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Tameness and Complexity in Quantum Field Theory and Gravity

This thesis demonstrates that diverse physical objects and theories in quantum field theory and gravity exhibit "tameness" within the framework of o-minimality, suggesting that effective theories of quantum gravity possess a fundamental description of finite complexity.

Original authors: Mick van Vliet

Published 2026-09-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mick van Vliet

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

The universe, as we understand it through the laws of physics, seems to operate on a principle of manageable order. While the mathematics used to describe the cosmos can be infinitely complex, containing wild, chaotic patterns that never repeat or settle, the physical world we observe appears to be far more disciplined. It is as if nature has a built-in limit on how much information it can hold in any given space. This idea is supported by the fact that we can only perform a finite number of measurements, and that even the most extreme objects, like black holes, have a strict cap on their entropy, or disorder. If the universe is indeed finite in its information, then the mathematical language we use to describe it should reflect that same finiteness. For decades, mathematicians have developed a specific framework called "tame geometry" to separate simple, well-behaved shapes from the chaotic, infinite ones. This framework asks a profound question: are the functions and shapes that govern our physical reality actually "tame," meaning they can be broken down into a finite number of simple pieces, or do they hide infinite complexity that we simply haven't detected yet?

A recent doctoral thesis from Utrecht University, titled Tameness and Complexity in Quantum Field Theory and Gravity, takes this mathematical question and applies it directly to the most fundamental theories of physics. The author, Mick van Vliet, investigates whether the equations describing quantum fields and gravity are inherently simple enough to be described with a finite amount of information. The research focuses on a specific branch of logic known as "o-minimality," which provides a rigorous way to define what it means for a mathematical object to be tame. In this context, a tame object is one that does not contain infinite spirals, endless oscillations, or fractal-like structures that repeat forever within a small space. Instead, a tame object can be sliced into a finite number of smooth, simple segments. The thesis explores whether the wavefunctions of particles, the behavior of fields, and the structure of the vacuum of space all fit into this category of simplicity.

The study begins by examining the basic building blocks of physics, such as the wavefunctions that describe the position of a particle in quantum mechanics. In many standard scenarios, these wavefunctions oscillate endlessly, which would normally make them "wild" and mathematically unmanageable. However, the research demonstrates that when these particles are bound in a stable state, their wavefunctions are actually tame. They can be described using a specific type of mathematical chain that limits their complexity. The author shows that the complexity of these wavefunctions grows in a predictable way as the energy of the particle increases, but it never spirals into infinity. This finding suggests that even in the quantum realm, where things often seem chaotic, there is an underlying structure that keeps the information content finite and controllable.

Moving beyond simple particles, the thesis tackles the more difficult problem of non-perturbative observables in quantum field theory. These are quantities that cannot be calculated by simply adding up small corrections, but require a full, often infinite, summation of interactions. In many models, these calculations involve functions that appear to have infinite complexity. The author uses advanced mathematical techniques, including a method called Borel resummation, to show that these infinite series can actually be reorganized into a form that is tame. The result is that these complex physical quantities, which seem to require an infinite amount of data to describe, can in fact be captured by a finite set of rules. This is a significant step because it implies that the "wild" behavior often associated with quantum corrections might be an illusion of the calculation method, rather than a true feature of nature.

The research then extends to the large-scale structure of the universe, specifically looking at cosmological correlators, which are the patterns imprinted on the early universe by quantum fluctuations. The author finds that the mathematical functions describing these patterns are also tame. They can be generated by a finite sequence of differential equations, meaning their complexity is bounded and predictable. This suggests that the intricate patterns we see in the cosmic microwave background are not the result of infinite, chaotic processes, but are governed by a finite, structured set of laws. The thesis further argues that this finiteness is not just a mathematical curiosity but a necessary condition for a theory to be consistent with gravity.

