Year: 2025
Type: preprint
Abstract: In this paper we consider self-interacting scalar quantum field theories over a $d$-dimensional Minkowski spacetime with various interaction Lagrangians which are suitable functions of the field. The interacting field observables are represented as power series over the free theory by means of perturbation theory. The object which is employed to obtain this power series is the time-ordered exponential of the interaction Lagrangian which is the $S$-matrix of the theory and thus itself a power series in the coupling constant of the theory. We analyze a regularization procedure which makes the $S$-matrix convergent to well-defined unitary operators. This regularization depends on two parameters. One describes how much the high-frequency contributions in the propagators are tamed and a second one describes how much the large-field contributions are suppressed in the interaction Lagrangian. We finally discuss the limits in which the parameters are removed in some lower-dimensional theories and for specific interaction Lagrangians. In particular, we show, by an abstract argument, that in three spacetime dimensions for a $ϕ^4_3$ theory one obtains sequences of unitary operators which are weakly-$*$ convergent to bounded operators in the limit of vanishing parameters. Although we lose control over the unitarity of the operator obtained in the limit and although the uniqueness of the limit is not under control, we notice that some basic causal properties are satisfied by the corresponding interacting fields. We furthermore observe that the very same limits, applied at fixed order, give back the ordinary perturbation theory results. Finally, we discuss a possible extension of these results to the case of a $ϕ^4_4$ theory.
Source: arXiv (Cornell University)
Author: Pinamonti, Nicola
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407806391