Ideas for Physics Thesis: 2 Year Project

[Jimmy84]

Im majoring in physics, and I would highly appreciate some ideas and some orientation about what I'm getting into.

Im beginning the third year of the major and I need ideas for a thesis. I consider that the thesis could take around 2 years.

I have been teaching myself the main subjects about the math major. Analysis, Abstract Algebra, Topology and some Functional Analysis. I am still in the process of studying these subjects.
Eventually I would like to study some Differential Geometry, some Algebraic Topology, and Quantum Mechanics as well.

The fields that I'm interested in per suing, "not now but in the future", are Topological quantum field theory and Gauge Theory I'm also interested in Category theory.

I would kindly appreciate any suggestion or idea about any possible undergraduate problems in those areas or in related areas.

Your comments and suggestions are very valuable to me.

Thanks.

 

[Jorriss]

Ultimately an undergraduate research project will require you to work with a faculty member in your department or a related department.

I would recommend looking into people have research you might interested in and contacting them.

 

[micromass]

You should really talk to a few of your professors. You'll have to cooperate with them eventually.
Asking for a topic to strangers on the internet is a weird thing to do.

 

[Jimmy84]

You should really talk to a few of your professors. You'll have to cooperate with them eventually.
Asking for a topic to strangers on the internet is a weird thing to do.

I understand. I just feel that I must be oriented beforehand about the subjects that I would like to study in the future and how to get there.

I would also like to propose a subject first, in order to have more freedom to per sue the direction that I would like to take.

 

[Jorriss]

Ok. What subjects have you liked so far?

Why are you interested in topological QFT if you haven't studied Quantum mechanics?

 

[Jimmy84]

Ok. What subjects have you liked so far?

My interest is mainly in mathematics, and in mathematical physics.

About mathematics, there are many subjects that I personally enjoy, but I can't follow them all at the same rate, and that is why I would like to select early some area to focus on.

I planned to teach myself some math major subjects such as abstract algebra, topology, differential geometry and functional analysis

I personally enjoy abstract algebra, and I wish to be able to learn later subjects like Algebraic Geometry, Homological Algebra, Categories, Algebraic Topology, Knot Theory, because I find these subjects interesting, and I would like to learn about them only because of personal interest.

Why are you interested in topological QFT if you haven't studied Quantum mechanics?

Topological QFT as far as I know, has to do with many mathematical subjects that I'm interested about, and I find axiomatic approaches to QFT intriguing. Furthermore I am interested in the possibility of maybe doing research in quantum computing. There are some papers that point to some applications of TQFT in quantum computing, though there is few research being done relating TQFT and quantum computing as far as I know, perhaps because TQFT is not a mainstream subject.

The subject I'm envisioning for the thesis does not has to do directly with TQFT but for a future access to that area. I am also considering subjects as quantum mechanics or classical mechanics for the thesis, but I would love doing something that would give me some access to per sue QFT or Mathematical physics.

I would like to have an idea for instance, whether there are plenty of accessible problems in subjects such as manifolds, bundles, lie groups, gauge transformations, general/algebraic topology, abstract algebra in undergraduates physics , or whether problems relating to those subjects are scarce and unaccessible to undergraduates.

Thanks.

 

[espen180]

Topological QFT as far as I know, has to do with many mathematical subjects that I'm interested about, and I find axiomatic approaches to QFT intriguing. Furthermore I am interested in the possibility of maybe doing research in quantum computing. There are some papers that point to some applications of TQFT in quantum computing, though there is few research being done relating TQFT and quantum computing as far as I know, perhaps because TQFT is not a mainstream subject.

The subject I'm envisioning for the thesis does not has to do directly with TQFT but for a future access to that area. I am also considering subjects as quantum mechanics or classical mechanics for the thesis, but I would love doing something that would give me some access to per sue QFT or Mathematical physics.

I would like to have an idea for instance, whether there are plenty of accessible problems in subjects such as manifolds, bundles, lie groups, gauge transformations, general/algebraic topology, abstract algebra in undergraduates physics , or whether problems relating to those subjects are scarce and unaccessible to undergraduates.

I think it would be possible for you to learn about TQFT for a thesis, but not to do research.
If you work hard and learn the neccesary manifold theory and differential geometry, it might be realistic for you to learn, for example, about Chern-Simmons theory, which is a Schwarz type TQFT with a rich supply of mathematics. However, I wouldn't call this undergraduate physics...

There is a wonderful little book by Kock: "Frobenius algebras and 2D Topological Quantum Field Theories", which completely characterises the 2D case. Maybe you should read this first to see if you like it. It you manage to get through it, there are papers by people like John Baez and Jacob Lurie about higher-dimensional TQFTs.

Since you are interested in this kind of thing, you must have seen the definition of an n-dimensional TQFT as a functor $\mathrm{Bord}_n^S\rightarrow \mathrm{Hilb}$ satisfying some axioms, so you realize that cobordism theory is a big part of TQFTs, so understanding $\mathrm{Bord}_n^S$ should be your goal after reading Kock's book. The neccesary prerequisite material for this is contained in the following books:
- "Differential Topology" - Hirsch
- "Morse Theory" - Milnor
- "From Calculus to Cohomology" by Madsen and Tornehave
- "A concise course in Algebraic Topology" by Peter May

There is a ton of math to get through to study this, but you can do it if you work hard. Once you learn some category theory, the nLab is a very valuable source of information about these things as well. http://ncatlab.org/nlab/show/HomePage

 

[yenchin]

You may try http://arxiv.org/abs/0810.0344.