Finally, the thesis connects these findings to the "swampland" program in theoretical physics, which attempts to distinguish between theories that can exist in a universe with gravity and those that cannot. The author proposes a "finite complexity conjecture," suggesting that any effective theory of quantum gravity must admit a description of finite complexity. In other words, if a theory requires an infinite amount of information to describe its low-energy behavior, it likely cannot be a consistent theory of our universe. The research provides evidence for this by showing that the spaces where the parameters of these theories live, known as moduli spaces, have a geometric structure that is tame. This implies that the vast landscape of possible physical theories is actually much smaller and more ordered than it appears, constrained by the requirement that nature must be describable with a finite amount of information.

The work does not claim to have solved every mystery in physics, nor does it suggest that all mathematical objects in physics are tame. The author acknowledges that there are scenarios, such as infinite oscillations over unbounded time, where the mathematics remains wild. However, the core finding is that the specific physical theories that describe our universe, particularly those involving bound states and effective descriptions of gravity, consistently exhibit this property of tameness. By using the tools of sharp o-minimality, the thesis provides a way to measure this complexity, turning the abstract idea of "finiteness" into a concrete, quantifiable property. This approach offers a new perspective on the language of physics, suggesting that the laws of nature are not just written in mathematics, but in a specific, highly structured dialect of mathematics that refuses to allow for infinite chaos. The ultimate conclusion is that the universe, in its deepest description, is finite in its complexity, and that this finiteness is a fundamental feature of reality that bridges the gap between quantum mechanics and gravity.

Based on the provided PhD thesis by Mick van Vliet, titled "Tameness and Complexity in Quantum Field Theory and Gravity," here is a detailed technical summary.

Problem Statement

The thesis addresses the fundamental question of the "language of physics." While mathematics allows for "wild" objects with pathological infinities (e.g., infinite oscillations in bounded regions, the Cantor set), physical laws appear to be constrained by finiteness. This finiteness is observed operationally (finite experimental data) and theoretically (e.g., the Bekenstein bound on entropy). However, standard quantum field theory (QFT) and effective field theories (EFTs) often involve infinite series of interactions and non-perturbative effects that seem to defy a finite description.

The core problem is to formalize the notion that physical theories should be describable with finite information. Specifically, the thesis investigates whether the mathematical structures underlying physical laws are tame (definable within an o-minimal structure) and, if so, how to quantify the complexity of these theories to distinguish between those consistent with quantum gravity and those that are not (the "Swampland").

Methodology

The thesis employs a rigorous mathematical framework derived from model theory and tame geometry, specifically o-minimality and its quantitative refinement, sharp o-minimality.

O-minimality: The author utilizes the theory of o-minimal structures, where definable sets in Euclidean space consist of finitely many connected components (cells). This excludes pathological objects like the topologist's sine curve or the set of integers.
Sharp O-minimality: To move beyond qualitative finiteness, the thesis adopts the theory of sharp o-minimality (developed by Binyamini and Novikov). This introduces a format-degree filtration $(F, D)$ for tame sets, providing explicit, computable bounds on geometric features (e.g., the number of connected components) based on the descriptive complexity of the defining functions.
Pfaffian Functions: A central technical tool is the class of Pfaffian functions, which satisfy triangular systems of first-order differential equations. These functions behave like polynomials regarding finiteness (Khovanskii's theorem) but include many transcendental functions relevant to physics (e.g., exponentials, restricted trigonometric functions).
Application to Physics: The methodology involves mapping physical objects (wavefunctions, Lagrangians, observables, moduli spaces) into these mathematical structures. The author analyzes whether these objects are definable in specific o-minimal structures (like $\mathbb{R}_{Pf}$ or $\mathbb{R}_{an,exp}$ ) and calculates their format and degree to establish complexity bounds.

Key Contributions and Results

Part I: Tame Geometry and Complexity

Formalization of Tameness: The thesis reviews the foundational theorems of o-minimality (Monotonicity, Cell Decomposition, Triangulation) and establishes the connection between logical definability and geometric finiteness.
Quantitative Framework: It introduces the format-degree filtration as a measure of complexity. For a tame set $X$ , the complexity is characterized by a spectrum of pairs $(F, D)$ . The thesis demonstrates that for Pfaffian functions, the number of connected components and topological complexity (Betti numbers) are bounded by explicit polynomials in the degree and exponential functions of the format.
Emergence of Simplicity: A key conceptual contribution is the demonstration that sequences of functions with diverging complexity (e.g., Taylor expansions) can converge to a limit with finite complexity due to the emergence of a simpler dual description (e.g., a differential equation).

