I have been working few months with the QFT limit of TGD. The idea which led to the realization of what QFT limit of TGD could be is simple.
- Only fermions are fundamental particles in quantum TGD and bosons are fermion-antifermion pairs with fermion and antifermion quantum numbers residing at the opposite 3-D light-like throats of wormhole contacts which are surfaces possessing Euclidian signature of induced metric and are glued to space-time sheets having Minkowskian signature of induced metric. Feynman diagrams can therefore be understood in terms of space-time topology and space-time metric. The interpretation of generalized Feynman diagrams differs dramatically from that for stringy diagrams since vertices are points where light-like 3-surfaces join together just like likes of ordinary Feynman diagram do. Stringy diagrams provide a space-time correlate for the propagation of particle along two different routes followed by fusion and interference.
- Only fermions are fundamental fields in TGD. This suggests that gauge bosons, which have components of induced spinor connection and projections of CP2 Killing vector field as classical geometric correlates, should emerge in some sense at QFT limit. In other words, the action for QFT approximating TGD contains nothing but Dirac action coupled to gauge potentials, and the bosonic action containing YM term plus infinite number of vertices defined by closed fermion loops is generated radiatively. This approach leads to a generalization of Feynman rules and in principle predicts all coupling constants and their evolution without any input parameters except CP2 size and quantum criticality. p-Adic mass calculations demonstrated already 15 years ago that one can understand the mysterious proton mass to Planck mass ratio and elementary particle mass scales and even masses number theoretically.
- An essential element of the approach is a formulation for UV cutoff. A cutoff in both mass squared and hyperbolic angle is necessary since Wick rotation does not make sense in TGD framework. By assuming a geometrically natural hyperbolic UV cutoff motivated by zero energy ontology one can understand the evolution of the standard model gauge couplings and reproduce correctly the values of fine structure constant at electron and intermediate boson length scales. Also asymptotic freedom follows as a basic prediction. Contrary to the original beliefs propagator generates a mass term unless the hyperbolic cutoffs for time-like and space-like gauge boson momenta are in a definite relation. One could criticize this relation and argue that perhaps super-conformal symmetries might help to get the cancelation with identical cutoffs. It seems that this is not the case.
The UV cutoff for the hyperbolic angle as a function of p-adic length scale is the ad hoc element of the model in its recent form. How to formulate quantitatively the quantum criticality in terms of the behavior of the hyperbolic cutoff as function of p-adic length scale became therefore the basic problem and led what might like a numerics inspired random walk -or perhaps better to say sleep walk - towards what I believe to the solution of the problem. During this kind of heavy numerical calculations one realizes how important it would be to have a colleague replicating the calculations. One can never be quite sure about signs and numerical factors.
- The process gradually led to an improved understanding of the notion of coupling constant evolution itself. The fermionic loop integral contains a propagator pole contributing imaginary part to the inverse propagator and numerical calculations demonstrated that this contribution is too large to be physically acceptable. Moreover, the sign of coupling strength becomes negative for fermion masses above certain critical mass defining the IR cutoff for the loop momenta. The only manner to avoid difficulties is to assume that loop momenta are always below the p-adic mass scale associated with the momentum of the gauge boson. The assumption eliminates the imaginary part of propagator and keeps coupling constant strength positive. This also gives precise content to the notion of coupling constant evolution since it assigns to the mass scape of p IR cutoff kmax such that for k > kmax coupling constant strength is positive. A nice geometric interpretation is possible in zero energy ontology: loop corrections corresponding to geometric details sufficiently smaller than the length scale assignable to the mass squared.
- The next idea was that perhaps one could fix the cutoff on hyperbolic angle (hyperbolic cutoff) by some naturally occurring condition. The first guess was that the sign of the coupling constant strength changes at either end of the p-adic half octave for the mass of gauge boson. The motivation to this idea could have come from the calculation of the momentum at which the sign changes for the model reproducing physically reasonable coupling constant evolution: at long length scales the sign indeed changes very near to the end of the half-octave. Unfortunately this did not work.
