Maths required for studying intermediate to advance physics

[henry407]

Hope someone can help me to list the important math topics required to study for a person interested in studying condense matter, solid state physics, intermediate quantum mechanics and particle physics. (I can't get a
Personally, i prefer to study math tool before studying the physics content. So i hope I can get a touch on those math topics related to those fields in physics as early as possible.
Thanks for your help!

 

[Arsenic&Lace]

-Differential, integral, and vector calculus
-Linear algebra (I've met many physicists who are terrible at this subject, don't be one of those!)
-Ordinary and Partial differential equations
-Elementary complex analysis

Group theory, topology, and differential geometry occur in small doses in some or all of these fields, but normally the doses are small enough that you needn't bother taking a course or reading a pure math textbook on these topics.

 

[ZombieFeynman]

I think, if you would like to do theory, by the time you are done with undergrad, your mathematics knowledge should be at the level of Lea, Boas or Byron and Fuller. I would supplement the traditional courses required for physics from mathematics (Calculus, Differential Equations and Linear Algebra) with [as A&L states above] complex variables, PDE, and numerical methods (particularly ones dealing with solving diff eq's and matrix methods...no one does that stuff by hand any more!). I strongly suggest taking a course which covers group theory, as formal knowledge of this subject will help GREATLY in condensed matter physics. The use of character tables and representation theory is extensive in many areas of CM. Lastly, I would take things which interest you.

 

[henry407]

Thanks Zombie and Arsenic. I'd like to ask do you got books recommendation in each field stated above?
(espeically Linear algebra, Elementary complex analysis, Group theory)
[Personally, I have read Linear Algebra:A modern introduction and Metric Space (springer publisher). Currently, I am reading Partial Differential Equations with Fourier Series and Boundary Value Problems; but I still think I didn't learn enough for Linear Algebra and Metric Space]
I'd like to read some formal introduction to each topics instead of a summary book (e.g. Boas). Or it is not necessary to cover comprehensively (too mathematically) on each topics in order to become a good theoretical physicist?
Moreover, I found that problems from those pure mathematics book is extremely difficult that I didn't even attempt to work it out. (e.g. The book about Metric Space) So, I'd like to ask, if I treat those math as a tool for physics, is it necessary to work out those problems in the pure math textbooks?

 

[Arsenic&Lace]

Not only is it not necessary, it might even be harmful. Mathematicians have a culturally different point of view with regards to mathematics from physicists; the attention to detail and rigor is not something which is characteristic of most theoretical physics. The end result is that, as an example, if you develop a mathematician's intuiton for series from a real analysis text, you'll need to unlearn it when you start thinking about series in, say, the context of quantum field theory.

As far as most return on investment is concerned, solving physics problems will get you further along then solving mathematics problems.

All this said, if you enjoy that stuff and find it interesting, it's not THAT dangerous to you. As long as you are aware that mathematicians approach mathematics in a very different way from physicists and you separate their perspectives, you should be fine. Some on these boards will argue that the attention to detail and clarity of thinking provided might be beneficial.

 

[micromass]

Thanks Zombie and Arsenic. I'd like to ask do you got books recommendation in each field stated above?
(espeically Linear algebra, Elementary complex analysis, Group theory)
[Personally, I have read Linear Algebra:A modern introduction and Metric Space (springer publisher). Currently, I am reading Partial Differential Equations with Fourier Series and Boundary Value Problems; but I still think I didn't learn enough for Linear Algebra and Metric Space]
I'd like to read some formal introduction to each topics instead of a summary book (e.g. Boas). Or it is not necessary to cover comprehensively (too mathematically) on each topics in order to become a good theoretical physicist?
Moreover, I found that problems from those pure mathematics book is extremely difficult that I didn't even attempt to work it out. (e.g. The book about Metric Space) So, I'd like to ask, if I treat those math as a tool for physics, is it necessary to work out those problems in the pure math textbooks?

Reading pure math books is not strictly necessary but can be very helpful. Certainly, if you don't know the rigorous theory behind linear algebra, you can still be a very good quantum physicists (for example). But I have found that the rigorous theory gives you a conceptual change in perspective that can be very useful. For example, somebody knowing rigorous differential geometry will approach general relativity from a different point of view, and will have a more founded knowledge of the material and might also find some concepts easier to grasp.

Also, mathematics is philosophically and aesthetically pleasing to a very high degree. But again: only do it if you really like it.

As for books, I like the free book "Linear Algebra done wrong": http://www.math.brown.edu/~treil/papers/LADW/LADW.html Many sections in there are quite relevant to physics.

