arXiv subjects

Boris Zilber

Publications and source records attributed to Boris Zilber.

At least 19 recordsLinked to original sources

arXiv2026

Axioms of Quantum Mechanics in light of Continuous Model Theory

We revisit Dirac's axiomatization of quantum mechanics and show that it can essentially be reformulated in a more familiar logical setting - continuous model theory. Our aim is twofold: (i) to present the Dirac--von Neumann formalism in a genuinely axiomatic manner suitable for logicians, and (ii) to exhibit a structural analogy between Hilbert spaces and Tarski's cylindric algebras, which were introduced in the program of algebraisation of first-order logic. Recall that the cylindric algebra $\mathfrak{C}(\mathbb{A})$ of a first order structure $\mathbb{A}$ allows to recover $\mathbb{A}$ up to elementary equivalence. For a general continuous structure $\mathbb{M}$, we introduce an analogue $\mathcal{B}(\mathbb{M})$ of the cylindric algebra of a first--order structure. Under natural tameness assumptions, $\mathcal{B}(\mathbb{M})$ takes the form of a (rigged) Hilbert space with operators, and $\mathbb{M}$ can be recovered from $\mathcal{B}(\mathbb{M}).$

math.LO↗

arXiv2026

Structural approximation and a Minkowski space-time lattice with Lorentzian invariance

We introduce a framework of structural approximation to represent Lorentz-invariant Minkowski space-time as the limit of finite cyclic lattices, each equipped with the action of a finite quasi-Lorentz group. This construction provides a discrete model preserving Lorentz symmetry and offers new insights into the algebraic and geometric structure of space-time.

physics.gen-ph↗

arXiv2026

Approximation of structures:local and global

We provide a mathematically rigorous definition of local approximation and demonstrate its applicability to some interesting classes of structures. In particular, we prove that any compact simple Lie group is locally approximated by finite groups. The definition and main examples are motivated by physics but the techniques are of model theory. Namely, we introduce the ultraproduct of emerging metric structures, which generalises the ultraproduct in metric model theory.

math.LO↗

arXiv2025

Dirac - von Neumann axioms in the setting of Continuous Model Theory

We recast the well-known axiom system of quantum mechanics used by physicists (the Dirac calculus) in the language of Continuous Logic. For the basic version of the axiomatic system we prove that along with the canonical continuous model the axioms have approximate finite models of large sizes, in fact the continuous model is isomorphic to an ultraproduct of finite models. We analyse the continuous logic quantifier corresponding to Dirac integration and show that in finite context it has two versions, local and global, which coincide on Gaussian wave-functions.

math.LO↗

arXiv2024

On the logical structure of physics

One of the main claims of the paper is that Dirac's calculus and broader theories of physics can be treated as theories written in the language of Continuous Logic. Establishing its true interpretation (model) is a model theory problem. The paper introduces such a model for the fragment which covers ``free theories'', that is physical theories with Gaussian (quadratic) potential. The model is pseudo-finite (equivalently, a limit of finite models), based on a pseudo-finite field in place of the field of complex numbers. The advantage of this unusual setting is that it treats the quantum and the statistical mechanics as just domains in the same model and explains Wick rotation as a natural transformation of the model corresponding to a shift in scales of physical units.

math.LO↗

arXiv2023

Physics over a finite field and Wick rotation

The paper develops an earlier proposition that the physical universe is a finite system co-ordinatised by a very large finite field $\mathrm{F}_\mathfrak{p}$ which looks like the field of complex numbers to an observer. We construct a place (homomorphism) $\mathrm{lm}$ from a pseudo-finite field $\mathrm{F}_\mathfrak{p}$ onto the compactified field of complex numbers in such a way that certain multiplicative subgroups $'\mathbb{R}'_+$ and $'\mathbb{S}'$ correspond to the polar coordinate system $\mathbb{R}_+$ and $\mathbb{S}$ of $\mathbb{C}.$ Thus $\mathrm{F}_\mathfrak{p},$ $'\mathbb{R}'_+$ and $'\mathbb{S}'$ provide co-ordinates for physical universe. We show that the passage from the scale of units in $'\mathbb{R}'_+$ to the scale of units of $'\mathbb{S}'$ corresponds to a multiplication (on the logarithmic scale) by a very large integer $\mathfrak{i}$ equal approximately to $\sqrt{\mathfrak{p}}.$ This provides an explanation to the phenomenon of Wick rotation. In the same model we explain the phenomenon of phase transition in a large finite system

math.LO↗

arXiv2023

A topological $L_{ω_1,ω}$-invariant

We suggest to look at formal sentences describing complex algebraic varieties together with their universal covers as topological invariants. We prove that for abelian varieties and Shimura varieties this is indeed a complete invariant, i.e. it determines the variety up to complex conjugation.

math.LO↗

arXiv2023

On the Theory of Specialisations of Regular Covers of Zariski Structures

In algebraic geometry specialisations and valuations play and important role. In this paper we start investigating analogous structures for Zariski structures. Specifically, we look into the existence and uniqueness properties of extensions of universal specialisations from a base Zariski structure to its regular cover. In the process we begin to uncover some structural properties of regular covers of Zariski structures, and also to uncover the type of topological properties necessary for a Zariski structure to have a "good" theory of specialisations. A subclass of Zariski structures is identified with a ``good'' theory of specialisations.

math.LO↗

arXiv2022

Canonical models of modular curves and the Galois action on CM-points

We use the theory of canonical models of Shimura varieties to describe the projective limit of the curves Y(N), all N, and its automorphism group. In particular we prove that the Galois group of Q(CM) over Q is an extension of a certain abelian group by a 2-element group, where Q(CM) stands for the the extension of Q by all the CM-points on all the curves Y(N).

