From Feynman rules to conserved quantum numbers, III

Abstract

The present article complements the earlier ones in this series. The first part contains various results on the constituent system Cκ(M) of a graph model M, and on its feasibility system Iη(M) (which comprises a number of identities that define the number conservation rules). Those results include the general form of the (particle) number conservation rules in models without explicit propagator mixing.A few types of graph models (including self-conjugate and quasi-normal models) are defined. Oversimplifying, a model M is self-conjugate if it has a certain (weak) combinatorial type of C-symmetry, and M is quasi-normal if Iη(M) fully determines for which multisets of coloured, unlinked half-edges there is a graph in M with that exact multiset. It is shown not only that every self-conjugate model is quasi-normal, but also that some extensions of self-conjugate models are still quasi-normal. Most conveniently, self-conjugate models (and even some extended models) may be recognized in polynomial time.These results seem to lead to the following conclusion: for many (possibly ‘most’ of the) consistent, relevant QFT models, a complete correlation function 〈F〉 is graphical — i.e. there exist Feynman diagram(s) for 〈F〉 — if and only if 〈F〉 is allowed by the number conservation rules (i.e. by a complete system of such rules). Since these rules can be computed in polynomial time then, for many QFT models, deciding whether 〈F〉 is graphical also takes polynomial time.