What advanced math courses would you recommend for grad physics?

[pantheid]

Hi guys, I'm currently an undergraduate, but I'm going for a simultaneous Bachelors/Masters in Physics. I've already taken advanced math courses in the following subjects: Combinatorics, Real Analysis, Partial Differential Equations, and Numerical Analysis, and I have also studied some Gamma functions on my own a lot. I intend to go into theoretical physics, specifically particle dynamics, and I'm wondering what further advanced math courses I should take. For example, is it worthwhile for me to take a year of Graduate Linear Algebra or Group Theory to better understand QM?

 

[Poley]

In addition to linear algebra and group theory, also consider complex analysis and differential geometry.

 

[retro10x]

I think complex analysis should be on the top of your list, and group theory would probably be right behind it

 

[jesse73]

Linear Algebra should be top of list (Heisenberg formulation)

 

[Vanadium 50]

This is a good example of why it is a good idea to look at the background of people who respond.

I have a PhD in physics. I have never taken a course in differential geometry nor group theory. I did just fine. If you were planning on working in GR, maybe a course in differential geometry would be helpful. For particle physics, all the group theory was worked out decades ago, and the amount of material needed is much less than one course.

What you really need is not so much a lot of material, but a high degree of facility with the basics. It will help if you can, e.g. deal with parabolic coordinates without thinking too hard about it, so you can focus on the physics of the Stark Effect and not where you dropped a minus sign.

 

[homeomorphic]

I have a PhD in physics. I have never taken a course in differential geometry nor group theory. I did just fine.

Doing just fine is hardly the point. Some people have different learning styles and preferences. If you have a more mathematical mindset, a couple more math classes might help such a person. If you check my background, I have a PhD in math, but from talking to physicists, there are certain very small parts of their field that I seem to have a better grasp of than at least the typical physics grad student, and not just from a more rigorous point of view, but I actually have more intuition than they do about those things. For example, Clifford algebras, which come up in dealing with fermions. Another thing that physicists might profit from that they don't seem to use too much is differential forms. It's not necessarily stuff you would cover in a standard course, though. Sometimes, differential forms are covered in differential geometry. But often, they are so badly taught that it defeats the purpose (for a proper treatment, one would need to turn to Arnold's book or A Geometric Approach to Differential forms). For Clifford algebras, too, if you learn it from, say, 90% of the books out there, there wouldn't be much point because they explain it so badly (decent treatments may be found in Penrose's book, The Road to Reality, for example). Even with the good books, you still have to think pretty hard for yourself about it to get what I consider to be a reasonable level of understanding.

It's easy for a physicist to think my approach is over-kill, but for me, it is certainly not. It's the way I have to do it, and if I have to do another way, I'm not interested. If you are like me, my brain just rejects certain things, until it can get more intuition. For example, the Dirac equation when I first saw it. I went and thought really hard about Clifford algebras and read lots of stuff and my brain no longer rejects it. Then again, if you are like me, you might not do very well in academia, so take it with a grain of salt. I'm now looking for work in industry, I think largely because of my insane compulsion to think too hard about stuff that is explained badly, just because I am offended by it, even if it isn't particularly relevant to my research.

You don't really need to take classes, though. You can study on your own because then you don't have to cover a whole course worth of stuff. The approach I would recommend is looking into more math on an as-needed basis, not necessarily dismissing it, but not spreading yourself too thin.

 

[jesse73]

This is a good example of why it is a good idea to look at the background of people who respond.

I have a PhD in physics. I have never taken a course in differential geometry nor group theory. I did just fine. If you were planning on working in GR, maybe a course in differential geometry would be helpful. For particle physics, all the group theory was worked out decades ago, and the amount of material needed is much less than one course.

What you really need is not so much a lot of material, but a high degree of facility with the basics. It will help if you can, e.g. deal with parabolic coordinates without thinking too hard about it, so you can focus on the physics of the Stark Effect and not where you dropped a minus sign.

Like the many posts you get in this section of people who supposedly self study and understand "advanced maths" like differential geometry and topology but have trouble with intro physics textbooks.

This forum wears you down after hearing so many HS students tell other HS students they need a PhD in math to start learning physics (only slight hyperbole). The other common case being that one understands math too well to understand physics.

 

[jesse73]

If people made a suggestion that went along the lines of "grad school QM commonly has textbook X which assumes an ease with math Y" it would seem more useful.

Example:

Grad school QM courses use Sakurai's book on QM which assumes an ease with linear algebra in its exposition and exercises therefore you should take that class if you have not.

It provides advice grounded on experience that touches on the specific coursework for that class.

 

[homeomorphic]

The other common case being that one understands math too well to understand physics.

