arXiv subjects

John C. Baez

Publications and source records attributed to John C. Baez.

At least 19 recordsLinked to original sources

arXiv2026

Three Generations in E7

Starting from the Standard Model Lie algebra $\mathfrak{g}_{\mathrm{SM}} = \mathfrak{sl}_3 \oplus \mathfrak{sl}_2 \oplus \mathbb{C}$ sitting inside the complex Lie algebra $\mathfrak{e}_7$, we show how to decompose $\mathfrak{e}_7$ into the direct sum of a Lie subalgebra containing $\mathfrak{g}_{\mathrm{SM}}$ and three 32-dimensional subspaces, each of which forms the same representation of $\mathfrak{g}_{\mathrm{SM}}$ as one generation of fermions and their antiparticles. The setting is due to Nasmith, and much of the mathematics is that underlying the $\mathrm{E}_7$ generation unification of Kugo and Yanagida. New features include the derivation, in which as many results as possible rely only on the embedding $\mathfrak{g}_{\mathrm{SM}} \subset \mathfrak{e}_7$, and also the description of each 32-dimensional subspace as a copy of the exterior algebra $Λ\mathbb{C}^5$.

math-ph↗

arXiv2026

Three-Dimensional Geometry in Exceptional Algebra

We review some topics in "exceptional mathematics'' from the perspective of 3-dimensional geometry: the octonions $\mathbb{O}$, the split octonions $\mathbb{O}'$, the bioctonions $\mathbb{O}_\mathbb{C} \cong \mathbb{C} \otimes_\mathbb{R} \mathbb{O}$, the complex Albert algebra $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$, and the complex form of the exceptional Lie algebra $\mathfrak{e}_6$. We show how to functorially build an algebra isomorphic to $\mathbb{O}$ from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to $\mathbb{O}'$ starting from a 3d real vector space equipped with a volume form, and one isomorphic to $\mathbb{O}_\mathbb{C}$ starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$ starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra $\mathfrak{e}_6$ and its subalgebra $\mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C})$.

math.RA↗

arXiv2026

Double Categories of Open Systems: the Cospan Approach

This is an overview of double categories of "open systems": systems that can interact with their environment. We focus on the variable sharing paradigm, where we compose open systems by identifying variables. This paradigm is often implemented using structured or decorated cospans. We explain this approach using three main examples: open Petri nets, open dynamical systems, and open Petri nets with rates. We compare the virtues of structured and decorated cospan double categories, and study their common features. We show that any symmetric monoidal structured or decorated cospan double category comes with maps from two simpler double categories: its "exoskeleton" and its "outer shell". Finally, we study the concept of "hypergraph double category", a kind of double category that should subsume structured and decorated cospans in a common framework for studying open systems in the variable sharing paradigm.

math.CT↗

arXiv2026

Jordan Pair Quantum Theory and the Standard Model

Jordan pairs and hermitian Jordan triples were discovered by mathematicians studying Jordan algebras, which describe the possible algebras of observables in quantum mechanics. We point out a striking correspondence between the doubly exceptional hermitian Jordan triple (the so-called ``bi-Cayley'' triple) and the structure of the Standard Model of particle physics. We also point out how ordinary quantum mechanics may be reformulated, and generalized, using hermitian Jordan triples.

math-ph↗

arXiv2026

The Standard Model Gauge Group from the Exceptional Jordan Algebra

We construct the Standard Model gauge group using the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$ and its automorphism group $\text{F}_4$. The group $\text{F}_4$ acts on pairs of Jordan subalgebras $X \subset B \subset \mathfrak{h}_3(\mathbb{O})$ with $X \cong \mathfrak{h}_2(\mathbb{C})$ and $B \cong \mathfrak{h}_3(\mathbb{C})$, and for any such pair the stabilizer of $X$ intersected with the identity component of the stabilizer of $B$ is isomorphic to the Standard Model gauge group. Since $\mathfrak{h}_2(\mathbb{C})$ is the Jordan algebra of observables of a qubit and $\mathfrak{h}_3(\mathbb{C})$ is the Jordan algebra of observables of a qutrit, we could say $\mathfrak{h}_3(\mathbb{O})$ is the Jordan algebra of observables of an 'octonionic qutrit'. In this language our result says roughly that the Standard Model gauge group is the group of symmetries of an octonionic qutrit that restrict to act as unitary operators on an ordinary qutrit and, within that, a qubit.

