A Modern Approach to Nonrenormalizable Models

Abstract

In the preceding chapter we discussed a sequence of quartic self-interacting relativistic scalar fields that range from super renormalizable models (for spacetime dimension n = 2, 3), to strictly renormalizable models (n = 4), and finally to nonrenormalizable models (n ≥ 5). In the super renormalizable cases, the perturbative solutions we outlined agree completely with the rigorous results obtained by more sophisticated, nonperturbative techniques. Thus these cases are unambiguous and are considered solved (from an existence point of view, at least). The strictly renormalizable case exhibits all the expected features of an acceptable perturbation analysis, but there is good reason to believe that it does not agree with the nonperturbative solution. That is, the nonperturbative solution is not equal to the one derived by a perturbation theory, and as renormalization group and Monte Carlo studies have indicated, the nonperturbative strictly renormalizable model leads to a free theory in the continuum limit. By summing a suitable set of infinitely many perturbation terms, even the perturbation theory can be made to suggest that the continuum limit is also a free theory. Moreover, when we examine the nonrenormalizable cases from a nonperturbative viewpoint and limit the set of counterterms to include only mass and coupling constant modifications, they too can be rigorously shown to approach free theories in the continuum limit.