Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Monday, October 20, 2008

Quick Update

Alas, busy days at the cafe make for boring blogging. But today was pleasantly quiet at work, with time for coffee and a chat with a visiting English professor. And I assured everybody that the 90 km/hr wind was merely a light spring breeze.


Blogger L. Riofrio said...

Hanging out at a cafe is a nice life. 90 km/hr? It would have been easy for a New Zealander to invent the aeroplane.

October 21, 2008 5:50 AM  
Blogger Kea said...

Yes, the wind regularly tops this during a nor'wester.

October 21, 2008 7:41 AM  
Blogger Matti Pitkanen said...

Just a little information bit about further application of category theoretic thinking to TGD. See my blog.

Symplectic QFT which emerges as an analog of conformal QFT allows an obvious application of category theoretical ideas. The algebra generated by the values of symplectic field at points of sphere. Product of fields at two different points is integral of values at various points of sphere: this is totally different from ordinary fusion rules. The product of field vanishes at the limit when the points co-incide.

One can develop quite concrete ansatz for the fusion rules from braided commutativity and associativity and symplectic invariance. Unfortunately my analytic integration skills are not enough to check whether the simplest ansatz works and I am too lazy to try to check the rules numerically.

Discretization of S^2 implied by a finite measurement resolution leads to an infinite hierarchy of finite-D nilpotent algebras with generators satisfying x^2=0 plus braided commutatitivity and associativity. The dimension of algebra corresponds to the number of braid strands. These algebras differ from Grassman algebras in that the product of two generators is sum of generators. It would be interesting to know whether this kind of algebras are part of standard mathematical literature.

October 22, 2008 12:55 AM  
Blogger Kea said...

Hi Matti. That does sound like the kind of algebra that Carl and I are trying to make sense of.

October 22, 2008 8:32 AM  
Blogger Matti Pitkanen said...

I try to find to the conditions explicit solutions for small values of N. The conditions sum_m CklmCklm=0 following from braided associativity and x^2=0 have interpretation as orthogonality conditions for wave function Cklm and its conjugate defined in discrete space of N points.
The proportionality to elements of cylic group Z_{N-2} represented as phases provides one solution.

October 22, 2008 5:05 PM  
Blogger Matti Pitkanen said...

I managed to construct a very general solution to the finite-D duality conditions in terms of phases of group Z_N-2.

The solution is based on the condition that complex conjugation sends function basis of N-1 functions to its orthogonal dual in Hilbert space sense. This condition does not leave very many options besides wave functions which are just Z_N-2 phases.

I understood also the geometric realization of the algebra assuming that the exponent of the area of symplectic triangle repesents the phase of Cklm as Bohm-Aharonov phase associated with charge Q.

October 22, 2008 11:54 PM  
Blogger Matti Pitkanen said...

It seems that I understand fusion algebras satisfactorily.

The continuum generalization of the discrete symplectic fusion algebra, whose structure constant I managed to discover, does not seem to exist. This in accordance with category theoretical wisdom and notion of finite measurement resolution having number theoretic braids as its representation. Also p-adicization disfavors its existence since the needed area integrals do not exist p-adically.

The purely category theoretical approach posing on the symplectic triangles very mild conditions wins the physics inspired approach in which the sides of triangles were assumed to be pieces of geodesics or their generalizations for charged particles. The partial symplecto-magnetic fluxes int A_mudx^mu for the sides are quantized as fractions 2pi/(N-2) for N-D algebra.

I found also the geometric realization for the hierarchy of fusion algebras.

There are excellent reasons to believe that fusion algebras form a planar operad since the increase of measurement resolution adds to a disk surrounding given point of S^2 defining element of algebra a set of points. Inclusion would induce algebra homomorphisms for this hierarchy of fusion algebras.

For cleaned up and shortened version of the construction see
my blog.

October 23, 2008 5:14 PM  
Blogger Kea said...

Wow, Matti. Powering along here! Unfortunately, I just don't have time to look at this now. Carl and I are writing a paper.

October 23, 2008 5:40 PM  

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