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arXiv:2212.06977

How Haag-tied is QFT, really?

Chris Mitsch, Marian J. R. Gilton, D. Freeborn

Published Dec 14, 2022• 0 citations
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Abstract

Haag's theorem cries out for explanation and critical assessment: it sounds the alarm that something is (perhaps) not right in one of the standard way of constructing interacting fields to be used in generating predictions for scattering experiments. Viewpoints as to the precise nature of the problem, the appropriate solution, and subsequently-called-for developments in areas of physics, mathematics, and philosophy differ widely. In this paper, we develop and deploy a conceptual framework for critically assessing these disparate responses to Haag's theorem. Doing so reveals the driving force of more general questions as to the nature and purpose of foundational work in physics.

DOI
10.48550/arxiv.2212.06977

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Haag's theorem cries out for explanation and critical assessment: it sounds the alarm that something is (perhaps) not right in one of the standard way of constructing interacting fields to be used in generating predictions for scattering experiments. Viewpoints as to the precise nature of the problem, the appropriate solution, and subsequently-called-for developments in areas of physics, mathematics, and philosophy differ widely. In this paper, we develop and deploy a conceptual framework for critically assessing these disparate responses to Haag's theorem. Doing so reveals the driving force of more general questions as to the nature and purpose of foundational work in physics.
Haag’s theorem cries out for explanation and critical assessment: It sounds the alarm that something is (perhaps) not right in one of the standard ways of constructing interacting fields to be used in generating predictions for scattering experiments. Viewpoints as to the precise nature of the problem, the appropriate solution, and subsequently-called-for developments in areas of physics, mathematics, and philosophy differ widely. In this paper, we develop and deploy a conceptual framework for critically assessing these disparate responses to Haag’s theorem. Doing so reveals the driving force of more general questions as to the nature and purpose of foundational work in physics.
Quantum field theory (QFT) is the physical framework that integrates quantum mechanics and the special theory of relativity; it is the basis of many of our best physical theories. QFT's for interacting systems have yielded extraordinarily accurate predictions. Yet, in spite of unquestionable empirical success, the treatment of interactions in QFT raises serious issues for the foundations and interpretation of the theory. This dissertation takes Haag's theorem as a starting point for investigatingthese issues. It begins with a detailed exposition and analysis of different versions ofHaag's theorem. The theorem is cast as a reductio ad absurdum of canonical QFT prior to renormalization. It is possible to adopt different strategies in response to this reductio: (1) renormalizing the canonical framework; (2) introducing a volume i.e., long-distance) cutoff into the canonical framework; or (3) abandoning another assumption common to the canonical framework and Haag's theorem, which is the approach adopted by axiomatic and constructive field theorists. Haag's theorem doesnot entail that it is impossible to formulate a mathematically well-defined Hilbert space model for an interacting system on infinite, continuous space. Furthermore, Haag's theorem does not undermine the predictions of renormalized canonical QFT; canonical QFT with cutoffs and existing mathematically rigorous models for interactions are empirically equivalent to renormalized canonical QFT. The final two chapters explore the consequences of Haag's theorem for the interpretation of QFT with interactions. I argue that no mathematically rigorous model of QFT on infinite, continuous space admits an interpretation in terms of quanta (i.e., quantumparticles). Furthermore, I contend that extant mathematically rigorous models for physically unrealistic interactions serve as a better guide to the ontology of QFT than either of the other two formulations of QFT. Consequently, according to QFT, quanta do not belong in our ontology of fundamental entities.
Haag's theorem is a classic no-go theorem. It rigorously demonstrates there is a logical problem with the interaction picture (IP), one of the most widely used modeling tools in quantum field theory (QFT). The significance of the theorem for the use of the IP in QFT has been the subject of long-running debate, focused around how ``worried'' we should be. In this paper, we argue for an alternative and opposite perspective on Haag's theorem, rejecting the `worry' framing in favor of emphasizing the no-go theorem's implications for model development.
The road on the foundations of science in general consists in (a) making precise what the assumptions are one makes resulting from our measurements (b) holding a “good” balance between theoretical assumptions and genericity of predictions (c) saying as precisely as possible what you mean. Unfortunately, recent work where these three criteria are met is scarce and I often encounter situations where physicists talk about different things in the same words or the other way around, identify distinct concepts (even without being aware of it), or introduce unnecessary hypothesis based upon a too stringent mathematical interpretation of some observation. In this work, I will be as critical as possible and give away those objections against modern theories of physics which have become clear in my mind and therefore transcend mere intuition. All these objections result from the use of unclear language or too stringent assumptions on the nature of reality. Next, we weaken the assumptions and discuss what I call process physics; it will turn out that Bell’s concerns do find a natural solution within this framework.
Haag’s theorem has been interpreted as establishing that quantum field theory cannot consistently represent interacting fields. Earman and Fraser have clarified how it is possible to give mathematically consistent calculations in scattering theory despite the theorem. However, their analysis does not fully address the worry raised by the result. In particular, I argue that their approach fails to be a complete explanation of why Haag’s theorem does not undermine claims about the empirical adequacy of particular quantum field theories. I then show that such empirical adequacy claims are protected from Haag’s result by the techniques that are required to obtain theoretical predictions for realistic experimental observables. I conclude by showing how Haag’s theorem is illustrative of a general tension between the foundational significance of results that can be obtained in perturbation theory and non-perturbative characterizations of the content of quantum field theory. 1 Introduction2 Haag’s Theorem and the Interaction Picture3 Earman and Fraser on the Success of Scattering Theory4 Haag’s Theorem and Empirical Adequacy5 Conclusion