As for group theory. Pure math books in group theory will sadly focus mostly on the finite groups and those are useful in physics, but not as useful as the infinite matrix groups. Furthermore, it will usually not focus on representation theory (at least a first course won't). A good book on the topic is Hall: https://www.amazon.com/dp/0387401229/?tag=pfamazon01-20
The bad thing about this book is that it requires familiarity with elementary group theory already, but I think you can learn that fairly quickly. Another nice book is https://www.amazon.com/dp/9971966573/?tag=pfamazon01-20 But I have found the exercises weird.

For complex analysis, the book by Flanigan should be enough: https://www.amazon.com/dp/0486613887/?tag=pfamazon01-20 The visual complex analysis book is also a good read for intuition: https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20

Finally, for PDE's, there is probably no better book than Strauss: https://www.amazon.com/dp/0470054565/?tag=pfamazon01-20

 

[henry407]

Thanks once again to Arsenic and micromass! Finally, I want to ask how much maths normally will the university teach and cover during undergraduate physics syllabus (Take the theoretical physics stream + honor stream)? I want to use the reference to check my progress.

 

[Arsenic&Lace]

Depends on the university. Mine had a two semester mathematical physics sequence, whose only real drawback was the linear algebra, which was too weak in my opinion. Some additional mathematics was covered as needed in upper division courses but not much (tensors spring to mind).

"theoretical physics" is an extremely broad term. Some sub-disciplines seem to use quite a bit of sophisticated mathematics you would not learn in a typical undergraduate sequence but might learn in the math department, for instance if you work on string theory. Others use scarcely any more mathematics than is taught at the undergraduate level, such as non-equilibrium statistical mechanics*.

*Perhaps there is a counter-example out there where somebody used differential topology to study non-equilibrium statistical mechanics, but my grasp of that field, as an example, is that the majority of the methods rarely get outside the realm of 19th century functional calculus. At any rate it's quite tame next to say, string theory.

 

[George Jones]

Finally, I want to ask how much maths normally will the university teach and cover during undergraduate physics syllabus (Take the theoretical physics stream + honor stream)? I want to use the reference to check my progress.

Depends on the university.

And the vintage.

The math requirements (all taught by the math department) for my honours physics B.Sc. were as follows

First-year: calculus 1 (also taken by math majors); calculus 2 (also taken by math majors).

Second-year: calculus 3 (also taken by math majors), ordinary differential equations (also taken by math majors); a two-semester linear algebra course (also taken by math majors) that had a closed book, no cheat-sheet, cumulative final exam.

Third-year: complex analysis (also taken by math majors); special functions and partial differential equations.

Fourth-Year: a two-semester applied math course that treated some material from previous courses at a more advanced level, that introduced new material (e.g., Green's functions and distributions), and that had a very difficult open-book cumulative exam. For the exam were allowed to bring in any texts and notes that we wanted, and we were allowed to write for as long as we wanted. Most of us wrote for six hours. After six hours, the prof took the students to the pub and bought the first round.

Those were the math requirements; I took more math for the fun of it. A few years after I went through, the required math content of the physics B.Sc. was relaxed somewhat.

 

[Arsenic&Lace]

An important point raised by Jones is that university is more flexible. What I described from my program are the minimum requirements, which are much feebler than his program's minimum requirements; I personally have taken a course in intro proof writing, linear algebra, two semesters of abstract algebra, advanced calculus, topology and more to fill out my degree. If you are judicious about what courses you take, you can obtain a better education.

As far as bang for your buck is concerned, courses in numerical mathematics/computer programming (particularly applied courses where you will actually program) are the most value (in my humble opinion) non-physics courses to take.

 

[Jorriss]

-Differential, integral, and vector calculus
-Linear algebra (I've met many physicists who are terrible at this subject, don't be one of those!)
-Ordinary and Partial differential equations
-Elementary complex analysis

Pretty much this and the amount you actually need to know about partial differential equations and complex analysis (to start intermediate level physics topics) is pretty small too.

 

[porcupine137]

Moreover, I found that problems from those pure mathematics book is extremely difficult that I didn't even attempt to work it out. (e.g. The book about Metric Space) So, I'd like to ask, if I treat those math as a tool for physics, is it necessary to work out those problems in the pure math textbooks?

You don't need to be able to carry out the detailed mathematical proofs and manage the HW of the type you'd see in Real Analysis classes (although some physics guys actually do like that and find that interesting as well).

One interesting book is Visual Complex Analysis (came across it on my own).

 

[ZombieFeynman]

One interesting book is Visual Complex Analysis (came across it on my own).

I like this book a lot, but would strongly suggest not using it as a first book. Instead I recommend a book like Fisher or Churchill.

 

[porcupine137]

I like this book a lot, but would strongly suggest not using it as a first book. Instead I recommend a book like Fisher or Churchill.

Yeah, I was thinking of it more as a supplement.

 

[porcupine137]

It varies so much school to school and person to person. The minimum requirements are often pretty low, perhaps just two course beyond Calc III.