math-ph↗

arXiv2026

Triangulations of the Sphere

Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $π/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}} = \mathbb{PC}^{10}_+/Γ$. Here $\mathbb{C}^{10}_+ = \{ v \in \mathbb{C}^{10} \vert \; Q(v) > 0\}$ for some quadratic form $Q$ of signature $(1,9)$ on $\mathbb{C}^{10}$, $\mathbb{PC}^{10}_+$ is its projectivization, and $Γ$ is a certain discrete group of linear transformations of $\mathbb{C}^{10}$ preserving both $Q$ and the lattice $\mathbb{E}^{10} \subset \mathbb{C}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $π/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.

math.MG↗

arXiv2026

Second Quantization for the Kepler Problem

The Kepler problem concerns a point particle in an attractive inverse square force. After a brief review of the classical and quantum versions of this problem, focused on their hidden $\text{SU}(2) \times \text{SU}(2)$ symmetry, we discuss the quantum Kepler problem for a spin-$\frac{1}{2}$ particle. We show that the Hilbert space $\mathcal{H}$ of bound states for this problem is unitarily equivalent, as a representation of $\text{SU}(2) \times \text{SU}(2)$, to the Hilbert space of solutions of the Weyl equation on the spacetime $\mathbb{R} \times S^3$. This equation describes a massless left-handed spin-$\frac{1}{2}$ particle. We then form the fermionic Fock space on $\mathcal{H}$ and show this is unitarily equivalent to the Hilbert space of a massless left-handed spin-$\frac{1}{2}$ free quantum field on $\mathbb{R} \times S^3$, again as representations of $\text{SU}(2) \times \text{SU}(2)$. By modifying the Hamiltonian of this free field theory, we obtain the well-known "Madelung rules". These give a reasonable approximation to the observed filling of subshells as we consider elements with more and more electrons, and match the rough overall structure of the periodic table.

math-ph↗

arXiv2026

The Inverse Cube Force Law

Newton's Principia is famous for its investigations of the inverse square force law for gravity. But in this book Newton also did something that remained little-known until fairly recently. He figured out what kind of central force exerted upon a particle can rescale its angular velocity by a constant factor without affecting its radial motion. This turns out to be a force obeying an inverse cube law! Here we discuss this and some other interesting features of the inverse cube force law.

physics.class-ph↗

arXiv2026

Motifs and Emergent Feedback in Labeled Graphs

In fields ranging from business to systems biology, directed graphs with edges labeled by signs are used to model systems in a simple way: the nodes represent entities of some sort, and an edge indicates that one entity directly affects another either positively or negatively. Multiplying the signs along a directed path of edges lets us determine indirect positive or negative effects, and if the path is a loop we call this a positive or negative feedback loop. Here we generalize this to graphs with edges labeled by a monoid, whose elements represent `polarities' possibly more general than simply "positive" or "negative". We study three notions of morphism between graphs with labeled edges, each with its own distinctive application: to refine a simple graph into a complicated one, to transform a complicated graph into a simple one, and to find recurring patterns called "motifs". We construct three corresponding symmetric monoidal double categories of "open" graphs. We also study feedback loops using a generalization of the homology of a graph to homology with coefficients in a commutative monoid. In particular, we describe the emergence of new feedback loops when we compose open graphs using a variant of the Mayer-Vietoris exact sequence for homology with coefficients in a commutative monoid.

math.CT↗

arXiv2026

Topological Crystals

Sunada's work on crystallography emphasizes the role of the "maximal abelian cover" of a graph $X$. This is a covering space of $X$ for which the group of deck transformations is the first homology group $H_1(X,\mathbb{Z})$. An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie along the edges. We prove that for any connected graph $X$ without bridges, there is a canonical embedding of the maximal abelian cover of $X$ into the vector space $H_1(X,\mathbb{R})$, called a "topological crystal". Crystals of graphene and diamond are examples of this construction. We prove that any symmetry of a graph lifts to a symmetry of its topological crystal. We also compute the density of atoms in a topological crystal. The key technical tools are a way of decomposing the 1-chain coming from a path in $X$ into manageable pieces, and the work of Bacher, de la Harpe and Nagnibeda on integral cycles and integral cuts.

math.AT↗

arXiv2026

Coxeter and Dynkin Diagrams

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or "ADE" Dynkin diagrams also classify finite subgroups of SU(2) and quivers with finitely many indecomposable representations. This introductory tour of Coxeter and Dynkin diagrams, based on the column This Week's Finds in Mathematical Physics, is made to accompany a series of lecture videos.

math.RT↗

arXiv2025

Dirichlet Species and Arithmetic Zeta Functions

Though Joyal's species are known to categorify generating functions in enumerative combinatorics, they also categorify zeta functions in algebraic geometry. The reason is that any scheme $X$ of finite type over the integers gives a "zeta species" $Z_X$, and any species $F$ gives a Dirichlet series $\widehat{F}$, in such a way that $\widehat{Z}_X$ is the arithmetic zeta function of $X$, a well-known Dirichlet series that encodes the number of points of $X$ over each finite field. Specifically, a $Z_X$-structure on a finite set is a way of making that set into a semisimple commutative ring, say $k$, and then choosing a $k$-point of the scheme $X$. This is an elaboration of joint work with James Dolan.

math.CT↗

arXiv2025

What is Entropy?

