Abstract
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.