But as others have said you'd want, beyond Calc III, for sure a course in linear algebra and one in differential equations. It certainly wouldn't hurt to take more (although they are often taught as pure math, even the linear and diff eq might be to some extent and even calc if you take the honors sections). It might not hurt to take a semester of abstract algebra and to take complex analysis and methods of mathematical physics and computational physics and various such courses as the latter two. You might want to study Lie Groups and such yourself.

Not that it matters and is necessarily the ideal for someone focusing purely on physics but various books used in the classes I took:
Apostol - HCalc I/II (one semester combined class, I forget, we probably didn't cover every last bit at the very end, I'd have to check to refresh my memory)
Marsden/Tromba - HCalc III
Linear Algebra - can't recall and it's not right here at the moment to check
Boyce - Elementary Differential Equations
Osborne - Complex Variables
Abstract Algebra I - again don't recall the title and it's not right here at the moment, not that it really matters and again many of these books and classes were more strictly serious math major oriented. It definitely wasn't aimed at physics majors. That said it's not the worst thing to get a general math major perspective at the beggings of group theory and such before going off and picking up more directly what you need, not required though by any means in this form though .
Marsden - Elementary Classical Analysis I (don't be mislead by the title ;) it's probably the trickiest material out of all the books mentioned)To a certain extent you might get farther, faster in physics spending more time going over and getting intro and intermediate level physics down inside out by doing problem after problem on your own after going through them all first and studying various bits of math from a more physics oriented viewpoint than a ton of pure math classes in various subjects (the mathy classes can be fun and can give you certain extra insights than some in physics miss, but OTOH, taking tons of those while others are cramming in more physics classes and self-direct studied of bits of math on your own just for physics and practicing problems a whole ton might put you behind in some ways and make it easier for the others to score super, super well on the physics GRE- speaking from experience; classes like Real Analysis can take up a ton of time).

 

[Arsenic&Lace]

Review papers on the mathematics relevant to theoretical disciplines in which you are interested can be a very inexpensive (time wise) way to learn what you need. Mermin for instance wrote this review of the topology which is relevant to condensed matter physicists:
http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.591

Attempting to filter out the bits of pure mathematics which might actually be helpful is often challenging, given their foreign cultural point of view. A typical pure mathematics course is built with fundamentally different assumptions about how the subject should be approached and as such will spend time answering questions which are likely of little interest to you. I recall in my first semester of abstract algebra a discussion of the Sylow theorems, which are not difficult to understand, but whose purpose is to aid in the objective of classifying finite groups, which I a). don't really care about and b) is not relevant to anything I might do with groups, and topics such as these will abound in a typical pure math class. If you are curious about such questions (and I admit that sometimes I am) you might enjoy it, in which case the fact that it isn't relevant to you isn't really a problem. If you aren't, there are probably good books which bridge the cultural divide you ought to read instead.

 

[homeomorphic]

The end result is that, as an example, if you develop a mathematician's intuiton for series from a real analysis text, you'll need to unlearn it when you start thinking about series in, say, the context of quantum field theory.

I don't think you really have to actively spend any time "unlearning." Just disregard. No need to unlearn. The only case in which it could hurt is if you are one of the people who are compulsive about rigor and can't compromise.

 

[micromass]

I recall in my first semester of abstract algebra a discussion of the Sylow theorems, which are not difficult to understand, but whose purpose is to aid in the objective of classifying finite groups, which I a). don't really care about and b) is not relevant to anything I might do with groups, and topics such as these will abound in a typical pure math class.

You really have a habit to pick out the worst examples possible. Sure, the Sylow theorems are not that useful for physics (or even for all of mathematics outside of group theory). But how much do the Sylow theorems really occur in a first course of abstract algebra? Not much, I'd say. The vast majority of the course is about definitions, theorems and techniques which are very much useful for physics and chemistry.

 

[Arsenic&Lace]

Fair enough homeomorphic, the tone I took was a bit dramatic. I still think the distinction between "series to a physicist" and "series to a mathematician" should be drawn sharply.

Micromass, I revisited the professor's homepage to look at the syllabus and I guess the department changed its rules. The course I took is now a topics course called group theory, since the course I took was devoted entirely group theory (there was not one mention of rings), and it spent quite a bit of time on topics such as the Sylow theorems, free groups, the isomorphism theorems, permutation groups, factor groups, that big abelian group theorem whose name I can't recall and probably other things (it was ages ago). So it is likely less that I pick the worst examples and rather that the course I took, while called introductory abstract algebra, was really a topics course on group theory, which resulted an unusual point of view regarding abstract algebra courses.

 

[homeomorphic]

Fair enough homeomorphic, the tone I took was a bit dramatic. I still think the distinction between "series to a physicist" and "series to a mathematician" should be drawn sharply.

Only if you are talking about the final result. There's no law at all that mathematicians can't do this or that until they are writing an actual proof.