math.AG↗

arXiv2022

Modular curves and their pseudo-analytic cover

We find a natural $L_{ω_1,ω}$-axiomatisation $Σ$ of a structure on the upper half-plane $\mathbb{H}$ as the covering space of modular curves. The main theorem states that $Σ$ has a unique model in every uncountable cardinal. The proof relies heavily on the theory of complex multiplication and the work on Langland's conjecture on the conjugation of Shimura varieties. We also use the earlier work on a related problem by C.Daw and A.Harris. The essential difference between the setting of this work and that of the current paper is that the former was in the language which named the CM-points of the modular curves while our results here are over $\mathbb{Q}.$

math.LO↗

arXiv2022

Non-elementary categoricity and projective locally o-minimal classes

Given a cover $\mathbb{U}$ of a family of smooth complex algebraic varieties, we associate with it a class $\mathcal{U},$ containing $\mathbb{U}$, of structures locally definable in an o-minimal expansion of the reals. We prove that the class is $\aleph_0$-homogenous over submodels and stable. It follows that $\mathcal{U}$ is categorical in cardinality $\aleph_1.$ In the one-dimensional case we prove that a slight modification of $\mathcal{U}$ is an abstract elementary class categorical in all uncountable cardinals.

math.LO↗

arXiv2021

A model theory section conjecture

We introduce the category of structures and interpretations which allows us to discuss some issues of Grothendieck's anabelian geometry in model-theory terms. Our main result is a formulation in terms of pure stability theory of a problem closely related to Grothendieck's section conjecture

math.LO↗

arXiv2021

Section and towers

We discuss the towers of finite étale covers which were essentially introduced by A.Tamagawa. The statement about correspondence between sections and cofinal towers is a folklore but perhaps not in a very explicit form. The last section explains how the "injectivity statement" of Grothendieck section conjecture fails for abelian varieties, which is also known in some form. The paper is based on an earlier article which was aimed to reinterpret anabelian setting in model theory terms.

math.AG↗

arXiv2020

Definability, interpretations and étale fundamental groups

The aim of the paper and of a wider project is to translate main notions of anabelian geometry into the language of model theory. Here we finish with giving the definition of the étale fundamental group $π^{et}_1(X,x)$ of a non-singular quasiprojective scheme over a field of characteristic 0.

math.LO↗

arXiv2019

Around logical perfection

In this article we present and describe a notion of "logical perfection". We extract the notion of "perfection" from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically. In the past two decades, categoricity notions have started to overlap with more classical notions of robustness and smoothness. These have been crucial in various parts of mathematics since the nineteenth century. We postulate and present the category of logical perfection. We draw on various notions of perfection from mathematics of the 19th and 20th centuries and then trace the relation to the concept of categoricity in power as a logical notion of what a "mathematically perfect" structure is.

math.LO↗

arXiv2017

A model theoretic Rieffel's theorem of quantum 2-torus

We defined a notion of quantum 2-torus $T_θ$ in "Masanori Itai and Boris Zilber, Notes on a model theory of quantum 2-torus $T_q^2$ for generic $q$, arXiv:1503.06045v1 [mathLO]" and studied its model theoretic property. In this note we associate quantum 2-tori $T_θ$ with the structure over ${\mathbb C}_θ= ({\mathbb C}, +, \cdot, y = x^θ),$ where $θ\in {\mathbb R} \setminus {\mathbb Q}$, and introduce the notion of geometric isomorphisms between such quantum 2-tori. We show that this notion is closely connected with the fundamental notion of Morita equivalence of non-commutative geometry. Namely, we prove that the quantum 2-tori $T_{θ_1}$ and $T_{θ_2}$ are Morita equivalent if and only if $θ_2 = {\displaystyle \frac{a θ_1 + b}{c θ_1 + d}}$ for some $ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in {\rm GL}_2({\mathbb Z})$ with $|ad - bc| = 1$. This is our version of Rieffel's Theorem in "M. A. Rieffel and A. Schwarz, Morita equivalence of multidimensional noncummutative tori, Internat. J. Math. 10, 2 (1999) 289-299" which characterises Morita equivalence of quantum tori in the same terms. The result in essence confirms that the representation $T_θ$ in terms of model-theoretic geometry \cite{IZ} is adequate to its original definition in terms of non-commutative geometry.

math.LO↗

arXiv2016

Model theory and geometry of representations of rings of integers

The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what geometry this object carries. A.Connes and C.Consani published recently an important paper which introduces a much more complex structure called {\em the arithmetic site} which includes $\mathrm{Spec}(\mathbb{Z}).$ Our approach is based on the generalisation of constructions applied by the first author for similar purposes in non-commutative (and commutative) algebraic geometry. The current version is quite basic. We describe a category of certain representations of integral extensions of $\Z$ and establish its tight connection with the space of elementary theories of pseudo-finite fields. From model-theoretic point of view the category of representations is a multisorted structure which we prove to be superstable with pregeometry of trivial type. It comes as some surprise that a structure like this can code a rich mathematics of pseudo-finite fields.

math.LO↗

arXiv2016

The semantics of the canonical commutation relation

We treat the canonical commutation relations and the conventional calculus based on it as an algebraic syntax of quantum mechanics and establish a geometric semantics of this syntax. This leads us to a geometric model, the space of states with the action of time evolution operators, which is a limit of finite models. The finitary nature of the space allows to give a precise meaning and calculate various classical quantum mechanical quantities.

math-ph↗