- The next guess was that the value of boson momentum at which the sign changes is as near as possible to the end of the mass squared octave. Tedious calculations in a rather arctic numerical environment demonstrated that one obtains a discrete set of coupling constant evolutions but that the hyperbolic cutoff is increasing as a function of k rather than decreasing as required by the coupling constant evolution in standard model. The increase can be understood as a positive feedback effect: the vanishing of the inverse of the coupling constant at given length scale requires a contribution, which increases as a function of the p-adic length scale since the inverse of the coupling constant itself increases. The attempts to modify the model to modify this behavior failed.
- The next idea was that perhaps p-adic fractality helps to assign the change of the sign at the ends of half octaves or to prime for which p-adic length scale is very near to that defined by the end of the half octave (p <≈ 2k). p-Adic fractals were one of the first ideas about p-adic physics and I learned quite recently that also mathematicians have discovered them. They are obtained by mapping reals to p-adics by the inverse of the canonical identification I (or a proper variant of it) performing the arithmetics, and map the result back to reals by I. I had not found any direct application except in the case of p-adic mass calculations where p-adic mass squared is mapped to its real counterpart.
The guess was obvious. Express M-matrix element a function of standard Lorentz invariants with dimensions of mass squared so that a very close connection with mass calculations is obtained. Map the invariants to their p-adic counterparts using the inverse of I, carry out the arithmetics defining the function in the p-adicity under question, and return to the reality using I. Maybe this could allow to achieve the cancelation at the end of the p-adic octave for mass squared. I do not believe this anymore but again a wrong idea led to what looks like a real increase in the understanding of quantum TGD and how p-adic and real physics relate at the level of M-matrix. One nice finding was that p-adic existence forces the loop masses to be above the mass of virtual gauge boson forced by purely physical conditions. It however seems that one must introduce transcendentals like log(2) and π so that an algebraically infinite-dimensional and basically non-algebraic extension of p-adic numbers is unavoidable.
- The p-adicization program for M-matrix involve a technical difficulty which led to a further progress. It is not possible to perform loop integrals in the p-adic context. All loop integrals must be carried out in the real context and the resulting functions must be p-adicized. For the bosonic vertices defined as purely fermionic loops this is not a problem but the situation changes for the expansion of the M-matrix involving both bosonic and fermionic lines inside loops. The same problem is encountered in the twistorialization and the solution of the problem is based on Cutkosky rules allowing unitarization of the tree amplitudes in terms of TT+ contribution involving only light-like momenta seems to be the only working option and requires that TT+ makes sense p-adically. This idea is actually very near to the original idea that only light-like momenta appear in loops so that twistorialization is elegant. TT+ indeed allows interpretation in terms of loops so that I was not after all totally silly. The p-adic existence of the analytic continuation of TT+ by dispersion relations poses strong constraints on otherwise not completely unique continuation.
- After these steps I was mature to realize how to formulate quantum criticality in such a manner that it could fix the hyperbolic cutoff and hence coupling constant evolution uniquely. The fermionic loops defining bosonic vertices vanish when the incoming momenta are massless. This is it! The condition emerges as a consistency condition: if the vanishing does not occur for on mass shell bosons, one obtains T-matrix expressible in terms of analytic continuation of TT+ and one does not have vertex identified as something irreducible anymore. The condition is suggested also by quantum criticality: the vanishing of vertices is very much analogous to the vanishing of higher functional derivatives of the action with respect to gauge fields at criticality (or derivatives of the potential function in Thom's catastrophe theory). Also the fact that only BFF vertex is fundamental vertex if bosonic emergence is accepted suggests the conditions. The vanishing of on mass shell N-vertices gives an infinite number of conditions on the hyperbolic cutoff as a 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. It is not yet clear whether dynamical symmetries, in particular super-conformal symmetries, are involved with the realization of the vanishing conditions or whether hyperbolic cutoff is all that is needed.