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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
-
jim stasheff -
John Baez