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jdstokes Jun26-07 05:14 AM

Mathematical recommendations for the study of theoretical physics
 
I think it's fair to say that a lot (perhaps most) of basic research in theoretical physics these days takes place within the pure mathematics departments of universities. I suspect the reason for this is that the amount of maths necessary to tackle these subjects could fill an entire career.

If I'm going to understand this field at any substantial level it looks like I'm going to have to take a heck of a lot of maths. To put things into context, I'm third undergrad at an Australian university majoring in physics/pure maths. I do a bit of research into the the theory of complex colloidal plasmas with a couple of publications to my name. At the moment I'm planning on doing Honours (fourth year) followed by a PhD in the same field. I'd like to be able to understand current research in the more theoretical areas such as strings and loop quantum gravity. My current mathematical knowledge is OK but not excellent. I've done PDEs, vector calculus and group theory at second year level. This year (third year), I've so far done metric spaces (topology) and field theory. I will be doing differential geometry and Lagrangian and Hamiltonian dynamics next semester. I've done no pure analysis courses (unless you count metrics).

The following pure mathematics courses (in addition to advanced third year courses) are offered in fourth year

Functional analysis
Partial differential equations
Algebraic topology
Algebraic geometry
Commutative algebra
Representations of the symmetric group

http://www.maths.usyd.edu.au/u/UG/HM/pure2007.pdf

I understand that functional analysis is basically quantum mechanics and that alg topology has some applications to LQG. I also understand how useful group theory can be in physics as I've encountered it in my research on plasma physics.

Would anyone be able to rate these in terms of their usefulness for theoretical physics? Please keep in mind that by majoring in physics I'm limiting the amount of maths I can do to probably 2 of these.

Thanks in advance

James.

ZapperZ Jun26-07 07:33 AM

Quote:

Quote by jdstokes (Post 1364708)
I think it's fair to say that a lot (perhaps most) of basic research in theoretical physics these days takes place within the pure mathematics departments of universities.

This is incorrect. If this is true, then there will hardly be any theorists in the physics departments at all these universities, and most of them are in the mathematics department. So how many are there at your school?

There's a difference between learning about the tools and using the tools to as a means to do something else. Physicists do the latter. Are there mathematicians that also address physics issues? Sure! But there are also mathematicians that dabbled in the financial market too! It doesn't mean that MOST of the people who study the mathematics of finance are in the mathematics dept!

You do need mathematics, that is a given. But you need them for a PURPOSE, as a tool for address various physics problems. This does not make you a mathematician. Theorists in condensed matter physics, atomic/molecular physics, nuclear physics, high-energy physics, etc. certainly do not call themselves at mathematicians, and they certainly do not work in the mathematics dept.

Zz.

jdstokes Jun26-07 08:40 AM

Hi ZapperZ,

Thanks for your reply.

I was careful in my wording when I said BASIC theoretical physics, ie the study of the interactions on their most fundamental level. I personally do not include theoreticians who study, eg goldstone modes in condensed matter or plasma instabilities in tokamaks in this category. With that said, I don't think my statement is incorrect.

James

ZapperZ Jun26-07 08:58 AM

Quote:

Quote by jdstokes (Post 1364803)
Hi ZapperZ,

Thanks for your reply.

I was careful in my wording when I said BASIC theoretical physics, ie the study of the interactions on their most fundamental level. I personally do not include theoreticians who study, eg goldstone modes in condensed matter or plasma instabilities in tokamaks in this category. With that said, I don't think my statement is incorrect.

James

Could you show me examples of schools that actually have most of the people doing this in the mathematics dept.?

The University of Chicago here has a huge group that study "basic interactions on their most fundamental level". You're welcome to look at the physics dept's website. I know this because many of the theorists in my division (high energy physics) also have joint appointments there, and many of UofC people also come here very often. They are ALL in the physics dept., not mathematics.

So I've given you one example that counters your statement. I don't think UofC is an exception, at least no here in the US. People like Steven Weinberg, Frank Wilczek, etc. (which I would assume you would consider as people fitting int your "study basic interactions" categories) are all in physics dept., not mathematics.

As an off-topic issue: I do not consider condensed matter physics as not "fundamental". Many arguments have been presented especially by Laughlin that the emergent phenonomena studied on condensed matter physics have indications that they are as fundamental as anything studied in particle/high energy physics. The existence of fractional charges clearly points to such arguments. Furthermore, the Higgs mechanism came directly out of condensed matter physics, and so did the principle of broken gauge symmetry that came out of Phil Anderson's work. These are as fundamental as anything you can point to in other fields of physics.

Zz.

jdstokes Jun26-07 09:51 AM

I think we're really discussing semantics more than anything. Certainly as you delve into the more mathematical problems facing theoretical physics at the moment you will encounter more people working directly in maths departments than physics. It's just a question of where you draw the line between what counts as physics rather than pure mathematics.

The phenomenon I'm describing is more prominant in schools where there is a defficiency of people studying pure theory in the sense that I'm talking about. I could probably fit my own school under that category (sydney university), where the only person (that I know of) working in supersymmetry theory is located in pure maths.

ZapperZ Jun26-07 09:59 AM

Quote:

Quote by jdstokes (Post 1364843)
I think we're really discussing semantics more than anything. Certainly as you delve into the more mathematical problems facing theoretical physics at the moment you will encounter more people working directly in maths departments than physics. It's just a question of where you draw the line between what counts as physics rather than pure mathematics.

The phenomenon I'm describing is more prominant in schools where there is a defficiency of people studying pure theory in the sense that I'm talking about. I could probably fit my own school under that category (sydney university), where the only person (that I know of) working in supersymmetry theory is located in pure maths.

So you count only those working in field theories, string, and supersymmetry as qualifying under "fundamental" work? Would people like Lisa Randall, Sundram, and Arkani-Hamed count? Randall is in the Physics Dept. at Harvard. In fact, a lot of theorists in that area at Harvard, Stony Brook, etc.. tend to be in the physics dept.

I would say that your school is more of the exception than the rule. That is why I disagree with your categorization that MOST of the people working in such area (which I disagree with as being the only one that would be considered as "fundamental") are in the mathematics dept. It certainly isn't true from my observation.

Zz.

jtbell Jun26-07 12:02 PM

I wouldn't be surprised if this varies from one country to another, depending on the different traditions of their university systems

Stingray Jun26-07 12:29 PM

It certainly does vary between countries. I know that in my field (relativity), almost everyone in the UK is in math departments. Very similar people in the US have jobs in physics departments. In either case, I doubt anyone would consider them mathematicians. It's just tradition.

SeReNiTy Jun27-07 12:02 AM

When i was at Monash University in Australia, all the cutting edge theory was done in the mathematics department.

At UCB its different however...

jdstokes Jun27-07 04:59 AM

Thanks everyone for their comments.

I would just like to ask, does anyone have any familiarity with the material taught in the classes I listed and would be able to provide their advice of the relevence to physics?

To ZapperZ,

It sounds like I might be describing an Australian pheomenon which led me to form those views.


Thanks again.

Locrian Jun27-07 09:25 AM

Quote:

Quote by jdstokes (Post 1364803)
I was careful in my wording when I said BASIC theoretical physics, ie the study of the interactions on their most fundamental level. I personally do not include theoreticians who study, eg goldstone modes in condensed matter or plasma instabilities in tokamaks in this category.

The above definition could definitely cause a lot of confusion. The idea that studying condensed matter is less basic (or not a part of) studying "interactions on their most fundamental level" is entirely subjective and impossible to defend rigorously.


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