quantum groups in physics

quantum groups in physics

[John Baez]

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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Re: quantum groups in physics

[jim stasheff]

John Baez wrote:
>
> 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!
Can you give me a clue as to how quantum groups enter
since

Chern-Simons theory can be
described in a purely algebraic way

for ordinary Lie groups

do you mean just that
Chern-Simons theory
described in a purely algebraic way
extends to quqantum groups?

jim






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re: quantum groups in physics

[John Baez]

Happy New Year!

I wrote:


> 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!
>>
>
Jim Stasheff wrote:


>  Can you give me a clue as to how quantum groups enter? -
> since
>
> Chern-Simons theory can be
> described in a purely algebraic way
>
> for ordinary Lie groups.
>

In my remark, I was speaking of Chern-Simons theory as a 3d field theory
with the Lagrangian

tr(A dA + (2/3) A^3)

Here A is a connection on some bundle with an ordinary Lie group as
structure group.

As a classical field theory, the solutions of Chern-Simons theory are just
flat connections.   But when you quantize it, quantum groups come in!  The
moduli space of flat connections has a symplectic structure, and when you
geometrically quantize it, the resulting Hilbert space has a nice
description in terms of the category of representations of the quantum group
associated to your original Lie group.

This is what Witten initiated with his famous paper "Quantum field theory
and the Jones polynomial".   For a really nice account, try this book:

Bojko Bakalov and Alexander Kirillov, Jr., Lectures on Tensor Categories and
Modular Functors, American Mathematical
Society, Providence, Rhode Island, 2001.  Preliminary version available
http://www.math.sunysb.edu/~kirillov/tensor/tensor.html

For many mathematicians physicists, this connection to field theory is a big
part of why quantum groups are interesting!

Best,
jb


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