⚛️ quantum physics

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

This paper proposes a realist, anti-wavefunction interpretation of quantum field theory grounded in an expanded ontology inspired by the renormalization group, using a non-interacting scalar field toy model to demonstrate how Planck's constant emerges as a temperature-dependent, observer-relative quantity flowing toward a fixed point.

Original authors: Gary Kapilevich

Published 2026-08-21

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CC BY 4.0

Original authors: Gary Kapilevich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the grand architecture of modern physics, two great pillars stand apart. One is quantum mechanics, the rulebook for the very small, where particles exist in clouds of probability and seem to vanish from one place to appear in another without crossing the space in between. The other is the theory of fields, which describes how forces like electromagnetism and gravity ripple through the universe. For decades, physicists have tried to merge these two into a single, seamless description of reality, a quest that has led to the development of quantum field theory. Yet, a deep mystery remains: why does the universe behave so strangely at the smallest scales? Why is there a fundamental limit to how much we can know about a particle's position and speed at the same time? This limit is governed by a tiny number known as Planck's constant, a value that seems to appear out of nowhere, dictating the rules of the quantum game. Most interpretations of quantum mechanics treat this constant as a fundamental, unchangeable feature of nature, or they rely on the existence of a wavefunction—a mathematical description of all possible states of a system—that collapses into a single reality only when observed.

A new perspective, proposed by physicist Gary Kapilevich, suggests that these strange quantum rules are not fundamental at all, but rather emerge from a deeper, more ordinary reality. Imagine a vast, classical universe made of smooth, continuous fields, much like the ripples on a pond, but existing in a state of constant thermal jiggling, like a hot gas. In this view, the universe is not inherently quantum; it is a classical system that becomes quantum only when viewed through a specific lens. This lens is provided by a mathematical tool called the renormalization group, which physicists use to understand how a system looks different depending on the scale at which it is observed. When you zoom out, you ignore the tiny, frantic details of individual atoms and see only the smooth flow of the water. Kapilevich's work asks a radical question: what if the "quantum" nature of our universe is simply the result of an observer who cannot see the full picture?

The paper introduces a "toy model," a simplified version of reality that strips away the complexity of interacting particles to focus on a single, non-interacting field. In this model, the universe is filled with a classical field in three spatial dimensions that has a high-energy limit and a low-energy limit. The key idea is that different observers exist at different points within this hierarchy of energy scales. One observer, let's call her Alice, might be situated at a scale where she can only see the low-energy, large-scale features of the field. To describe the physics she sees, she must mathematically "integrate out" or ignore the high-energy, tiny-scale fluctuations that she cannot access. This process of ignoring the small details is standard in physics, but Kapilevich proposes that when an observer does this in a specific way, something remarkable happens. The classical field she is left with begins to behave exactly like a quantum field.

The mechanism relies on a specific set of conditions. First, the observer must ignore a vast range of high-energy modes, effectively cutting off the view of the universe at a certain scale. Second, the system must be in a state where the influence of the ignored parts is felt from "very far away," a condition the author calls the radiation zone limit. Third, the system must settle into a state of self-similarity, where the physics looks the same regardless of how much you zoom in or out, a state known as a fixed point. When these conditions are met, the temperature of the classical system, which usually just measures how hot the field is, transforms into what we recognize as Planck's constant. In this framework, the quantum uncertainty we observe is not a fundamental property of nature, but a consequence of the observer's ignorance of the high-energy modes they have integrated out. The "collapse" of a quantum wavefunction, the moment a probability becomes a definite reality, is reinterpreted as the observer gaining information about a state that was actually determined by interactions with the hidden, high-energy parts of the universe.

The paper explores this idea by constructing a mathematical model of a non-interacting scalar field, a simple type of field that does not have the complications of particle collisions. By applying the renormalization group to this three-dimensional field, the author shows that the resulting equations for the field's behavior, under specific limiting conditions, match those of a quantum field theory in four-dimensional spacetime. The speed of light and Planck's constant, usually treated as fixed universal numbers, are proposed here as values that could depend on the temperature of the system and the specific scale at which the observer is looking. However, the author explicitly notes that in this simplified non-interacting toy model, the speed of light is not actually renormalized and is set to a constant value; the proposal for its flow is a speculative argument intended for future generalization to interacting theories. If the observer were to integrate out even more modes, moving further up the energy ladder, the effective Planck's constant would change, and the system would eventually look classical again. This suggests a universe where quantum mechanics is a local phenomenon, a perspective-dependent view of a deeper, deterministic reality.

