Abstract
Haag’s theorem is traditionally viewed as a no-go theorem for the mainstream physicists’ approach to interacting quantum field theory, i.e. the interaction picture and its attendant methods of perturbation theory. Mainstream quantum field theory employs the interaction picture to model interactions. In this interaction picture, interacting fields are modeled as perturbations of free fields. Once the fundamental assumptions of this approach are made mathematically precise, it follows from these assumptions that the putatively interacting field must in fact be unitarily equivalent to the free field. This result, demonstrating that the interacting field is equivalent to the free field, is called Haag’s theorem. Thus, much of the philosophical literature interprets Haag’s theorem as a classic no-go result: mainstream physicists’ methods for modeling interactions are a no-go because of the fundamental assumptions of the interaction picture. And yet, mainstream physicists’ methods (making use of the interaction picture, perturbation theory, and regularization and renormalization techniques) have proved to be highly successful at modeling interactions by empirical standards. In recent work, [Duncan, 2012] and [Miller, 2018] explain this success by appealing to the calculational detail of regularization and renormalization techniques, arguing that these techniques invariably violate one or another of the assumptions that go into Haag’s theorem. Thus, regularization and renormalization seem to provide an evasion strategy for Haag’s theorem, as well as an explanation for the empirical success of mainstream methods. In light of these developments, this paper presents an alternative to the no-go interpretation of Haag’s theorem: Haag’s theorem is rather a howto theorem. The two readings are distinguished by the status taken by the fundamental assumptions for the theorem. While on a no-go reading these assumptions are strictly immutable, on a how-to reading they are subject to revision. The central consequence of the assumptions’ change in status reveals itself when we consider the empirical success of mainstream models of interaction. On the no-go reading, one is tempted to dismiss this success as a mirage precisely because they controvert the theorem’s assumptions. In contrast, on the how-to reading, the success is taken as evidence that the assumptions require revision. In short: no-go entails no success because assumptions are true; how-to entails success but only if some assumption is false. Thus, the latter reading, but not the former, leads naturally to questions of how precisely the assumptions must be modified in order for the theorem to be evaded. It is in this sense that it is a how-to reading. Thus, a how-to reading relies upon the attitude of opportunism at work in what [R´edei and St¨oltzner, 2006] call “soft axiomatisation.” By way of conclusion, we offer some reflections as to the general methodological and philosophical implications of adjudicating between no-go and how-to interpretations of theorems such as this.