 

[homeomorphic]

Also, apparently, some mathematicians DO work with non-convergent series. That's not my area of expertise, but maybe analytically continuing to get the Riemann zeta function would be an example. So, even in math, it's misleading to think that convergent series are the only ones that you can do anything with.

 

[Arsenic&Lace]

Only if you are talking about the final result. There's no law at all that mathematicians can't do this or that until they are writing an actual proof.

Sure, mathematicians aren't enslaved by physicists and engineers, the point I'm making is that exporting the style of thinking developed in a pure math course to work in theoretical physics seems like a bad idea to me. But it should be pointed out that there probably isn't anybody in this discussion who is qualified to verify or denounce such a viewpoint; there are sub-fields in theory which seem to be dominated by a mathematical mindset (e.g. string theory), and famous figures who expressed skepticism about such a mindset (see: Feynman lectures, I'll quote the proper page when I find my old copy), but such a debate, *eh hem*, is unlikely to go anywhere.

 

[homeomorphic]

Sure, mathematicians aren't enslaved by physicists and engineers, the point I'm making is that exporting the style of thinking developed in a pure math course to work in theoretical physics seems like a bad idea to me.

My point is that you can decide to export or not export as you see fit. Then, you can reap any rewards and not pay any costs. Maybe that's easier said than done, though.

 

[Arsenic&Lace]

My point is that you can decide to export or not export as you see fit. Then, you can reap any rewards and not pay any costs. Maybe that's easier said than done, though.

I will humbly opine then that I think it is easier said than done, since the influence can be subtle. But that is for another thread, if at all, since it is not a debate for which a conclusive argument one way or another is fathomable (to me anyway).

 

[porcupine137]

I don't think you really have to actively spend any time "unlearning." Just disregard. No need to unlearn. The only case in which it could hurt is if you are one of the people who are compulsive about rigor and can't compromise.

Yeah I really can't say I had to unlearn anything at all from any of my more pure math classes. As you say, so long as you don't get bogged down and OCD that every last little step in physics has to be put in the strictest terms and get lost in the math and forget to see the physics I don't see an issue at all. Sure, plenty here and there may be of relatively help much of the time, but I certainly don't think it hurts (other than taking time away from physics).

 

[henry407]

Thanks for all your guys reply. Actually, I am now more confused after the last bit of the discussion... But, indeed, this is a question without a conclusive answer. So, I'd like to ask the following: sometimes I found the proof in topic of metric space is redundant and unnecessary; the result itself is intuitive and "obvious"( especially those about A map to B and bahbahbah property remains the same). In this case, is this a "good" habit for physicist to ignore the "minor" details of the vigorous proof or a harmful habit which will cause me losing some important knowledge for my future career? Thanks for everyone's contribution to this post !

 

[WannabeNewton]

Most of the time results are not intuitive. So you can either take things for granted and just focus on the physics or take a detour to understand the result more rigorously. It is up to you really. It is a matter of time vs. satisfaction. It doesn't hurt to look at things more rigorously from time to time but if you focus on it too much then you won't get anywhere. It also depends heavily on the subject at hand. If youre learning GR then it doesn't take much time to verify results rigorously because the necessary math is very easy at the learning stage. But if youre learning QFT then you would basically have no choice but to accept all the hand-waving and rug sweeping.

Coming back to the topic at hand, most of the necessary math for a basic foundation in physics you will learn naturally as an undergrad. The only extra stuff which will be absolutely necessary in GR, QFT, EM etc. that you might not be forced to learn as an undergrad and which are usually not taught well in the classes themselves are methods of contour integration and residues, Green's functions, and the physicsts' level of representation theory of Lie groups. If you get that stuff down you'll be just fine. The physics is infinitely more fun than the math so just focus on that and learn what you need as you go along.

 

[porcupine137]

Coming back to the topic at hand, most of the necessary math for a basic foundation in physics you will learn naturally as an undergrad. The only extra stuff which will be absolutely necessary in GR, QFT, EM etc. that you might not be forced to learn as an undergrad and which are usually not taught well in the classes themselves are methods of contour integration and residues, Green's functions,

Sometimes the math department Complex Variable classes do OK by contour integration, residues, analytic continuation, conformal transformations.

 

[porcupine137]

One thing I should also add is that in the more advanced topics of physics like field theory and such that a lot of the stuff, I believe has actually never been rigorously proven yet by anyone in pure math terms so if you try to build it up in pure math terms at a certain level you may find yourself VERY bogged down . If you tried to do that you'd have found yourself left decades behind the guys making all the radical advancements in physics.

 

[WannabeNewton]

Sometimes the math department Complex Variable classes do OK by contour integration, residues, analytic continuation, conformal transformations.

Hi porcupine! I should have been clearer. I meant that said topics are not taught properly in the physics courses in which they predominantly appear.