I don't think there really is such a thing, except that if you spend too much time on math, you're not going to have enough time for physics. That's really the only issue, I think--it's just hard to find the time to learn both. I don't think it's the case that one is going to actively obstruct your understanding of the other. They just deal with somewhat independent subject matter, but occasionally, they do work together if you know where to look.

Some people also might spend too much time being rigorous, but that could also be viewed as an end in itself. To most physicists, making things rigorous does not seem terribly relevant, but it's a potentially interesting mathematical problem for its own sake.

I suppose in my own case, you could say I understand physics too well to do math, and math too well to do physics, if you just look at it purely as a time-management problem. That's why I intend my work (as a hobby) to be only expository.

It is a perfectly viable path to only take the minimum amount of math required--if it was really necessary, they'd just make it a requirement, but that's not necessarily the right thing for each person. They have to find their own path. Telling people not to study math, just because they are a physicist might be like telling them not to study music, even if there were no relevance to physics. It just isn't going to hurt, as long as you don't spend so much time on it that it becomes a distraction. So, I think the advice is just not to get carried away with it, rather than to avoid advanced math, if you have an interest in it.

 

[jesse73]

I don't think it exists either but it is still an excuse I have heard here before in some variant .

 

[jgens]

This forum wears you down after hearing so many HS students tell other HS students they need a PhD in math to start learning physics (only slight hyperbole).

No need to be melodramatic. All the of the mathematics topics here are pretty standard in undergraduate math curricula.

If people made a suggestion that went along the lines of "grad school QM commonly has textbook X which assumes an ease with math Y" it would seem more useful.

I agree because this allows the OP to figure out which suggestions are most suitable.

 

[jesse73]

No need to be melodramatic. All the of the mathematics topics here are pretty standard in undergraduate math curricula.

The point is you don't need to double major in math to be a physics major or grad student. The majority of advice is misguided by assuming you can take the relevant math course for every single topic that will come up in your physics course or misguided in believing other physicists have done the same.

As a physics student especially as an undergraduate you don't know exactly which field you will end up in and therefore don't know which math classes will be useful and at what depth topics will be useful. You could spend your time learning differential geometry and group theory but then end up doing stat phys based research where stochastic calculus and other completely different set of math is useful. You will inevitably have to learn the math on the fly through your physics instructor or own your own.

 

[WannabeNewton]

Grad school QM courses use Sakurai's book on QM which assumes an ease with linear algebra in its exposition and exercises therefore you should take that class if you have not.

That's a terrible example. The mathematics in Sakurai is a joke- I found it hard not to burn the book after the first chapter. You can't compare that to a graduate class in linear algebra.

Anyways, OP if you are truly interested in pure math then take group theory, differential geometry etc. but unless you go into mathematical physics those classes will be of little to no avail. Take a look through "Advanced Linear Algebra"-Roman and compare it to a decently mathematically rigorous QM book like Ballentine and see how much material actually overlaps (hint: very little). If pushed, you could take functional analysis (which is what QM is built on) if you ever find yourself frustrated at the hand-wavy nature of various QM books when it comes to mathematics (seriously how hard is it to explain the difference between bounded and unbounded operators?) but again if you're a pragmatist then this won't be an optimal choice. OTOH if you ever end up wanting to study the theory of gauge fields, a working knowledge of algebraic and differential topology would be very useful but one could argue this borders on mathematical physics anyways (apart from the classics like Kobayashi/Nomizu and Spivak, you could check out Bredon "Topology and Geometry" if you want a good pure math treatment of algebraic and differential topology).

But if your main interest is physics then just learn the math on the fly; it's easier than you might think-the physics is the hard part. You can't take a pure math course for every math topic that disseminates throughout a physical theory, not unless you're Von Neumann or Poincare.

 

[jesse73]

That's a terrible example. The mathematics in Sakurai is a joke- I found it hard not to burn the book after the first chapter. You can't compare that to a graduate class in linear algebra.

Which is great because I wasnt comparing Sakurai/GradQM to a graduate course in linear algebra but rather a graduate course in quantum mechanics (QM).

Well I seem to have got the impression that OP was interested in knowing the math for grad QM because the OP mentioned explicitly QM in the context of grad physics which although he mentions larger ambitions. He should focus on the task closer with respect to time. As long as he goes to grad school he will have to take grad QM but he might not necessarily follow through and stay in high energy.

 

[WannabeNewton]

Which is great because I wasnt comparing Sakurai/GradQM to a graduate course in linear algebra but rather a graduate course in quantum mechanics (QM).

Ah! My apologies. I totally agree then.

Well I seem to have got the impression that OP was interested in knowing the math for grad QM because the OP mentioned explicitly QM in the context of grad physics which although he mentions larger ambitions he should focus on the task closer with respect to time.

Definitely. To reiterate, the physics is the hard part- the math can be learned contextually without as much effort. Like my QM professor always says, leave the boring stuff to the mathematicians