Mythic Power Series: the mathematical structure arising from renormalized perturbation theory

Description
Perturbation theory in renormalizable quantum field theory is an expansion in powers of a mythical quantity – the “renormalized couplant” – that does not exist, at least not uniquely.  There are infinitely many renormalized couplants, corresponding to different renormalization schemes, which are all a priori on an equal footing.  In principle, physical predictions of the theory are independent of which renormalized couplant is used (“RG invariance”).  However, finite-order approximants created by truncating the perturbation series are not invariant.  I argue that mythic power series are not just ordinary power series with an annoying, minor complication, but a new mathematical structure of great richness.  The proper definition of the partial sum of a mythic power series, I argue, requires an “optimization” procedure to make the approximant locally RG invariant.  I present an analytic solution to the resulting “optimization equations” and outline a practical algorithm.  The deep question of convergence of mythic power series (the conditions under which the sequence of optimized approximants approaches a finite limit) is touched upon.
Organised by Maurizio Consoli