- 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.
- 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.
- 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.
- 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.
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