Tuesday, October 07, 2008

Category theory and quantum TGD: possible other applications

It is not difficult to imagine also other applications of category theory in TGD framework besides those discussed in previous two postings. Here I only briefly mention some potential applications.

1. TGD variant for the category nCob

John Baez has suggested that quantum field theories could be formulated as functors from the category of n-cobordisms to the category of Hilbert spaces [6,7,8]. In TGD framework light-like 3-surfaces containing the number theoretical braids define the analogs of 3-cobordisms and surface property brings in new structure. The motion of topological condensed 3-surfaces along 4-D space-time sheets brings in non-trivial topology analogous to braiding and not present in category nCob.

Intuitively it seems possible to speak about one-dimensional orbits of wormhole throats and -contacts (fermions and bosons) in background space-time (homological dimension). In this case linking or knotting are not possible since knotting is co-dimension 2 phenomenon and only objects whose homological dimensions sum up to D-1 can get linked in dimension D. String like objects could topologically condense along wormhole contact which is string like object. The orbits of closed string like objects are homologically co-dimension 2 objects and could get knotted if one does not allow space-time sheets describing un-knotting. The simplest examples are ordinary knots which are not allowed to evolve by forming self intersections. The orbits of point like wormhole contact and closed string like wormhole contact can get linked: a point particle moving through a closed string is basic dynamical example. There is no good reason preventing unknotting and unlinking in absolute sense.

2. Number theoretical universality and category theory

Category theory might be also a useful tool to formulate rigorously the idea of number theoretical universality and ideas about cognition. What comes into mind first are functors real to p-adic physics and vice versa. They would be obtained by composition of functors from real to rational physics and back to p-adic physics or vice versa. The functors from real to p-adic physics would provide cognitive representations and the reverse functors would correspond to the realization of intentional action. The functor mapping real 3-surface to p-adic 3-surfaces would be simple: interpret the equations of 3-surface in terms of rational functions with coefficients in some algebraic extension of rationals as equations in arbitrary number field. Whether this description applies or is needed for 4-D space-time surface is not clear.

At the Hilbert space level the realization of these functors would be quantum jump in which quantum state localized to p-adic sector tunnels to real sector or vice versa. In zero energy ontology this process is allowed by conservation laws even in the case that one cannot assign classical conserved quantities to p-adic states (their definition as integrals of conserved currents does not make sense since definite integral is not a well-defined concept in p-adic physics). The interpretation would be in terms of generalized M-matrix applying to cognition and intentionality. This M-matrix would have values in the field of rationals or some algebraic extension of rationals. Again a generalization of Connes tensor product is suggestive.

3. Category theory and fermionic parts of zero energy states as logical deductions

Category theory has natural applications to quantum and classical logic and theory of computation [7]. In TGD framework these applications are very closely related to quantum TGD itself since it is possible to identify the positive and negative energy pieces of fermionic part of the zero energy state as a pair of Boolean statements connected by a logical deduction, or rather- quantum superposition of them. An alternative interpretation is as rules for the behavior of the Universe coded by the quantum state of Universe itself. A further interpretation is as structures analogous to quantum computation programs with internal lines of Feynman diagram would represent communication and vertices computational steps and replication of classical information coded by number theoretical braids.

4. Category theory and hierarchy of Planck constants

Category theory might help to characterize more precisely the proposed geometric realization of the hierarchy of Planck constants explaining dark matter as phases with non-standard value of Planck constant. The situation is topologically very similar to that encountered for generalized Feynman diagrams. Singular coverings and factor spaces of M4 and CP2 are glued together along 2-D manifolds playing the role of object and space-time sheets at different vertices could be interpreted as arrows going through this object.

References

[1] The chapter Nuclear String Model of "p-Adic length scale Hypothesis and Hierarchy of Planck constants".
[2] C. Rogers and A. Hoffnung (2008), Categorified symplectic geometry and the classical string.
[3] Operad theory.
[4] Planar algebra.
[5] D. Bisch, P. Das, and S. K. Gosh (2008), Planar algebra of group-type subfactors.
[6] Multicategories.
[7] J. Baez (2007), Quantum Quandaries.
[8] J. Baez and M. Stay (2008), Physics, topology, logic and computation: a Rosetta Stone.
[9] J. Baez (2008), Categorifying Fundamental Physics.
[10]Planar Algebras, TFTs with Defects.
[11] M. D. Sheppeard (2007), Gluon Phenomenology and a Linear Topos, thesis.

The article Category Theory and Quantum TGD gives a summary of the most recent ideas about applications of category theory in TGD framework. See also the new chapter Category Theory and TGD of "Towards S-matrix".

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