Monday, April 23, 2007

Quantum criticality, hierarchy of Planck constants, and category theory

I have used some time to ponder the problem of whether category theoretic structures might have direct topological counterparts at space-time level with objects possibly identified as the space-time correlates of elementary particles and morphisms as the space-time correlates of their (bound state) interactions.

I became again aware of this problem as I was playing with various interpretations for what darkness interpreted in terms of nonstandard value of Planck constant and the modification of imbedding space obtained by gluing together H→ H/Ga×Gb, Ga and Gb discrete subgroups of SU(2) associated with Jones inclusions along their common points. Ga and Gb could be restricted to be cyclic and thus leaving the choice of quantization axis invariant. A book like structure results with different copies of H analogous to the pages of the book. Probably brane people work with analogous structures.

The most conservative form of darkness is that only field bodies of particles are dark and that particle space-time sheet to which I assign the p-adic prime p characterizing particle corresponds to its em field body. Also Compton length as determined by em interaction would characterize this field body. Compton length would be completely operational concept. This option is implied by a strong hypothesis that elementary particles are maximally quantum critical meaning that they belong to subspace of H left invariant by all groups Ga×Gb leaving quantization axis invariant so that all dark variants of particle identified as 2-D partonic surface would be identical.

The implication would be that particle possess field body associated with each interaction and an extremely rich repertoire of phases emerges if these bodies are allowed to be dark and characterized by p-adic primes. Planck constant would be assigned with a particular interaction of particle rather than particle. This conforms with the formula of gravitational Planck constant hbargr= GMm211, whose dependence on particle masses indeed forces the assignment of this constant to the gravitational field body.

What I realized is that if elementary particles are maximally quantum critical they would be analogous to objects and the field bodies mediating interactions between them would be analogous to morphisms. The basic structures of category theory would have direct implementation at the level of many-sheeted space-time. This looks nice. On the other hand, the basic composition of morphisms f(A→B)f(B→C)=f(A→C) for morphisms would mean that particle interactions also obey composition law. The hierarchy of p-adic primes and Planck constants and composition law would imply hierarchy of interactions and one could not speak about single universal interaction between particles. As a matter fact, one cannot speak about uni-directional arrow: rather, a bi-directional arrow would be in question.


At 5:44 PM, Blogger Kea said...

Matti, I'm afraid you've lost me here, but I appreciate the attempt to incorporate categories into your framework. Where are the higher dimensional morphisms here?

At 11:14 PM, Blogger Matti Pitkanen said...

Dear Kea,

the picture is rather metaphorical and it might well be that morphism is not sufficiently general notion (I might of course understand morphism in too restricted sense). Bi-directional arrow describing for what it is to form a bound state by some interaction would be the proper mental image now.

We have objects and bonds between them realized as particle 3-surfaces and topological field quanta connecting them and serving as correlates for bound state formation. Particles and their interactions<--->objects and morphisms.

I am not sure what you mean with higher-dimensional morphisms so that I make a guess.

a) TGD leads to hierarchies (say infinite primes defining infinite hierarchy of second quantization in which many particle states of previous level becomes particles of new level, hierarchy of space-time sheets with systems of previous level become particle like structure of new level). This would suggest a hierarchy of categories with categories of previous level objects at the next level.

b) One might think that a collection of objects and their morphisms defining physical system as a category at a given level of hierarchy becomes object at the next level of hierarchy. Elementary particles connected by flux quanta forming bound state appearing particle at the next level would be connected in turn by topological field quanta to another particles of this kind.


At 4:55 PM, Blogger Kea said...

OK, (a) is the sort of thing I have in mind.


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