The development of renormalization group methods for particle physics: Formal analogies between classical statistical mechanics and quantum field theory

Analogies between classical statistical mechanics (CSM) and quantum field theory (QFT) played a pivotal role in the development of renormalization group (RG) methods for application in the two theories. This paper focuses on the analogies that informed the application of RG methods in QFT by Kenneth Wilson and collaborators in the early 1970's (Wilson and Kogut 1974). The central task that is accomplished is the identification and analysis of the analogical mappings employed. The conclusion is that the analogies in this case study are formal analogies, and not physical analogies. That is, the analogical mappings relate elements of the models that play formally analogous roles and that have substantially different physical interpretations. Unlike other cases of the use of analogies in physics, the analogical mappings do not preserve causal structure. The conclusion that the analogies in this case are purely formal carries important implications for the interpretation of QFT, and poses challenges for philosophical accounts of analogical reasoning and arguments in defence of scientific realism. Analysis of the interpretation of the cutoffs is presented as an illustrative example of how physical disanalogies block the exportation of physical interpretations from from statistical mechanics to QFT. A final implication is that the application of RG methods in QFT supports non-causal explanations, but in a different manner than in statistical mechanics

The development of renormalization group methods for particle physics: Formal analogies between classical statistical mechanics and quantum field theory

Analogies between classical statistical mechanics (CSM) and quantum field theory (QFT) played a pivotal role in the development of renormalization group (RG) methods for application in the two theories. This paper focuses on the analogies that informed the application of RG methods in QFT by Kenneth Wilson and collaborators in the early 1970's (Wilson and Kogut 1974). The central task that is accomplished is the identification and analysis of the analogical mappings employed. The conclusion is that the analogies in this case study are formal analogies, and not physical analogies. That is, the analogical mappings relate elements of the models that play formally analogous roles and that have substantially different physical interpretations. Unlike other cases of the use of analogies in physics, the analogical mappings do not preserve causal structure. The conclusion that the analogies in this case are purely formal carries important implications for the interpretation of QFT, and poses challenges for philosophical accounts of analogical reasoning and arguments in defence of scientific realism. Analysis of the interpretation of the cutoffs is presented as an illustrative example of how physical disanalogies block the exportation of physical interpretations from from statistical mechanics to QFT. A final implication is that the application of RG methods in QFT supports non-causal explanations, but in a different manner than in statistical mechanics

Haag's Theorem and the Interpretation of Quantum Field Theories with Interactions

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

The non-miraculous success of formal analogies in quantum theories

The Higgs model was developed using purely formal analogies to models of superconductivity. This is in contrast to historical case studies such as the development of electromagnetism, which employed physical analogies. As a result, quantum case studies such as the development of the Higgs model carry new lessons for the scientific realism--anti-realism debate. I argue that, by breaking the connection between success and approximate truth, the use of purely formal analogies is a counterexample to two prominent versions of the 'No Miracles' Argument (NMA) for scientific realism, Psillos' refined explanationist defence of realism and the Argument from History of Science for structural realism (Frigg and Votsis 2011). The NMA is undermined, but the success of the Higgs model is not miraculous because there is a naturalistically acceptable explanation for its success that does not invoke approximate truth. I also suggest some possible strategies for adapting to the counterexample for scientific realists who wish to hold on to the NMA in some form

Justifying the use of purely formal analogies in physics

Recent case studies have revealed that purely formal analogies have been successfully used as a heuristic in physics. This is at odds with most general philosophical accounts of analogies, which require analogies to be physical in order to be justifiably used. The main goal of this paper is to supply a philosophical account that justifies the use of purely formal analogies in physics. Using Bartha’s (2010) articulation model as a starting point, I offer precise definitions of formal and physical analogies and propose a new submodel of analogical reasoning that accounts for the successful use of purely formal analogies in the development of renormalization group methods for use in particle physics in the early 1970’s (Fraser 2020). Two distinctive features of this new applied mathematics submodel for analogical reasoning are that the conclusion of the argument from analogy includes both an entire model (and not only a hypothesis or a prediction) and the construction procedure for this model. A third important difference from arguments from physical analogy is that only the prima facie plausibility of the conclusion is established, and not stronger types of plausibility associated with confirmation. The use of purely formal analogies is justified because they are suited to supporting conclusions of this sort. Formulating a general philosophical account of analogies that covers purely formal analogies also serves two additional purposes: (1) to highlight the features of this case that are novel in the context of examples of analogies traditionally considered by philosophers and (2) to establish that physicists should not automatically dismiss purely formal analogies when evaluating heuristics for the development of new models

Eliminating the ‘impossible’: Recent progress on local measurement theory for quantum field theory

Arguments by Sorkin and Borsten, Jubb, and Kells establish that a natural extension of quantum measurement theory from non-relativistic quantum mechanics to relativistic quantum theory leads to the unacceptable consequence that expectation values in one region depend on which non-selective measurement is performed in a spacelike separated region. Sorkin labels such scenarios `impossible measurements'. We explicitly present these arguments as a no-go result with the logical form of a reductio argument and investigate the consequences for measurement in quantum field theory (QFT). Sorkin-type impossible measurement scenarios clearly illustrate the moral that Microcausality is not by itself sufficient to rule out superluminal signalling in relativistic quantum theories that use L{\"u}ders' rule. We review three different approaches to formulating an account of measurement for QFT and analyze their responses to the `impossible measurements' problem. Two of the approaches are recent proposals for measurement theories for QFT: a measurement theory based on detector models proposed in Polo-G{\'o}mez, Garay, and Mart{\'i}n-Mart{\'i}nez and a measurement framework for algebraic QFT proposed in Fewster and Verch. Of particular interest for foundations of QFT is that they share common features that may hold general morals about how to represent measurement in QFT, in spite of being much different in spirit and in details such as the form taken by the state update rules. Careful attention to the dynamics is an important component of both strategies for responding to the `impossible measurements' problem. Both also abandon the traditional operational interpretation of a local algebra of observables $\mathcal{A}(O)$ as representing possible operations carried out in region $O$. Their respective state update rules cannot be literally interpreted as representing a physical change of state of the system upon measurement that occurs in any region of spacetime. The third response to the `impossible measurements' problem that we examine is the histories-based approach that is preferred by Sorkin. While there are open questions about how to address the `impossible measurements' problem using this approach, it is much different in spirit than the detector models approach and the FV framework yet also shares some common features. We hope that this paper lays the groundwork for productive dialogue among the many communities of physicists and philosophers who are working on theoretical, practical, and interpretative issues surrounding the treatment of local measurements in QFT

Note on episodes in the history of modeling measurements in local spacetime regions using QFT

The formulation of a measurement theory for relativistic quantum field theory (QFT) has recently been an active area of research. In contrast to the asymptotic measurement framework that was enshrined in QED, the new proposals aim to supply a measurement framework for measurements in local spacetime regions. This paper surveys episodes in the history of quantum theory that contemporary researchers have identified as precursors to their own work and discusses how they laid the groundwork for current approaches to local measurement theory for QFT