F eb 1 99 5 p - Adic Field Theory limit of TGD is free of UV divergences Matti Pitkänen

The p-adic description of Higgs mechanism in TGD framework provides excellent predictions for elementary particle and hadrons masses ([email protected] 9410058-62). The gauge group of TGD is just the gauge group of the standard model so that it makes sense to study the p-adic 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 p-adic norm not larger than one so that the contribution of loop momentum squared with p-adic norm p converges as p for boson loop. The existence of the action exponential forces number theoretically the decomposition of action into free and interacting parts. The free part is of order O(p) and must vanish (and does so by equations of motion) and interaction part is at most of order O( √ p) p-adically. p-Adic coupling constants are of form g √ p: their real counterparts are obtained by canonical identification between p-adic and real numbers. The discretized version of Feynmann rules of real theory should give S-matrix elements but Feynmann rules guarantee unitarity in formal sense only. The unexpected result is the upper bound Lp = L0/ √ p (L0 ∼ 10 √ G) for the size of p-adic convergence cube from the cancellation of infrared divergences so that p-adic field theory doesn’t make sense above length scale Lp.