What good is a no-go theorem?

The perils of excessive idealization in constructing the underlying mathematical framework for fundamental physical theories are illustrated with some examples taken from relativistic quantum field theory: the triviality issue for standard model field theories, the nonexistence of the S-matrix in quantum electrodynamics, and Haag’s no-go theorem for the interaction picture formulation of relativistic field theories. It is argued that in all cases, known physical limitations of the theory, once taken into account, remove the apparent failure of the formalism, allowing phenomenologically relevant calculations to be made.

The tension between axiomatic, mathematically rigorous formulations of quantum field theory (AQFT) and Lagrangian quantum field theory (LQFT) as employed in the Standard Model of particle physics has been much discussed by philosophers and physicists alike. While debate was heated in the last decade (Fraser 2011, Wallace 2011), a sort of peaceful coexistence has been achieved. In the case of Haag’s theorem, [Duncan, 2012] and [Miller, 2018] offer similar characterizations of this coexistence: while Haag’s theorem undermines the interaction picture for a fully-Poincar´e QFT satisfying the Wightman axioms, LQFTs employ regularization and renormalization techniques that break Poincar´e invariance (among other things). Though the AQFT theorems are useful, they do not directly apply to LQFT, and philosophers must use new methods to better understand LQFT as used in practice. Algebraic/axiomatic methods may still lead to important insights into LQFT, but they are only useful insofar as they provide insight to LQFT [Wallace, 2006]. In this talk I will articulate and defend an attitude of Cautious Optimism for the relationship between AQFT and LQFT, and use Haag’s theorem as a test case. The Cautious Optimist thinks that AQFT (or some suitable modification) captures the essence of relativistic QFT. While it may be difficult or even impossible to construct exact models of the realistic interactions described with LQFT, the framework of AQFT provides insight and guidance to understanding QFT more broadly. Haag’s theorem in particular seems to provide a major obstacle to Cautious Optimism, more so than the failed attempts to explicitly relate AQFT and LQFT, as it is an explicit no-go theorem for using the interaction picture. I will show how one can reconcile the Duncan and Miller style solution to the dilemma of Haag’s theorem with a Cautious Optimism regarding the relationship between AQFT and LQFT. This requires a specific interpretation of regularization and renormalization techniques as calculational tools, rather than representing something physical about the Standard Model. This strategy preserves the relevance of AQFT results like the CPT theorem, and justifies the conventional understanding of QFT as the only way to construct relativistic quantum theory of particle interactions [Malament, 1996, Fraser, 2008] Finally, I end by discussing the value of Haag’s theorem for understanding the necessity of unitarily inequivalent representations in QFT.

Haag’s theorem is a valuable result for foundations of physics because it supplies direct information about relativistic QFT, which is a framework theory. Constructing theories within this framework (e.g., lattice QCD, φ 2 4 , the Standard Model) using the wide variety of strategies found in mainstream and mathematical physics is a way of indirectly learning about the foundations of the framework theory. Philosophers and physicists studying the foundations of QFT need to piece together information from both direct and indirect sources. However, directly informative sources, such as Haag’s theorem, do have some advantages. Like other ‘no go’ theorems in physics, Haag’s theorem takes the logical form of a reductio argument. This means that there is a clear negative lesson about the wrong way to represent relativistic quantum systems, and also a set of possible positive lessons about the right way(s) to represent relativistic quantum systems. As with any reductio, the possible positive lessons are delimited by the choices of either rejecting one (or more) of the premises of the theorem or biting the bullet and accepting the apparently unacceptable conclusion. This clear logical picture is complicated by the fact that there are different versions of Haag’s theorem which appear to rest on different sets of premises. The most well-known version of the theorem is the one proven in [Hall and Wightman, 1957] using the Wightman axiomatization. The essence of this version is that in relativistic QFTs with interactions, vacuum polarization necessarily occurs. However, there are other versions of Haag’s theorem that are presented as generalizing the theorem beyond relativistic QFT. [Emch, 1972] proves a version of Haag’s theorem within the algebraic framework that is based on [Streit, 1969]. Streit remarks that this generalization “essentially consists in dropping not only locality [i.e., microcausality] but relativistic covariance altogether” (674). Another example is [Schrader, 1974]’s Euclidean version of Haag’s theorem within Euclidean field theory, which in some circumstances can be interpreted as classical statistical mechanics [Guerra et al., 1975]. What should we make of these results? As I will explain, there is a respect in which Haag’s theorem can be regarded as a deep general result about the representation of symmetries in framework theories that goes beyond relativistic QFT. I will also clarify the relativistic premises that are needed to prove Haag’s theorem within relativistic QFT. One reason that this clarification is important is that relativistic principles pose more than one obstacle to constructing QFTs with interactions. Distinguishing the obstacles can help to inform theory construction strategies.

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.