# Arahnov - Bohm effect and U(1) bundles

1. Nov 10, 2013

### lavinia

I read in a differential geometry paper that Maxwell's equations can be formulated in terms of a connection on a Hermitian line bundle on Minkowski space.

I understand the derivation of the field strength 2 form,the proof that Maxwell's equations say that its exterior derivative is zero and its codifferential is the current density 1 form, and that there must exist a gauge potential whose exterior derivative equals the field strength.

But how does the Arahnov-Bohm effect make this interpretation preferable? How is it reflected in this mathematical formulation?

Last edited: Nov 10, 2013
2. Nov 10, 2013

### atyy

The Aharonov-Bohm effect can be predicted by coupling electrons and the gauge field. The Aharonov-Bohm phase is a gauge invariant quantity (Eq 7 of http://arxiv.org/abs/0711.4697 ).

3. Nov 11, 2013

### dextercioby

Lavinia, what paper are you reading ?

4. Nov 11, 2013

### lavinia

It is "Vector Bundles with a Connection" by Chern

It appears in

Math. Sci. Research Inst. (1987) 1-23

I have it in the collected works of Chern.

5. Nov 11, 2013

### lavinia

Thanks for the reference. Is there a free version of this paper?

6. Nov 11, 2013

### atyy

I think there's a link to the free version of the paper at the top right of that page (in the "Download" box).

The Aharonov-Bohm effect is a quantum effect. So for classical EM, we don't need the vector potential. The vector potential or gauge field is just a redundant way of describing electric and magnetic fields. The electric and magnetic fields are "physical" or gauge invariant, but the vector potential is not "physical", since it changes according to gauge. Physical quantities must be gauge invariant or "geometric". In quantum EM, there are these non-local effects like the Aharonov-Bohm phase. The integral is gauge invariant but nonlocal (since it integrates over space). We could attempt to use these gauge-invariant nonlocal loops to describe physics, but in order to preserve a manifestly local description (especially one manifestly consistent with special relativity), it is more convenient to use the gauge field, even though it is not "physical".

I linked to the paper only because it's free, but there are probably better references in standard textbooks, since that paper is mainly about anyons. Let me see if I can find more standard references. Googling suggests
http://www.ece.rice.edu/~kono/ELEC563/2005/AharonovBohm.pdf
http://www-dft.ts.infn.it/~resta/fismat/lez_berry.pdf
http://iopscience.iop.org/1751-8121/43/35/350301

You can also try looking up Wilson loops (I think mathematicians call these "holonomies").

Last edited: Nov 11, 2013
7. Nov 11, 2013

### dextercioby

Thanks, too bad it's not online. But I bet it's a good read.

8. Nov 12, 2013

### lavinia

I find Chern's papers super hard to read. They are terse and leave almost all computatuion to the reader.

But I found these lectures well presented

http://empg.maths.ed.ac.uk/Activities/GT/Lect1.pdfhttp://empg.maths.ed.ac.uk/Activities/GT/Lect1.pdf [Broken]

Just change the Lect1 to Lect2 and 3 and 4 to get the whole thing.

I am told that Physicists invented the idea of gauge potential and Mathematicians the idea of connection on a principal fiber bundle independently and at some point one of each were talking, maybe Simons and Yang, and the mathematician said "Oh. You are talking about a connection,"

Last edited by a moderator: May 6, 2017