p-Adic Field Theory limit of TGD is free of UV divergences

الأصل · EN

The p-adic description of Higgs mechanism in TGD framework provides excellent predictions for elementary particle and hadrons masses (hep-th@xxx.lanl.gov 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 p-adic 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 momentum squared with p-adic norm pᵏ converges as p⁻²ᵏ⁻² 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(p⁰) 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 10⁴√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.