Thursday, May 17, 2007

Category theoretical formulation of quantum TGD in nutshell

For long time it has been clear that category theory might provide a fundamental formulation of quantum TGD. The problem has been that category theory seems to postulate quite too many objects. The reading of Quantum Quandaries by John Baez helped to see the situation in all its simplicity.

  1. Topological Quantum Field theories have extremely simple formulation as a functor from the category of cobordisms (topological evolutions between n-1-manifolds by connecting n-manifold) to the category of Hilbert spaces assignable to n-1-manifolds.

  2. Since light-like partonic 3-surfaces correspond to almost topological QFT, with the overall important "almost" coming just from the light-likeness in the induced metric, the theory is non-trivial physically and nothing of the beauty of TQFT as a functor is lost. Cobordisms are however replaced by what I have christened Feynmann cobordisms generalizing the Feynman diagrams to 3-D context: the ends of light-like 3-manifolds meet at the vertices which correspond to 2-dimensional partonic surfaces.

  3. Also the counterparts of ordinary string diagrams having interpretation as ordinary cobordisms are possible but have nothing to do with particle reactions: the particle simply decomposes into several pieces and spinor fields propagate along different routes. This is the space-time correlate for what happens in double slit experiment when photon travels along two different paths simultaneously.

  4. The intriguing results is that for n<4-dimensional cobordisms unitary S-matrix exists only for trivial cobordisms. I wonder whether string theorists have considered the possible catastrophic consequences concerning the non-perturbative dream about the unique stringy S-matrix. In the zero energy ontology of TGD S-matrix appears as time-like entanglement coefficients and need not be unitary. I have already proposed that p-adic thermodynamics and thermodynamics in general could be regarded as an exact part of quantum theory in this framework and the basic mathematics of hyper-finite factors provides strong technical support for this idea. It could be that one cannot require unitarity in the case of Feynman cobordisms and that only the condition that S-matrix for a product of Feynman cobordisms is a product of S-matrices for composites. Hence the time parameter in S-matrix can be replaced with complex time parameter with imaginary part in the role of temperature without losing the product structure. p-Adic thermodynamics and particle massication might be topological necessities in this framework.

For more details see the chapter Construction of Quantum Theory: S-Matrix of "Towards S-Matrix".

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