arXiv subjects

David Chester

Publications and source records attributed to David Chester.

At least 19 recordsLinked to original sources

arXiv2026

Gauge-covariant Hamilton equations for teleparallel equivalents of general relativity

We construct a gauge-covariant Hamiltonian formulation for the metric and symmetric teleparallel equivalents of general relativity. When treating torsion and nonmetricity as generalized velocities, the conjugate momenta are proportional to their respective superpotentials. The quadratic actions give regular Legendre maps on these field-strength fibers. For both theories, the second Hamilton equation reproduces the Einstein field equations. In the reduced ADM sector, the hypersurface generators obey the standard deformation algebra. The regular field-strength map changes the Legendre description but does not remove the gravitational constraints or alter the classical physics.

gr-qc↗

arXiv2025

Three Dixon-Rosenfeld Planes

Rosenfeld postulated ``generalized'' projective planes, which exploit a correspondence between rank-one idempotents of Jordan algebras $\mathfrak{J}_3(\mathbb{A})$ and points of projective planes $\mathbb{A}P^2$. The isometry groups of the generalized projective planes (which were later defined rigorously as homogeneous spaces) are entries of the Tits-Freudenthal magic square. Given recent interest in the Dixon algebra $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, we extend Rosenfeld's approach and present three new coset manifolds. These "Dixon-Rosenfeld planes" have isometry algebras that are obtained from Tits' magic formula and involve all tensorial components of the Dixon algebra. We show that these are the only three planes obtainable with Tits' formula that preserve the analogy with Rosenfeld's planes. These non-simple Lie algebras generalize $\mathfrak{f}_{4},\mathfrak{e}_{6},\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$ for the octonionic plane $\mathbb{O}P^{2}$ and in the octonionic Rosenfeld planes $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$, $\left(\mathbb{H}\otimes\mathbb{O}\right)P^{2}$ and $\left(\mathbb{O}\otimes\mathbb{O}\right)P^{2}$. We finally investigate the relationships between the isometry algebras of the Dixon-Rosenfeld planes and the exceptional Lie algebras.

math-ph↗

arXiv2025

Preons, Braid Topology, and Representations of Fundamental Particles

In particle phenomenology, preon models study compositional rules of standard model interactions. In spite of empirical success, mathematical underpinnings of preon models in terms of group representation theory have not been fully worked out. Here, we address this issue while clarifying the relation between different preon models. In particular, we focus on two prominent models: Bilson-Thompson's helon model, and Lambek's 4-vector model. We determine the mapping between helon model particle states and representation theory of Lie algebras. Braided ribbon diagrams of the former represent on-shell states of spinors of the Lorentz group. Braids correspond to chirality, and twists, to charges. We note that this model captures only the $SU(3)_c\times U(1)_{em}$ sector of the standard model. We then map the twists of helon diagrams to the weight polytope of $SU(3)_c \times U(1)_{em}$. The braid structure maps to chiral states of fermions. We also show that Lambek's 4-vector can be recovered from helon diagrams. Alongside, we introduce a new 5-vector representation derived from the weight lattice. This representation contains both, the correct interactions found in 4-vectors and the inclusion of chirality found in helons. Additionally, we demonstrate topological analogues of CPT transformations in helon diagrams. Interestingly, the braid diagrams of the helon model are the only ones that are self-consistent with CPT invariance. In contrast to field-theoretic approaches, the compositional character of preon models offers an analogous particle-centric perspective on fundamental interactions.

physics.gen-ph↗

arXiv2024

Flipped $SU(5)$ GUT with conformal gravity from a single supermultiplet

The Grassmann envelope is used to find the $\mathcal{N}=1$ `superquasiconformal' algebra in $D=10+1$. The adjoint representation of this algebra is found to contain $\mathfrak{su}_{2,2}\oplus \mathfrak{u}_{1}\oplus \mathfrak{su}_{5}$ as a submaximal subalgebra, giving a spectrum of conformal gravity with flipped $SU_{5}\times U_{1}$ GUT. Combining Yang-Mills theory with MacDowell-Mansouri gravity over the conformal group is found to recover the Einstein-Hilbert action with a cosmological constant. An action for the theory is presented, which contains the gauge bosons of gravity and GUT, three generations of fermions in an efficient manner, and a new Higgs sector. By using the superalgebra for the entire multiplet, the theory gauges a non-supersymmetric subalgebra without introducing superpartners.

