How to Define Generalized Feynman Diagrams

Abstract

Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution.There are however several formidable looking challenges left. 1. One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta. 2. One must define the functional integral also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally. In this article a proposal giving excellent hopes for achieving these challenges is discussed.