pAdic Field Theory limit of TGD is free of UV divergences
Abstract
The padic description of Higgs mechanism in TGD framework provides excellent predictions for elementary particle and hadrons masses (hepth@xxx.lanl.gov 941005862). The gauge group of TGD is just the gauge group of the standard model so that it makes sense to study the padic counterpart of the standard model as a candidate for low energy effective theory. Momentum eigen states can be constructed purely number theoretically and the infrared cutoff implied by the finite size of the convergence cube of padic square root function leads to momentum discretization. Discretization solves ultraviolet problems: the number of momentum states associated with a fixed value of the propagator expression in the loop is integer and has padic norm not larger than one so that the contribution of momentum squared with padic norm $p^{k}$ converges as $p^{2k2}$ for boson loop. The existence of the action exponential forces number theoretically the decomposition into free and interacting parts. The free part is of order $O(p^0)$ and must vanish (and does so by equations of motion) and interaction part is at most of order $O(\sqrt{p})$ padically. pAdic coupling constants are of form $g\sqrt{p}$: their real counterparts are obtained by canonical identification between padic and real numbers. The discretized version of Feynmann rules of real theory should give Smatrix elements but Feynmann rules guarantee unitarity in formal sense only. The unexpected result is the upper bound $L_p=L_0/\sqrt{p}$ ($L_0\sim 10^4\sqrt{G}$) for the size of padic convergence cube from the cancellation of infrared divergences so that padic field theory doesn't make sense above length scale $L_p$.
 Publication:

arXiv eprints
 Pub Date:
 December 1994
 arXiv:
 arXiv:hepth/9412103
 Bibcode:
 1994hep.th...12103P
 Keywords:

 High Energy Physics  Theory
 EPrint:
 22 pages,latex. Reason for revision:correction in the definition of fermion propagator