hep-th↗

arXiv2023

Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

We generalize Koopman-von Neumann classical mechanics to poly-symplectic fields and recover De Donder-Weyl theory. Comparing with Dirac's Hamiltonian density inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman-von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize space-time, energy-momentum, frequency-wavenumber, and the Fourier conjugate of energy-momentum. We clarify how 1st and 2nd quantization can be found by simply mapping between operators in classical and quantum commutator algebras.

hep-th↗

arXiv2023

Jordan algebras over icosahedral cut-and-project quasicrystals

In this paper we present a general setting for aperiodic Jordan algebras arising from icosahedral quasicrystals that are obtainable as model sets of a cut-and-project scheme with a convex acceptance window. In these hypothesis, we show the existence of an aperiodic Jordan algebra structure whose generators are in one-to-one correspondence with elements of the quasicrystal. Moreover, if the acceptance window enjoys a non-crystallographic symmetry arising from H2, H3 or H4 then the resulting Jordan algebra enjoys the same H2, H3 or H4 symmetry. Finally, we present as special cases some examples of Jordan algebras over a Fibonacci-chain quasicrystal, a Penrose tiling, and the Elser-Sloane quasicrystal.

math.RA↗

arXiv2023

Dixon-Rosenfeld Lines and the Standard Model

We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Three different Lie groups are found as isometry groups of these coset manifolds using Tits' formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal-Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within $\mathbb{C}\otimes\mathbb{H}\otimes J_{2}(\mathbb{O})$, $\mathbb{O}\otimes J_{2}(\mathbb{C}\otimes\mathbb{H})$, and $\mathbb{C}\otimes\mathbb{O}\otimes J_{2}(\mathbb{H})$.

hep-th↗

arXiv2023

Three Fibonacci-Chain Aperiodic Algebras

Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered structures and in the construction of some integrable models in quantum mechanics. Starting from the works of Patera and Twarock we present three aperiodic algebras based on Fibonacci-chain quasicrystals: a quasicrystal Lie algebra, an aperiodic Witt algebra and, finally, an aperiodic Jordan algebra. While a quasicrystal Lie algebra was already constructed from a modification of the Fibonacci chain, we here present an aperiodic algebra that matches exactly the original quasicrystal. Moreover, this is the first time to our knowledge, that an aperiodic Jordan algebra is presented leaving room for both theoretical and applicative developments.

math.RA↗

arXiv2022

A magic approach to octonionic Rosenfeld spaces

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find 7 pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld's framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation \`a la Rosenfeld.

math.RA↗

arXiv2022

On the Operator Origins of Classical and Quantum Wave Functions

We investigate operator algebraic origins of the classical Koopman-von Neumann wave function $\psi_{KvN}$ as well as the quantum mechanical one $\psi_{QM}$. We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $\psi_{KvN}$ and $\psi_{QM}$ are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle ${\cal E}$ follows from this. $\psi_{KvN}$ and $\psi_{QM}$ are shown to be sections in ${\cal E}$. The difference between $\psi_{KvN}$ and $\psi_{QM}$ originates from a quantization map interpreted as "twisting" of sections over ${\cal E}$. We also show that the Schr\"{o}dinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schr\"{o}dinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.

math-ph↗

arXiv2022

Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing

We show that quasicrystals exhibit anyonic behavior that can be used for topological quantum computing. In particular, we study a correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of a class of quasicrystals, which includes the one dimensional Fibonacci chain and the two dimensional Penrose tiling. A possible encoding on tiling spaces of topological quantum information processing is also discussed.

quant-ph↗

arXiv2022

Character varieties and algebraic surfaces for the topology of quantum computing

It is shown that the representation theory of some finitely presented groups thanks to their $SL_2(\mathbb{C})$ character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of $SL_2(\mathbb{C})$ character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link $H$, whose character variety is a Del Pezzo surface $f_H$ (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups as well as that of the fundamental group for the singular fibers $\tilde{E}_6$ and $\tilde{D}_4$ contain $f_H$. A surface birationally equivalent to a $K_3$ surface is another compound of their character varieties.

quant-ph↗

arXiv2022

Octonionic Planes and Real Forms of G2, F4 and E6

In this work we present a useful way to introduce the octonionic projective and hyperbolic plane through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of G2, F4and E6 groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.

math.RA↗

arXiv2022

Warm Dark Matter from Higher-Dimensional Gauge Theories

Warm dark matter particles with masses in the keV range have been linked with the large group representations in gauge theories through a high number of species at decoupling. In this paper, we address WDM fermionic degrees of freedom from such representations. Bridging higher-dimensional particle physics theories with cosmology studies and astrophysical observations, our approach is two-folded, i.e., it includes realistic models from higher-dimensional representations and constraints from simulations tested against observations. Starting with superalgebras in exceptional periodicity theories, we discuss several symmetry reductions and we consider several representations that accommodate a high number of degrees of freedom. We isolate a model that naturally accommodates both the standard model representation and the fermionic dark matter in agreement with both large and small-scale constraints. This model considers an intersection of branes in $D=27+3$ in a manner that provides the degrees of freedom for the standard model on one hand and 2048 fermionic degrees of freedom for dark matter, corresponding to a $\sim$2 keV particle mass, on the other. In this context, we discuss the theoretical implications and the observable predictions.

