On the coordinate (in)dependence of the formal path integral

When path integrals are discussed in quantum field theory, it is almost always assumed that the fields take values in a vector bundle. When the fields are instead valued in a possibly-curved fiber bundle, the independence of the formal path integral on the coordinates becomes much less obvious. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the "formal" or "Feynman-diagrammatic" path integral construction. We then outline a proof of the following claim: the formal path integral does not depend on the choice of coordinates, but only on a choice of fiberwise volume form. Our outline is an honest proof when the formal path integral is defined without ultraviolet divergences.Comment: 9 pages, uses Tik

Surreal numbers with derivation, Hardy fields and transseries: a survey

The present survey article has two aims: - To provide an intuitive and accessible introduction to the theory of the field of surreal numbers with exponential and logarithmic functions. - To give an overview of some of the recent achievements. In particular, the field of surreal numbers carries a derivation which turns it into a universal domain for Hardy fields