arXiv subjects

Mick van Vliet

Publications and source records attributed to Mick van Vliet.

8 recordsLinked to original sources

arXiv2026

Tameness and Complexity in Quantum Field Theory and Gravity

Finiteness appears to play a fundamental role in the mathematical structures underlying the laws of physics. Tame geometry, defined by the theory of o-minimality, provides a framework for making this idea precise. In this thesis we study the concept of tameness and finiteness of complexity across applications in quantum field theory and quantum gravity, and demonstrate that many classes of physical functions and theories are tame. To quantify these ideas, we use the novel theory of sharp o-minimality to measure the complexity of physical objects. The analysis covers wavefunctions in quantum mechanics, classical field configurations, non-perturbative observables in zero-dimensional quantum field theories, cosmological correlation functions, and Lagrangians of effective field theories. Finally, we argue that the emergence of tameness in physics aligns with manifestations of finiteness in quantum gravity, converging towards the idea that effective theories of quantum gravity admit a description of finite complexity. The thesis contains an introductory review for readers who wish to get familiar with tame geometry and its applications to physics.

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arXiv2026

Tame Complexity of Effective Field Theories in the Quantum Gravity Landscape

Effective field theories consistent with quantum gravity obey surprising finiteness constraints, appearing in several distinct but interconnected forms. In this work we develop a framework that unifies these observations by proposing that the defining data of such theories, as well as the landscape of effective field theories that are valid at least up to a fixed cutoff, admit descriptions with a uniform bound on complexity. To make this precise, we use tame geometry and work in sharply o-minimal structures, in which tame sets and functions come with two integer parameters that quantify their information content; we call this pair their tame complexity. Our Finite Complexity Conjectures are supported by controlled examples in which an infinite Wilsonian expansion nevertheless admits an equivalent finite-complexity description, typically through hidden rigidity conditions such as differential or recursion relations. We further assemble evidence from string compactifications, highlighting the constraining role of moduli space geometry and the importance of dualities. This perspective also yields mathematically well-defined notions of counting and volume measures on the space of effective theories, formulated in terms of effective field theory domains and coverings, whose finiteness is naturally enforced by the conjectures.

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arXiv2025

On the Complexity of Quantum Field Theory

We initiate a study of the complexity of quantum field theories (QFTs) by proposing a measure of information contained in a QFT and its observables. We show that from minimal assertions, one is naturally led to measure complexity by two integers, called format and degree, which characterize the information content of the functions and domains required to specify a theory or an observable. The strength of this proposal is that it applies to any physical quantity, and can therefore be used for analyzing complexities within an individual QFT, as well as studying the entire space of QFTs. We discuss the physical interpretation of our approach in the context of perturbation theory, symmetries, and the renormalization group. Key applications include the detection of complexity reductions in observables, for example due to algebraic relations, and understanding the emergence of simplicity when considering limits. The mathematical foundations of our constructions lie in the framework of sharp o-minimality, which ensures that the proposed complexity measure exhibits general properties inferred from consistency and universality.

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arXiv2025

Tame Embeddings, Volume Growth, and Complexity of Moduli Spaces

Quantum gravity is expected to impose constraints on the moduli spaces of massless fields that can arise in effective quantum field theories. A recent proposal asserts that the asymptotic volume growth of these spaces is severely restricted, and related to the existence of duality symmetries. In this work we link this proposal to a tameness criterion, by suggesting that any consistent moduli space should admit a tame isometric embedding into Euclidean space. This allows us to promote the volume growth constraint to a local condition, and give the growth coefficient a geometric interpretation in terms of complexity. We study the implications of this proposal for the emergence of dualities, as well as for the curvature and infinite distance limits of moduli spaces.

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arXiv2025

A Reduction Algorithm for Cosmological Correlators: Cuts, Contractions, and Complexity

Cosmological correlators are fundamental observables in an expanding universe and are highly non-trivial functions even at tree-level. In this work, we uncover novel structures in the space of such tree-level correlators that enable us to develop a new recursive algorithm for their explicit computation. We begin by formulating cosmological correlators as solutions to GKZ systems and develop a general strategy to construct additional differential operators, called reduction operators, when a GKZ system is reducible. Applying this framework, we determine all relevant reduction operators, and show that they can be used to build up the space of functions needed to represent the correlators. Beyond relating different integrals, these operators also yield a large number of algebraic relations, including cut and contraction relations between diagrams. This implies a significant reduction in the number of functions needed to represent each tree-level cosmological correlator. We present first steps to quantify the complexity of our reduction algorithm by using the Pfaffian framework. While we focus on tree-level cosmological correlators, our approach provides a blueprint for other perturbative settings.

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arXiv2024

Taming non-analyticities of QFT observables

Many observables in quantum field theories are involved non-analytic functions of the parameters of the theory. However, it is expected that they are not arbitrarily wild, but rather have only a finite amount of geometric complexity. This expectation has been recently formalized by a tameness principle: physical observables should be definable in o-minimal structures and their sharp refinements. In this work, we show that a broad class of non-analytic partition and correlation functions are tame functions in the o-minimal structure known as $\mathbb{R}_{\mathscr{G}}$ - the structure defining Gevrey functions. Using a perturbative approach, we expand the observables in asymptotic series in powers of a small coupling constant. Although these series are often divergent, they can be Borel-resummed in the absence of Stokes phenomena to yield the full partition and correlation functions. We show that this makes them definable in $\mathbb{R}_{\mathscr{G}}$ and provide a number of motivating examples. These include certain 0-dimensional quantum field theories and a set of higher-dimensional quantum field theories that can be analyzed using constructive field theory. Finally, we discuss how the eigenvalues of certain Hamiltonians in quantum mechanics are also definable in $\mathbb{R}_{\mathscr{G}}$.

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arXiv2024

Complexity in Tame Quantum Theories

Inspired by the notion that physical systems can contain only a finite amount of information or complexity, we introduce a framework that allows for quantifying the amount of logical information needed to specify a function or set. We then apply this methodology to a variety of physical systems and derive the complexity of parameter-dependent physical observables and coupling functions appearing in effective Lagrangians. In order to implement these ideas, it is essential to consider physical theories that can be defined in an o-minimal structure. O-minimality, a concept from mathematical logic, encapsulates a tameness principle. It was recently argued that this property is inherent to many known quantum field theories and is linked to the UV completion of the theory. To assign a complexity to each statement in these theories one has to further constrain the allowed o-minimal structures. To exemplify this, we show that many physical systems can be formulated using Pfaffian o-minimal structures, which have a well-established notion of complexity. More generally, we propose adopting sharply o-minimal structures, recently introduced by Binyamini and Novikov, as an overarching framework to measure complexity in quantum theories.

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arXiv2024

Structure and Complexity of Cosmological Correlators

Cosmological correlators capture the spatial fluctuations imprinted during the earliest episodes of the universe. While they are generally very non-trivial functions of the kinematic variables, they are known to arise as solutions to special sets of differential equations. In this work we use this fact to uncover the underlying tame structure for such correlators and argue that they admit a well-defined notion of complexity. In particular, building upon the recently proposed kinematic flow algorithm, we show that tree-level cosmological correlators of a generic scalar field theory in an FLRW spacetime belong to the class of Pfaffian functions. Since Pfaffian functions admit a notion of complexity, we can give explicit bounds on the topological and computational complexity of cosmological correlators. We conclude with some speculative comments on the general tame structures capturing all cosmological correlators and the connection between complexity and the emergence of time.

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