New techniques for worldline integration (Q2244749)

This article reviews the formulation of quantum field theories like quantum electrodynamics or scalar quantum electrodynamics (scalars and photons) in the worldline formalism. The worldline formalism dates back to Feynman, with two papers from 1950 and 1951, respectively. The article by Edwards, Mata, Müller and Schubert summarises the current state-of-the-art and addresses open problems. The worldline formalism has the advantage that it offers a nice compact integral representation for the amplitude, the ``Bern-Kosower master formula'' of eq. (3.2). The authors address then the open problem on how to process this formula further without re-expanding it into ordered sectors. They first discuss the case of a polynomial integrand, where a systematic method can be given. As a spin-off this method leads to interesting relations among Bernoulli numbers (the Miki relations and the Faber-Pandharipande-Zagier relations). However, the master formula of Bern-Kosower involves an exponential. In principle one could expand the exponential. Each term of the expansion is then a polynomial and could be treated with the method above. However, it is not obvious if the infinite sum from the expansion can be re-arranged in a closed form, spoiling the initial advantage of a short compact expression. For this reason, the authors advocate a different route: Expansion in inverse derivatives. In order to make this well-defined, they first take out the zero mode. In the last section of this article they report on the current state-of-the-art of the technique of the expansion in inverse derivatives.