Perturbative classical and quantum field theory

TL;DR

This paper explores classical phi^{p+1} field theory using Butcher series for perturbative solutions and demonstrates how to formally derive the interacting quantum field from this classical expansion.

Contribution

It introduces a method to explicitly compute and prove convergence of classical solutions and connects classical perturbative expansions to quantum fields via Butcher series.

Findings

Perturbative solutions converge for small coupling constants.

Heisenberg's interacting quantum field can be formally derived from classical expansions.

Provides a bridge between classical and quantum field theories using series methods.

Abstract

In a first part we study the phi^{p+1}--field theory from the classical point of view. Using Butcher series we compute explicitly the perturbative expansion of the solutions and we prove that this expansion converges if the coupling constant is small enough. Then we show that we can formally recover the Heisenberg's interacting quantum field directly from this expansion. In other words we write the Heisenberg interacting field as a Butcher series.