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arXiv:2608.19458

Renormalization group ontology in quantum foundations: a non-interacting spin-0 toy model

Gary Kapilevich

quant-phhep-thSubmitted Aug 19, 2026• Announced Aug 21, 2026• 0 citations
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Abstract

A realist, anti-psi interpretation of quantum field theory is introduced, based on an expanded ontology that was inspired by the renormalization group. A goal will be to work with a classical non-interacting scalar field "toy model" in the canonical ensemble and show that an observer in this expanded ontology, under certain limiting conditions, will see a QFT. The inverse renormalization group (IRG) will be key in both interpreting this expanded ontology and furnishing novel examples of QFTs. This will lead to a definition of Planck's constant that flows as we head towards a fixed point, is expressed in terms of a temperature, and depends on the observer being studied in this expanded ontology. Our hope is that this approach will lead both to insights into quantum foundations and to prescriptions for developing new QFTs.

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19 pages

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A realist, anti-psi interpretation of quantum field theory is introduced, based on an expanded ontology that was inspired by the renormalization group. A goal will be to work with a classical non-interacting scalar field "toy model" in the canonical ensemble and show that an observer in this expanded ontology, under certain limiting conditions, will see a QFT. The inverse renormalization group (IRG) will be key in both interpreting this expanded ontology and furnishing novel examples of QFTs. This will lead to a definition of Planck's constant that flows as we head towards a fixed point, is expressed in terms of a temperature, and depends on the observer being studied in this expanded ontology. Our hope is that this approach will lead both to insights into quantum foundations and to prescriptions for developing new QFTs.
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.
We show how one can reproduce the state evolution known from interacting QFT from an ontological deterministic theory. This is an important step in finding the reality that underlies QFT.In previous work ontic theories behind free QFTs were found, for all QFTs that occur in the Standard Model (spin-0, spin-1/2 & spin-1). In this paper we extend the ontological theory behind QFT to the case with interactions. We do this by starting out from the ontic theories found for free QFTs and see what happens if we try to include interactions (known from QFT). This will lead us to the conjecture that the ontic theory behind any QFT with interaction is, at the basis, the corresponding classical field theory. We study such a classical field theory for the case of a real scalar field with φn-interaction. We determine the evolution of the probability density of the ontic fields and cast it into a formalism known from QFT. We will focus on the simplest case of self-interaction, to illustrate our procedure. We find that we can reproduce the probability evolution from real scalar QFT with (self-)interaction, for this simplest case.
Introduction: Conceptual issues in quantum field theory Part I. Philosophers' Interests in Quantum Field Theory: 1. Why are we philosophers interested in quantum field theory 2. Quantum field theory and the philosopher Part II. Three Approaches to the Foundations of Quantum Field Theory: 3. The usefulness of a general theory of quantized fields 4. Effective field theory in condensed matter physics 5. The triumph and limitations of quantum field theory 6. Comments Discussions Part III. Does Quantum Field Theory Need a Foundation: 7. Does quantum field theory need a foundation? Part IV. Mathematics, Statistical Mechanics and Quantum Field Theory: 8. Renormalization group theory: its basis and formulation in statistical physics 9. Where does quantum field theory fit into the big picture? 10. The unreasonable effectiveness of quantum field theory 11. Comment: the quantum field theory of physics and of mathematics Part V. Quantum Field Theory and Spacetime: Introduction 12. Quantum field theory and spacetime: formalism and reality 13. Quantum field theory of geometry 14. 'Localization' in quantum field theory: how much of QFT is compatible with what we know about spacetime 15. Comments VI. 16. What is quantum field theory and what did we think it was? 17. Comments Discussions Part VII.Renormalization Group: 18. What is fundamental physics? A renormalization group perspective 19. Renormalization group: an interesting yet puzzling idea Part VIII. Non-Abelian Gauge Theory: 20. Gauge fields, gravity and Bohm's theory 21. Is the Aharonov-Bohm effect local? Discussions Part IX. The Ontology of Particles or Fields: 22. The ineliminable classical face of quantum field theory 23. The logic of quanta 24. Do Feynman diagrams endorse a particle ontology? 25. On the ontology of QFT Part X. Panel Discussion.
Philosophical interpretations of theories generally presuppose that a theory can be presented as a consistent mathematical formulation that is interpreted through models. Algebraic quantum field theory (AQFT) can fit this interpretative model. However, standard Lagrangian quantum field theory (LQFT), as well as quantum electrodynamics and nuclear physics, resists recasting along such formal lines. The difference has a distinct bearing on ontological issues. AQFT does not treat particle interactions or the standard model. This paper develops a framework and methodology for interpreting such informal theories as LQFT and the standard model. We begin by summarizing two minimal epistemological interpretation of non-relativistic quantum mechanics (NRQM): Bohrian semantics, which focuses on communicables; and quantum information theory, which focuses on the algebra of local observables. Schwinger's development of quantum field theory supplies a unique path from NRQM to QFT, where each step is conceptually anchored in local measurements. LQFT and the standard model rely on postulates that go beyond the limits set by AQFT and Schwinger's anabatic methodology. The particle ontology of the standard model is clarified by regarding the standard model as an informal modular theory with a limited range of validity.
After a brief review of the definition and properties of the quantum effective Hamiltonian action, we describe its renormalization flow by a functional renormalization group equation. This equation can be used for a nonperturbative quantization and study of theories with bare Hamiltonians that are not quadratic in the momenta. As an example, the vacuum energy and gap of quantum mechanical models are computed. Extensions of this framework to quantum field theories are discussed. In particular, one possible Lorentz-covariant approach for simple scalar field theories is developed. Fermionic degrees of freedom, being naturally described by a first order formulation, can easily be accommodated in this approach.
We study the non relativistic limit of a model of fermions interacting through a Chern-Simons field, from a perspective that resembles the Wilson Renormalization Group (WRG) approach. The main difference of our approach to WRG is that instead of working in the Euclidean space, we directly work in Minkowski space. For the nonrelativistic limit this procedure admits a more direct interpretation of the results. As we will see, this modified WRG approach solves some difficulties, and gives a new understanding of non relativistic models that are not clear in the usual "classical" nonrelativistic (CNR) limit found in most text books. The method is general and our choice of the Chern-Simons as the gauge field is only for the sake of simplicity of illustration.