Tuesday, June 23, 2009

p-Adicization, twistor program, and quantum criticality

Just a brief note (strongly updated!) about the recent situation concerning bosonic emergence and QFT limit of TGD. There is now a very attractive overall view about how p-adic and real physics are fused together and how p-adic fractality emerges when real Lorentz invariants - typically mass squared for subsystem- are mapped to their p-adic counterparts of a suitable variant of canonical identification which in its simplest form reads as &sum xnpn → &sum xnp-n. One can say that quantum criticality, bosonic emergence, number theoretic universality, p-adic fractality, and twistor program seem to be very intimately inter-related in TGD Universe.

Loops are the problem of the p-adicization program as also twistor program. In twistorialization the problem can e overcome by using Cutkosky rules which means that one adds to the tree diagram TT+ contribution for on mass shell intermediate states allowing unitarization. Since this contribution involves only massless intermediate states twistorialization is possible. This is actually what I suggested earlier (only light-like loop momenta are allowed in twistor context) without properly realizing the connection with Cutkosky rules! If TT+ makes sense also p-adically, p-adicization and p-adic fractalization are possible.

Unitarization by Cutkosky rules does not make sense for fermionic loops defining the bosonic vertices as becomes clear by considering B→FFbar→B loop for massless particles. Furthermore, if these vertices were non-vanishing for on mass shell momenta (massless) unitarity would force the introduction of TT+ contribution and one could not speak about vertices anymore. Therefore it seems that the fermionic loops defining bosonic vertices vanish when the bosons are on mass shell. These conditions would generalize the quantum criticality condition and hopefully fix completely the vertices. It also means that only BFF vertex is non-vanishing for on mass shell particles as is natural since Dirac action coupled to gauge bosons is the basic action principle. The vanishing of on mass shell N-vertices gives an infinite number of conditions on the hyperbolic cutoff as function of the integer k labeling p-adic length scale at the limit when bosons are massless and IR cutoff for the loop mass scale is taken to zero. For a finite cutoff kmax the number of vanishing vertices is finite and correspond to some maximum value Ncr analogous to the order of perturbation theory and identifiable as characterization of the finite measurement resolution.

Whether the vanishing of the fermionic loops defining vertices is achieved by fixing the hyperbolic cutoff is not clear, and one can wonder whether dynamical on mass shell symmetries -in particular various super-conformal symmetries - could be involved. The first checks suggests that super-symmetry cannot lead to the vanishing of the on mass shell vertices and that hyperbolic cutoff and the non-trivial relation between time-like and space-like hyperbolic cutoffs are necessary.

To me it seems that TGD has forced a rather dramatic simplicification of the very notion of quantum field theory. If so, then the mere assumption about the existence of QFT limit (to say nothing about the assumption that this limit is GUT or a minimally supersymmetric version of standard model (MSSM) for which Lebensraum is shrinking continually) would have led competing unified theorists to a fatal side track.

A more detailed representation can be found from the last section of the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix". I also extracted from the chapter a short piece of text explaining in more detail the ideas discussed here and in the previous posting.


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