Regularized interacting scalar quantum field theories

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 which describes how much the large field contributions are suppressed in the interaction Lagrangian. We finally discuss how to remove the parameters in lower dimensional theories and for specific interaction Lagrangians. In particular, we show that in three spacetime dimensions for a $ϕ^4_3$ theory one obtains sequences of unitary operators which are weakly-$*$ convergent to suitable unitary operators in the limit of vanishing parameters. The coefficients of the asymptotic expansion in powers of the coupling constant of all the possible limit points coincide and furthermore agree with the predictions of perturbation theory. Finally we discuss how to extend these results to the case of a $ϕ^4_4$ theory were the final results turns out to be very similar to the three dimensional case.