Part II: Quantum Field Theory

Wavefunctions and Classical Configurations: The thesis proves that bound-state wavefunctions in one-dimensional quantum mechanics with polynomial potentials are tame. Specifically, they are definable in the structure $\mathbb{R}_{Pf}$ . The complexity of the $n$ -th excited state scales linearly with $n$ (the number of nodes), while the format remains constant.
Non-Perturbative Observables: In zero-dimensional QFTs, the author analyzes non-perturbative observables. Using Borel resummation, it is shown that the resummed asymptotic expansions of observables in various models are Gevrey functions. Since Gevrey functions are definable in an o-minimal structure, these non-perturbative quantities admit a description of finite complexity in these cases, despite originating from infinite perturbative series.
Cosmological Correlators: Tree-level wavefunction coefficients in cosmological QFTs (specifically for conformally coupled scalars) are shown to be captured by Pfaffian chains. This allows for the calculation of their complexity, linking the number of external legs and the order of the interaction to the format and degree.
Intrinsic Complexity of QFT: The thesis proposes defining the complexity of a QFT via the complexity of its Lagrangian and observables, suggesting that symmetries and renormalization group flows can reduce this complexity.

Part III: Quantum Gravity and the Swampland

Tameness Conjecture: Building on Grimm's proposal, the thesis argues that all field spaces, coupling functions, and parameter spaces in effective theories of quantum gravity must be definable within an o-minimal structure.
Moduli Spaces: The geometry of moduli spaces in highly supersymmetric theories (e.g., $N>8$ supergravity) is shown to be tame. The thesis references the remarkable result by Bakker, Klingler, and Tsimerman that the period maps in Hodge theory are definable in $\mathbb{R}_{an,exp}$ .
Tame Isometric Embeddings: A significant result is the link between quantum gravity constraints and geometry. The thesis argues that consistency with quantum gravity implies that moduli spaces must admit a tame isometric embedding into Euclidean space. This excludes spaces with volume growth faster than Euclidean space (e.g., hyperbolic spaces), aligning with the Distance Conjecture.
Finite Complexity Conjecture: The thesis culminates in the proposal of the finite complexity conjecture, suggesting that effective field theories consistent with quantum gravity admit a description of finite complexity.
- It defines a counting measure for the number of distinct EFTs in the landscape, weighted by the volume of their field spaces.
- Using the bounds from sharp o-minimality, it derives that the number of EFTs up to a cutoff $\Lambda$ scales as $n(\Lambda) \lesssim a |\log \Lambda|^b$ , where $a$ and $b$ depend on the complexity of the theory.
- This provides a quantitative realization of the Swampland program, suggesting that the "infinite" Wilsonian data of an EFT is actually encoded in a finite amount of information via differential constraints.

Significance and Claims

The thesis claims to provide a unifying mathematical language for the finiteness principles observed in quantum gravity.

Language of Physics: It posits that o-minimality is a strong candidate for the "language of physics," as it precisely captures the transition from infinite mathematical possibilities to finite physical realities.
Quantitative Swampland: By introducing sharp o-minimality, the work moves the Swampland program from qualitative criteria to quantitative bounds. It claims that the "finite complexity" of quantum gravity effective theories is not just a philosophical idea but a mathematically rigorous constraint that can be calculated.
Emergent Simplicity: The thesis highlights that physical theories often exhibit "emergent simplicity," where infinite series or complex configurations are governed by simple underlying differential structures (Pfaffian chains), allowing for finite descriptions even in the presence of non-perturbative effects.
Limitations: The author is modest about the scope, acknowledging that not all physical functions are tame (e.g., oscillations on unbounded domains are not). The claim is restricted to the subset of theories and functions that describe consistent low-energy limits of quantum gravity.

In summary, the thesis demonstrates that the mathematical structures of quantum field theory and quantum gravity are not arbitrary but are constrained by tameness, and that sharp o-minimality provides the necessary tools to quantify this constraint, offering a new perspective on the finiteness of the quantum gravity landscape.