astro-ph.CO↗

arXiv2022

Conjugation Matters. Bioctonionic Veronese Vectors and Cayley-Rosenfeld Planes

Motivated by the recent interest in Lie algebraic and geometric structures arising from tensor products of division algebras and their relevance to high energy theoretical physics, we analyze generalized bioctonionic projective and hyperbolic planes. After giving a Veronese representation of the complexification of the Cayley plane $\mathbb{O}P_{\mathbb{C}}^{2}$, we present a novel, explicit construction of the bioctonionic Cayley-Rosenfeld plane $\left( \mathbb{C}\otimes \mathbb{O}\right) P^{2}$, again by exploiting Veronese coordinates. We discuss the isometry groups of all generalized bioctonionic planes, recovering all complex and real forms of the exceptional groups $F_{4}$ and $E_{6}$, and characterizing such planes as symmetric and Hermitian symmetric spaces. We conclude by discussing some possible physical applications.

math-ph↗

arXiv2020

Monstrous M-theory

In $26+1$ space-time dimensions, we introduce a gravity theory whose massless spectrum can be acted upon by the Monster group when reduced to $25+1$ dimensions. This theory generalizes M-theory in many respects and we name it Monstrous M-theory, or M$^{2}$-theory. Upon Kaluza-Klein reduction to $25+1$ dimensions, the M$^{2}$-theory spectrum irreducibly splits as $\mathbf{1}\oplus\mathbf{196,883}$, where $\mathbf{1}$ is identified with the dilaton, and $\mathbf{196,883}$ is the dimension of the smallest non-trivial representation of the Monster. This provides a field theory explanation of the lowest instance of the Monstrous Moonshine, and it clarifies the definition of the Monster as the automorphism group of the Griess algebra, by showing that such an algebra is not merely a sum of unrelated spaces, but descends from massless states for M$^{2}$-theory, which includes Horowitz and Susskind's bosonic M-theory as a subsector. Further evidence is provided by the decomposition of the coefficients of the partition function of Witten's extremal Monster SCFT in terms of representations of $SO_{24}$, the massless little group in $25+1$; the purely bosonic nature of the involved $SO_{24}$-representations may be traced back to the unique feature of $24$ dimensions, which allow for a non-trivial generalization of the triality holding in $8$ dimensions. Last but not least, a certain subsector of M$^{2}$-theory, when coupled to a Rarita-Schwinger massless field in $26+1$, exhibits the same number of bosonic and fermionic degrees of freedom; we cannot help but conjecture the existence of a would-be $\mathcal{N}=1$ supergravity theory in $26+1$ space-time dimensions.

hep-th↗

arXiv2020

Beyond the standard model with six-dimensional spinors

6D spinors with $Spin(3,3)$ symmetry are utilized to efficiently encode three generations of matter. $E_{8(-24)}$ is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati-Salam, $SU(5)$, and $Spin(10)$ grand unified theories are found when a single generation is isolated. For spacetime symmetries, $Spin(4,2)$ may be used for conformal symmetry, $AdS_5\rightarrow dS_4$, or simply broken to $Spin(3,1)$ of Minkowski space. Another class of representations finds $Spin(2,2)$ and can give $AdS_3$ with various GUTs. An action for three generations of fermions in the Majorana-Weyl spinor ${\bf 128}$ of $Spin(4,12)$ is found with $Spin(3)$ flavor symmetry inside $E_{8(-24)}$. The ${\bf 128}$ of $Spin(4,12)$ can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane $E_{8(-24)}/Spin(4,12)= (\mathbb{O}_s\times\mathbb{O})\mathbb{P}^2$.

physics.gen-ph↗

arXiv2019

Holographic Code Rate

Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation rule protect quantum information stored in the bulk from errors on the boundary provided the code rate is less than one. Hyperbolic geometry bounds the holographic code rate and guarantees quantum error correction for codes grown with any inflation rule on all regular hyperbolic tessellations in a class whose size grows exponentially with the rank of the perfect tensors for rank five and higher. For the tile completion inflation rule, holographic triangle codes have code rate more than one but all others perform quantum error correction.

quant-ph↗