Abstract
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.
