linear algebra, which will at least require the correct definition of a vector space and some facts about matrix diagonalisation. This is usually rated "pure math". Every $21^{st}$ century course on quantum mechanics I have ever heard of has a huge fail rate because the students are not taught a sufficient amount of linear algebra. You have been warned.

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Abstract algebra (Group and representation theory),Topology and functional analysis are very interesting and useful. More than learning a bunch of stuff, just as there is a "physics way" of thinking, there is also a 'math way" of thinking which is good to acquire. Believe me, the two are quite different!

Although you may find it difficult and superfluous at first, you develop an appreciation for the need for rigor. It is hard to put in words, but just stick with it and hopefully you will understand what I am trying to convey.

In summary, I would urge you to try to study pure Mathematics to expand your thinking.

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There is definitely plenty of math you need to learn that will not be in your required math coursework and will only be given a shaky basis if it is covered at all in the physics curriculum. But I wouldn't say its necessary to sit through lots of math classes or wade through math textbooks (especially since a lot of math is taught in a terrible fashion).

For example, I was never actually taught topology in a class, nor has any physics student I know. But the idea of a theoretical physics who doesn't know any topology is absurd. So you'll probably have patch these gaps yourself when the time comes.

Of course learning more math is never a bad idea, and if you enjoy learning theoretical physics you will probably enjoy learning pure math. Just do what comes naturally.

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If you are going to be a theoretical physicist, abstract algebra (groups theory) is an absolute must. –  Antillar Maximus Nov 8 '11 at 23:24
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For some things such as category theory, most theoretical physicists can do without (sorry John Baez). However, the less mathematics you understand, the harder it is to advance into many realms of physics. For example, it would be near impossible to study, say, canonical quantum gravity without knot theory. Even classical theoretical physicists need to be well versed in symplectic topology and jet manifolds.

However, this applies only to theoretical physics. I would also like to state that your education in mathematics need not be a technical. A lot of higher mathematics (like algebraic topology) is very intuitive and feels more like an art than a science so don't stress out. :)

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