On 7/17/06, Jason Holt <jason@lunkwill.org> wrote:
Oh yeah, and maybe Mike Stay could also briefly summarize quantum gravity for us in a way that makes it clear just how that principle works. I was wondering about that, too.
I could explain topological quantum field theories, which are a toy version; they give quantum gravity in 2D. In ~2001 the guys at Microsoft's Project Q showed that they can be efficiently simulated by the quantum circuit model, and conversely that there's a TQFT universal for quantum computation. Right now they're working on braiding anyons. A TQFT is a tensor-product-preserving functor from 2Cob to Hilb. 2Cob is the category of 1-dimensional manifolds (tensor products of circles) and 2-d cobordisms between them (caps, cups, pants, and upside-down pants). Hilb has hilbert spaces and linear transformations between them. A functor maps objects to objects and morphisms to morphisms such that composition is preserved. See http://www.math.ucr.edu/home/baez/qg-winter2001/qg15.1.html for a very gentle intro. You build the functor by triangulating the cobordism, taking the poincare dual to get a bunch of Y's or inverted Y's, and then interpreting those as multiplication or comultiplication, repsectively; the centre of this algebra is the part that's immune to changes by Pachner moves, and since Pachner moves can take you between any two triangulations, the functor is triangulation independent. See http://www.math.ucr.edu/home/baez/qg-fall2004/ for lecture notes on how that all works. -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike