quantum groups in physics

[John Baez <john.c.baez@gmail.com>]

Andre Joyal writes:

The theory of quantum groups is mathematically very interesting but it has
> no applications that I know to real quantum physics...


The fractional quantum Hall effect is a strange effect that occurs when a
thin film of superconducting material is put in a transverse magnetic
field.  The 1998 Nobel Prize in physics was awarded for its discovery and
explanation:

http://nobelprize.org/nobel_prizes/physics/laureates/1998/press.html

I think it's becoming pretty widely accepted that Chern-Simons theory is a
good description of the fractional quantum Hall effect.  See for example:

http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_2002/files/vetsigian.pdf<http://guava.physics.uiuc.edu/%7Enigel/courses/569/Essays_2002/files/vetsigian.pdf>

This is a question that experiment will ultimately decide.

On the other hand, it's been known in theoretical physics ever since the
work of Witten, Reshetihkin and Turaev that Chern-Simons theory can be
described in a purely algebraic way using quantum groups!

So, people interested in the fractional quantum Hall effect are learning
about quantum groups.  But interestingly, more important than the quantum
group itself is its category of representations, which is a modular tensor
category.

So in fact we're seeing a nice interplay between experimental condensed
matter physics and work on quantum groups and modular tensor categories.
But this is not really surprising, since quantum groups and modular tensor
categories arose from work on physics.

Attempts to use these ideas to build quantum computers are still
speculative:

http://en.wikipedia.org/wiki/Topological_quantum_computer

I can give lots more references if anyone wants.

Best,
jb


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