The model also addresses the famous "Wigner's friend" thought experiment, which questions whether a person inside a sealed room can be in a quantum superposition while an outside observer sees them as having a definite state. In this new framework, the answer depends entirely on where the observers are situated in the energy hierarchy. If the outside observer has access to more information about the field's modes than the person inside, they might see a classical, deterministic world. If they both lack access to the same high-energy modes, they will both see a quantum world, but their descriptions of reality might differ based on exactly which modes they have ignored. The paper argues that there is no single, objective wavefunction that describes the entire universe for everyone. Instead, there is a "Matryoshka universe," a nested structure where one observer's high-energy physics is another observer's low-energy observable universe.

Crucially, the author emphasizes that this is a realist interpretation. It does not rely on the idea that reality is created by observation, nor does it require the existence of multiple parallel universes. Instead, it posits that a single, classical reality exists, but our access to it is limited by the energy scales we can probe. The quantum weirdness we see is the shadow cast by the parts of the universe we cannot see. The paper acknowledges that this is a simplified model using a non-interacting field, and that the full complexity of the real world, with its interacting particles and gravity, has not yet been fully worked out. In fact, the paper explicitly excludes general relativity and notes that applying the formalism to gravity remains an open question. However, it provides a concrete pathway for how quantum mechanics and special relativity could emerge from a classical statistical system. By treating Planck's constant not as a fundamental constant but as a variable that flows with the observer's perspective, the work offers a fresh way to think about the foundations of physics. It suggests that the divide between the classical and the quantum is not a wall in nature, but a horizon of knowledge, determined by how much of the universe's hidden machinery an observer is able to see.

Technical Summary: Renormalization Group Ontology in Quantum Foundations

Problem Statement
The paper addresses the foundational question of how quantum field theory (QFT) and the specific constants of nature (Planck's constant $\hbar$ and the speed of light $c$ ) might emerge from a classical, realist framework. Current interpretations often rely on the wavefunction ( $\psi$ ) as a fundamental ontological entity (psi-ontology) or require non-unitary dynamics (e.g., collapse models). The author proposes an alternative "anti-psi" ontology where the universe is described by a classical field, but the nature of the physics observed depends on the observer's position within a Renormalization Group (RG) hierarchy. The central problem is to demonstrate how a non-interacting, non-relativistic classical scalar field in a canonical ensemble can, under specific limiting conditions and observer constraints, manifest as a QFT in Minkowski spacetime.

Methodology
The paper employs a "toy model" approach using a non-interacting scalar field in three-dimensional space with a canonical ensemble (temperature $T$ ). The methodology relies on the following structural elements:

RG Hierarchy and Nested Observers: The model posits a hierarchy of momentum modes $\Lambda' \ll \Lambda_1 \ll \dots \ll \Lambda$ . Observers are situated at specific scales within this hierarchy.
- UV Observers (e.g., Alice): Integrate out high-energy modes ( $\Lambda > \Lambda_1$ ) using standard Renormalization Group (RG) flow.
- IR Observers (e.g., Bob): Integrate out low-energy modes ( $\Lambda' < \Lambda_1$ ) using the Inverse Renormalization Group (IRG).
- The core hypothesis is that one observer's "UV field" (after integrating out high modes) is another observer's "IR field" (after integrating out low modes), creating a "Matryoshka universe" where high-energy physics for one is the observable universe for another.
Three Postulates for Emergent QFT: To transition from the classical ensemble to a QFT, the author introduces three specific postulates applied to the field:
- P1: Integrate out high-energy modes up to a cutoff $\Lambda_1$ .
- P2: Take the "radiation zone limit" where the low-energy cutoff $\Lambda'$ is much smaller than $\Lambda_1$ , which is much smaller than the high-energy cutoff $\Lambda$ ( $\Lambda' \ll \Lambda_1 \ll \Lambda$ ).
- P3: Allow RG flow toward a self-similar fixed point.
Analytical Continuation and Parameter Flow: The paper utilizes the relationship between a Euclidean classical field and a Minkowski QFT via Wick rotation. Crucially, it proposes that Planck's constant $\hbar$ is not fundamental but emergent, defined as proportional to the temperature $T$ at the fixed point ( $\hbar \propto T$ ). Similarly, the speed of light $c$ is treated as a renormalized parameter that decreases as modes are integrated out. The paper explicitly notes that while it defines heuristic expressions for $\hbar$ and $c$ (e.g., $\hbar \approx T\gamma/\epsilon$ ), it does not derive these for the non-interacting model nor provide a method for consistently renormalizing them in the presence of interactions; these are presented as speculative arguments for future generalization.
Active vs. Passive Pictures:
- Passive Picture: Two different observers (Alice and Bob) view the same underlying field but integrate out different modes, leading to different effective theories (e.g., different fixed points).
- Active Picture: A single observer models two distinct fields by integrating out either UV or IR modes from a general theory.

