Quantum field theory accurately describes the world on the finest scales to which we have empirical access. There has been significant disagreement, however,about which mathematical structures ought to be taken as constitutive of the theory, and thus over which structures should serve as the basis for its interpretation. Perturbative methods allow for successful empirical prediction but require mathematical manipulations that are at odds with the canonical approach to interpreting physical theories that has been passed down from the logical positivists. Axiomatic characterizations of the theory, on the other hand, have not been shown to admit empirically interesting models. This dissertation shows how to understand the empirical success of quantum field theory by reconsidering widely held commitments about how physical meaning accrues to mathematical structure.
Quantum field theory accurately describes the world on the finest scales to which we have empirical access. There has been significant disagreement, however,about which mathematical structures ought to be taken as constitutive of the theory, and thus over which structures should serve as the basis for its interpretation. Perturbative methods allow for successful empirical prediction but require mathematical manipulations that are at odds with the canonical approach to interpreting physical theories that has been passed down from the logical positivists. Axiomatic characterizations of the theory, on the other hand, have not been shown to admit empirically interesting models. This dissertation shows how to understand the empirical success of quantum field theory by reconsidering widely held commitments about how physical meaning accrues to mathematical structure.
A standard approach towards interpreting physical theories proceeds by first identifying the theory with a set of mathematical objects, where such objects are defined according to mathematicians’ standards of rigor. In making this identification, philosophers rule out the relevance of many inferential methods that physicists use, as these often do not meet mathematicians’ standards of rigor. Philosophers thus sanitize physical theories of all math- ematically messy or ambiguous parts before interpreting them. My dissertation argues against this sanitized approach towards interpreting theories using the example of quantum field theory (QFT). When we look at the details of QFT, we find that the mathematical objects it requires differ according to the specific systems the theory is being applied to in ways that advocates of the sanitized approach do not anticipate. Furthermore, the mathematical objects required for successful application are still being developed in some applicational contexts, so it would be unwise to determine in advance which objects constitute the theory. During this ongoing developmental process, physicists interpret the mathematics using strategies that violate the standards of pure mathematics. In contrast to the sanitized approach, these strategies are more sensitive to the ways in which the mathematics required for the relevant contexts is still under development. I argue that these strategies are not merely instrumental. They suggest alternative approaches to interpretation that philosophers should take into account.
Quantum field theory (QFT) can be analyzed in terms of its mathematical structure, its conceptual system for physical descriptions, or its basic ontology. The analysis can be done logically or historically. In this chapter, only the genesis of the conceptual foundations of QFT relevant to its basic ontology will be treated; no discussion of its mathematical structures or its epistemological underpinnings will be given. Some conceptual problems, such as those related to probability and measurement, will be discussed, but only because of their relevance to the basic ontology of QFT, rather than their intrinsic philosophical interest. Here, by basic ontology I mean the irreducible entities that QFT is invented to describe. The often mentioned candidates for the basic ontology of QFT, in fact of the physical world, are the discrete particle and the continuous field. Another possible candidate (the spacetime point) has also been suggested recently (Redhead, 1983). Since the aim of this chapter is to analyze the historical process in which the conceptual foundations of QFT were laid down, rather than the logical structure of QFT which philosophers of the present day treat, no discussion of the last possibility will be given. The content of this chapter involves the formation and interpretation, in a roughly chronological order, of the concepts of the wave function, quantization, quantization of the field, the vacuum, interactions between fields, and renormalization. The first two topics will be discussed in relation to the quantization of the field, with their role being taken as the starting point of the conceptual development of QFT.
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.
This manuscript revisits quantum entanglement from a quantum field–theoretic perspective,with the aim of clarifying its conceptual status rather than proposing a new theory. The central claim is that quantum entanglement need not be understood as a mysteriousnonlocal phenomenon.When quantum fields are taken as the primary physical entities, entanglement followsnaturally from the non-separable structure of global field excitations.What appears puzzling in particle-based descriptions is thus revealed to be an expectedfeature of field-based ontology combined with reduced descriptions. The paper emphasizes a structural reading of the wavefunction as a description of fieldexcitations, and interprets entanglement as a consequence of restricting access to aglobally defined state.Locality is preserved at the level of operators, and all empirical predictions of quantumfield theory remain unchanged. No new dynamics, interpretations, or empirical claims are introduced.Instead, the work makes explicit a conceptual perspective that is already implicit instandard quantum field theory but is rarely articulated in a unified and accessible form.Formal details supporting the argument are provided in an appendix, while the main textfocuses on conceptual clarity. This work is intended as a contribution to the foundations of quantum theory, clarifyingthe relationship between entanglement, locality, and field-theoretic structure.
An Interpretive Introduction to Quantum Field Theory (Teller 1995; hereafter IIQFT) supersedes most of my prior work on quantum field theory. The gossip mill has described this book as a popularization of the most elementary parts of Bjorken and Drell (1965), which into the 1980s was the most widely used quantum field theory text. As with any good caricature, there is a great deal of truth in this comparison. Like Bjorken and Drell, who published in 1965, IIQFT presents the theory largely as it existed in the 1950s. But in order to see aspects of structure and interpretation more clearly, IIQFT presents the theory stripped of all the details needed for application. IIQFT also does not treat contemporary methods, such as the functional approach, and important recent developments, especially gauge theories and the renormalization group. Nonetheless it is hoped that by laying out the structure of the theory's original form in the 1950s, much of which survives in contemporary versions, and by developing a range of ways of thinking about that theory physically, one does essential ground work for a thorough understanding of what we have today. Why do we use the term ‘quantum field theory’? A good fraction of the work done in IIQFT aims to clarify the appropriateness, accurate development, and limitations of the application of the epithet ‘field’, as well as examination of alternatives. While called a ‘field theory’, quantum field theory (QFT) is also taken to be our most basic theory of ‘elementary particles’.
Abstract Quantum field theory (QFT) is the union of quantum mechanics and Einstein’ s special theory of relativity. QFT is the most fundamental physical theory and provides the answers to questions about the basic structure of the physical world. However, the way QFT describes the world is so different from the way our senses present it to us that it requires an overhaul of philosophic categories and arguments. How can we know the microscopic world without a satisfactory measurement theory? What are the general conditions of the world that make possible such knowledge? What are the presuppositions of physical theories? Dr.Auyang argues that the current relational substantial debate on space-time is inadequate and the observer-observed distinction in quantum interpretations illusory because they are hampered by philosophical constructs too narrow to accommodate the complicated structures of modern philosophical theories. Modern physical theories have incorporated many presuppositions of empirical knowledge, and consequently the theories inform us about the general nature of the world and out relation to it, in addition to information about elementary particles. The conceptual structure of QFT is extracted and articulated in commonsense terms, and used to answer many longstanding philosophical questions.