Differential Geometry in Physics

[trustinlust]

Hi guys,

what are the fields of theoretical physics (if any) -besides General Relativity, String Theory, Quantum Gravity...- where Differential Geometry and Tensorial Calculus currently find strong application?

Thanx

 

[Andy Resnick]

continuum mechanics, fluid dynamics, physical optics...

 

[BenTheMan]

http://iopscience.iop.org/0305-4470/33/1/102/

I don't know how popular this stuff is among finance guys. This author also has a book by the same name which you can find on amazon.

 

[Monocles]

It's everywhere in condensed matter, too.

 

[maverick280857]

It's everywhere in condensed matter, too.

Monocles, can you please elaborate on where and how it enters in condensed matter physics, as I too am interested to know.

 

[Monocles]

I will illustrate with an example of the quantum hall effect. The quantum hall effect, of course, demonstrates the quantization of resistance at low temperatures in a 2-dimensional substrate that is pierced by a perpendicular magnetic field. Why is this? Well, without going into too much detail, there are essentially two parameters you can vary in a quantum hall system. So, your quantum hall system has a two-parameter family of Hamiltonians. Both of these parameters happen to be periodic, i.e. if you slowly increase one parameter, you will eventually obtain the same system. Thus, your family of Hamiltonians forms a parameter space that is actually a torus.

Now, to each point of your torus you glue the associated eigenspace. To simplify matters, we set temperature to zero so that all of the occupied eigenstates lie below a certain energy, the Fermi energy, which allows us to consider the eigenspace to be finite-dimensional by just deleting all of the unoccupied eigenstates. We need some other 'niceness' requirements as well to ensure that the dimension of the eigenspace is constant over the parameter space, that there is no degeneracy of eigenvalues, etc.

Gluing the eigenspace to each point in the parameter space creates a complex vector bundle. In order to take into account the Aharonov-Bohm effect that causes a wavefunction's phase to change non-trivially when moving through a magnetic field, we require the vector bundle to be twisted (which manifests as the Chern character of the fiber bundle). Using the Kubo formula, it can be shown that it is precisely this twisting of the vector bundle that creates the quantization of resistance in the quantum hall system.

There are some major problems with this interpretation, but that's why we look at effects like these from more than one point of view. We can also look at the quantum hall effect from the point of view of Laughlin, which emphasizes the physical geometry of the system (which misses the point that the quantization of resistance comes from the structure of the parameter space, but makes evident the need for the localization of states). There is also the point of view of non-commutative geometry, which from what I understand clears up the problems, but it is beyond what I know so I can't explain it :)

I am sorry if that is too much jargon - I do not know how much differential geometry you know. But that is one beautiful example of the application of differential geometry in condensed matter systems. Unfortunately I'm an undergraduate so I have had very little exposure to other examples - I wish I could point you to more! My confidence that there are other examples comes primarily from condensed matter professors telling me so.

 

[maverick280857]

I am sorry if that is too much jargon - I do not know how much differential geometry you know. But that is one beautiful example of the application of differential geometry in condensed matter systems. Unfortunately I'm an undergraduate so I have had very little exposure to other examples - I wish I could point you to more! My confidence that there are other examples comes primarily from condensed matter professors telling me so.

Monocles, that was very helpful. I am about to begin grad school, and I do not know differential geometry as much (), and far less about its applications other than some mathematical aspects of quantum field theory. I've just begun reading 'The Geometry Of Physics' by Ted Frankel though, and that seems to be quite an interesting book. Thank you again for the description

 

[trustinlust]

Thanks everybody for the replies.

I have expecially appreciated the mention of Monocles about the application of Differential Geometry in the domain of the Condensed Matter Physics, and the use of this mathematic tool -besides the standard use of quantum field theory- in the study of quantum proprieties of matter. If you know some other details about the most "hot" current research subjects in this field, l'd like to know them.

Thanks

 

[element4]

Thanks everybody for the replies.

I have expecially appreciated the mention of Monocles about the application of Differential Geometry in the domain of the Condensed Matter Physics, and the use of this mathematic tool -besides the standard use of quantum field theory- in the study of quantum proprieties of matter. If you know some other details about the most "hot" current research subjects in this field, l'd like to know them.

Thanks

Right now, THE hottest subject in condensed matter physics is the so called "Topological Insulators". They are in some way, very related to the Quantum Hall Effect which Monocycles mentioned. There are MANY reasons why these materials are so exciting and I could not do them any justice right now. If I find more time later, I shall elaborate.

But in this field different aspects of differential geometry, algebraic/differential topology and even (topological) K-theory is routinely** used. You might want to read this piece from Nature: http://www.nature.com/news/2010/100714/full/466310a.html".

** = saying "routinely" might be an exaggeration.

 

[element4]

I've just begun reading 'The Geometry Of Physics' by Ted Frankel though, and that seems to be quite an interesting book.

As a first book on these subjects, I can recommend you https://www.amazon.com/dp/9810217293/?tag=pfamazon01-20. This book is in some sense much easier to read, much more intuitive and much better written than Frankel (in my opinion). And it covers lots of ground very quickly. Having read this book, it would be much easier to study Frankel (or more advanced books).

In general, read everything written by John Baez. It's always amazingly clear, funny and rewarding.

 

[TaylorRatliff]

Plasma physics - especially magnetic confinement for fusion.

 

[Monocles]

I want to echo the suggestion to read everything by John Baez! His column, This Week in Mathematical Physics, is absolutely incredible. It's my goal to eventually read all of it. You can learn mountains of cool stuff from it!

 

[maverick280857]

I want to echo the suggestion to read everything by John Baez! His column, This Week in Mathematical Physics, is absolutely incredible. It's my goal to eventually read all of it. You can learn mountains of cool stuff from it!

Yeah, I've been following it for some time. Couldn't study differential geometry 'formally', so have only a smattering of it. I'll check out the other book by Baez.

 

[Amanheis]

Classical mechanics can be described in terms of symplectic manifolds. See the freely available book by Abraham and Marsden, Foundations of Mechanics:
http://caltechbook.library.caltech.edu/103/