Key Results
The paper derives the following results within the non-interacting scalar toy model:

Emergence of the Propagator: By applying the postulates to the Green's function, the author shows that the propagator for the UV field (Alice's perspective) converges to the form of a modified Bessel function of the second kind, $K_1(\mu r)$ , which reproduces the causal structure of a non-interacting scalar QFT propagator.
Definition of $\hbar$ and $c$ : The paper proposes heuristic expressions where $\hbar \approx T\gamma/\epsilon$ and $c \approx c_0 \gamma^{1/4}$ (where $\gamma$ is a field scaling factor and $\epsilon$ is a small parameter from contour integration). In the non-interacting limit, $\gamma$ is constant, and the paper sets $\hbar$ and $c$ to 1. The framework speculates that in interacting theories, wavefunction renormalization would cause $\hbar$ and $c$ to flow, decreasing as more modes are integrated out, but the paper explicitly states it does not derive these flows or provide a consistent renormalization method for interacting fields.
Fixed Points and Correlation Lengths: The analysis identifies different fixed points based on the fine-tuning of the bare mass $\mu_0$ $μ_{0}$ :
- Gaussian Fixed Point: Achieved when $\mu_0$ is tuned to the lower rescaled cutoff, yielding a divergent correlation length and standard QFT behavior.
- Massless Fixed Point: Achieved when $\mu_0$ is tuned to the lower cutoff $\Lambda'$ , resulting in a different propagator structure ( $J_0$ Bessel functions).
- Fixed Point at Infinity: Achieved when $\mu_0$ is tuned to the upper cutoff $\Lambda$ , yielding a propagator behavior similar to the UV field but with different scaling conditions.
Commutativity of RG and IRG: The paper investigates whether applying RG and IRG operations commutes (i.e., $IRG \cdot RG = RG \cdot IRG$ ). It establishes criteria for when two distinct applications of RG operations lead to the "same" effective QFT (same fixed point, same $\hbar$ , same $c$ , and consistent time scales). For the non-interacting model, it demonstrates that under specific scaling choices, the order of integration does not alter the resulting fixed point or the effective QFT. However, the paper explicitly states that it does not have a concrete example to realize the possibility of RG acting as a group (where multiple applications return to a Gaussian fixed point with specific $\hbar$ flow) and presents this only as a thought experiment.
Resolution of the Measurement Problem: The paper argues that wavefunction collapse is not a physical dynamical process but an update of information for a non-deterministic observer. The "state" of a system is determined by interactions with the integrated-out modes (UV or IR) which the observer cannot access. Collapse merely reveals information about these hidden interactions.

Significance and Claims
The paper claims to offer a realist, anti-psi interpretation of QFT that unifies the emergence of quantum mechanics and special relativity from a classical statistical framework. Its primary contributions are:

Ontological Expansion: It proposes an ontology where the "universe" is an expanded hierarchy of modes, allowing for the possibility that an observer's "observable universe" is merely the IR limit of another observer's high-energy physics.
Emergent Constants: It provides a mechanism where fundamental constants ( $\hbar, c$ ) are not fixed inputs but flow parameters dependent on the observer's information (modes integrated out) and the temperature of the underlying classical ensemble, though the specific flow for interacting theories remains speculative.
Democratization of Scales: The framework suggests that high-energy physics and low-energy cosmology are symmetric perspectives within the RG hierarchy, rather than a hierarchy where one is "fundamental" and the other "effective."
Modest Scope: The author explicitly acknowledges the limitations of the work. The results are derived strictly for a non-interacting scalar field. The paper does not claim to have solved the renormalization of $\hbar$ and $c$ for interacting theories (like $\phi^4$ ) or to have fully incorporated gravity. The discussion of RG as a group (rather than a semigroup) and the specific behavior of interacting fields are presented as speculative directions for future research.

In summary, the paper posits that QFT is an emergent phenomenon arising from a classical field theory when an observer lacks access to specific modes of the RG hierarchy, with Planck's constant serving as a measure of the temperature at the self-similar fixed point of that hierarchy.

Technical Summary: Renormalization Group Ontology in Quantum Foundations

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