Description
Title: Interacting regularised scalar quantum field theories: convergence of the S-matrix
Abstract: We discuss the construction of self-interacting scalar quantum field theories on d-dimensional Minkowski spacetime, focusing on a class of interaction Lagrangians given by suitable functions of the scalar field. Using perturbation theory, we express interacting field observables as formal power series expansions over the free theory. Central to this construction is the time-ordered exponential of the interaction Lagrangian—the S-matrix—which itself is a power series in the coupling constant.
We introduce a regularization procedure that renders the S-matrix convergent to well-defined unitary operators. This regularization involves two parameters: one controlling the suppression of high-frequency modes in the propagators, and another regulating large field contributions in the interaction Lagrangian. We analyze the removal of these regularization parameters in lower-dimensional theories and for specific interaction Lagrangians.
In particular, we show that for a φ4 theory in three spacetime dimensions, form sequences of regularized S-matrices obtained scaling the regularization parameters to zero we can extract convergent subsequences in the weak-* topology. The asymptotic expansions of all the possible limit points of the extracted subsequences agree and match the predictions of perturbation theory. We also outline how these results extend to the four-dimensional case, where similar behavior is observed.