This short book is an elementary course on entropy, leading up to a calculation of the entropy of hydrogen gas at standard temperature and pressure. Topics covered include information, Shannon entropy and Gibbs entropy, the principle of maximum entropy, the Boltzmann distribution, temperature and coolness, the relation between entropy, expected energy and temperature, the equipartition theorem, the partition function, the relation between expected energy, free energy and entropy, the entropy of a classical harmonic oscillator, the entropy of a classical particle in a box, and the entropy of a classical ideal gas.

cond-mat.stat-mech↗

arXiv2025

Getting to the Bottom of Noether's Theorem

We examine the assumptions behind Noether's theorem connecting symmetries and conservation laws. To compare classical and quantum versions of this theorem, we take an algebraic approach. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether's theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics this mapping is multiplication by $\sqrt{-1}$. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a "dynamical correspondence", and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. We show this expresses a relation between quantum and statistical mechanics, closely connected to the principle that "inverse temperature is imaginary time".

math-ph↗

arXiv2025

Groupoid Cardinality and Random Permutations

If we treat the symmetric group $S_n$ as a probability measure space where each element has measure $1/n!$, then the number of cycles in a permutation becomes a random variable. The Cycle Length Lemma describes the expected values of products of these random variables. Here we categorify the Cycle Length Lemma by showing that it follows from an equivalence between groupoids.

math.CT↗

arXiv2025

Tannaka Reconstruction and the Monoid of Matrices

Settling a conjecture from an earlier paper, we prove that the monoid $\mathrm{M}(n,k)$ of $n \times n$ matrices in a field $k$ of characteristic zero is the "walking monoid with an $n$-dimensional representation". More precisely, if we treat $\mathrm{M}(n,k)$ as a monoid in affine schemes, the 2-rig $\mathrm{Rep}(\mathrm{M}(n,k))$ of algebraic representations of $\mathrm{M}(n,k)$ is the free 2-rig on an object $x$ with $Λ^{n+1}(x) \cong 0$. Here a "2-rig" is a symmetric monoidal $k$-linear category that is Cauchy complete. Our proof uses Tannaka reconstruction and a general theory of quotient 2-rigs and 2-ideals. We conclude with a series of conjectures about the universal properties of representation 2-rigs of classical groups.

math.RT↗

arXiv2025

Hoàng Xuân Sính's Thesis: Categorifying Group Theory

During what Vietnamese call the American War, Alexander Grothendieck spent three weeks teaching mathematics in and near Hanoi. Hoàng Xuân Sính took notes on his lectures and later did her thesis work with him by correspondence. In her thesis she developed the theory of "Gr-categories", which are monoidal categories in which all objects and morphisms have inverses. Now often called "2-groups", these structures allow the study of symmetries that themselves have symmetries. After a brief account of how Hoàng Xuân Sính wrote her thesis, we explain some of its main results, and its context in the history of mathematics.

math.CT↗

arXiv2024

The Hexagonal Tiling Honeycomb

The hexagonal tiling honeycomb is a beautiful structure in 3-dimensional hyperbolic space. It is called {6,3,3} because each hexagon has 6 edges, 3 hexagons meet at each vertex in a Euclidean plane tiled by regular hexagons, and 3 such planes meet along each edge of this honeycomb. It also appears naturally in algebraic geometry. If $\mathbb{E}$ denotes the Eisenstein integers, the Néron-Severi group of the abelian surface $\mathbb{C}^2/\mathbb{E}^2$ is isomorphic to the lattice $\mathfrak{h}_2(\mathbb{E})$ consisting of $2 \times 2$ hermitian matrices with Eisenstein integer entries. The points $A \in \mathfrak{h}_2(\mathbb{E})$ with $\mathrm{tr}(A) \gt 0$ and $\det(A) \gt 0$ come from ample line bundles on $\mathbb{C}^2/\mathbb{E}^2$, and among these points, those with $\det(A) = 1$ correspond to principal polarizations. But these points are precisely the centers of the hexagons in the hexagonal tiling honeycomb!

math.HO↗