A brief history of Euclidean quantum field theory: successes, failures, a long period of oblivion and new perspectives

by Prof. Rodolfo Figari

Wednesday 9 Apr 2025, 16:00 → 17:00 Europe/Rome

0M03

Constructive quantum field theory (CQFT) was formulated between the late 1950 and the late 1970 as an attempt to find specific examples of non-linear quantum fields that could be described mathematically within the framework of quantum mechanics. Based on a suggestion by Symanzik, several authors were able to prove that, in the case of bosons, the vacuum expectation values of field products, when analytically continued (t $\rightarrow$ it) into the Euclidean region, can be expressed as random distributions on Euclidean space with locality properties reducing, in the one-space-dimensional case  to the ordinary Markoff property. The convenience of dealing with the Euclidean group, with its positive-definite scalar product, instead of the Lorentz group is evident, and has been used by several authors, in different contexts.  In this discussion, I will briefly mention the successes in constructing non-trivial models of relativistic  bosons interacting through polynomial interactions in both a $1+1$ dimensional Minkowski space and a $1+1$ dimensional de Sitter Universe. I will also touch upon the lack of success in a similar project in $3+1$ space-time dimensions, which led to a decline in interest in boson CQFT. Analyzing recent results on the formal non-relativistic limit of field theories I will try to convince the audience that the quartic terms in the Lagrangian densities used to describe the interaction among bosons are not defined and that different renormalizations can resolve the ultraviolet divergences.