arXiv subjects

Klee Irwin

Publications and source records attributed to Klee Irwin.

At least 19 recordsLinked to original sources

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

All Hurwitz Algebras from 3D Geometric Algebras

Hurwitz algebras are unital composition algebras widely known in algebra and mathematical physics for their useful applications. In this paper, inspired by works of Lesenby and Hitzer, we show how to embed all seven Hurwitz algebras (division and split) in 3D geometric algebras, i.e. $\mathcal{G}\left(p,q\right)$ with $p+q=3$. This is achieved studying the even subalgebra, which is always of quaternionic or split-quaternionic type, and the algebra generated by the pseudoscalar which is always of complex or split-complex type. Reversion, inversion and Clifford conjugation correspond to biquaternionic, complex and quaternionic conjugation respectively. Octonionic algebras, i.e., octonions and split-octonions, are obtained in two ways introducing two different products, replicating a variation of the Cayley-Dickson process. Octonionic conjugation is achieved in the geometric algebra formalism through the involution defined by the full grade inversion.

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

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

Quasicrystalline Spin Foam with Matter: Definitions and Examples

In this work, we define quasicrystalline spin networks as a subspace within the standard Hilbert space of loop quantum gravity, effectively constraining the states to coherent states that align with quasicrystal geometry structures. We introduce quasicrystalline spin foam amplitudes, a variation of the EPRL spin foam model, in which the internal spin labels are constrained to correspond to the boundary data of quasicrystalline spin networks. Within this framework, the quasicrystalline spin foam amplitudes encode the dynamics of quantum geometries that exhibit aperiodic structures. Additionally, we investigate the coupling of fermions within the quasicrystalline spin foam amplitudes. We present calculations for three-dimensional examples and then explore the 600-cell construction, which is a fundamental component of the four-dimensional Elser-Sloane quasicrystal derived from the E8 root lattice.

gr-qc↗

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

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 à la Rosenfeld.

math.RA↗

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↗

arXiv2020

Space, Matter and Interactions in a Quantum Early Universe. Part I : Kac-Moody and Borcherds Algebras

We introduce a quantum model for the Universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac-Moody Lie algebra $\mathbf{e_9}$. We investigate Kac-Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.

gr-qc↗

arXiv2020

Space, Matter and Interactions in a Quantum Early Universe. Part II : Superalgebras and Vertex Algebras

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra $\textbf{${\mathfrak g}_{\mathsf u}$}$ that extends $\mathbf{e_9}$. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn $\textbf{${\mathfrak g}_{\mathsf u}$}$ into a Lie superalgebra $\textbf{$\mathfrak {sg}_{\mathsf u}$}$ with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on $\textbf{$\mathfrak {sg}_{\mathsf u}$}$, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular {\it resonant scattering}. Finally, we complete the model by merging the local $\textbf{$\mathfrak {sg}_{\mathsf u}$}$ into a vertex-type algebra.

gr-qc↗

arXiv2020

Aspects of aperiodicity and randomness in theoretical physics

In this work we explore how the heat kernel, which gives the solution to the diffusion equation and the Brownian motion, would change when we introduce quasiperiodicity in the scenario. We also study the random walk in the Fibonacci sequence. We discuss how these ideas would change the discrete approaches to quantum gravity and the construction of quantum geometry.

math-ph↗

arXiv2020

Informationally complete characters for quark and lepton mixings

A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters $κ$ of a finite group $G$. Here we introduce a $d$-dimensional Hilbert space with $d=cc(G)$, the number of conjugacy classes of $G$. Groups under consideration should follow two rules, (a) the character table contains both two- and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measurements under the action of a $d$-dimensional Pauli group over the characters of these representations. Groups with small $d$ that satisfy these rules coincide in a large part with viable ones derived so far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing matrices. Groups leading to physical $CP$ violation are singled out.

hep-ph↗

arXiv2020

Quantum computation and measurements from an exotic space-time R4

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group $G$ with relations. A valid subgroup $H$ of index $d$ in $G$ leads to a 'magic' state $\left|ψ\right\rangle$ in $d$-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite 'contextual' geometry. In the present work, we choose $G$ as the fundamental group $π_1(V)$ of an exotic $4$-manifold $V$, more precisely a 'small exotic' (space-time) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean $\mathbb{R}^4$). Our selected example, due to to S. Akbulut and R.~E. Gompf, has two remarkable properties: (i) it shows the occurence of standard contextual geometries such as the Fano plane (at index $7$), Mermin's pentagram (at index $10$), the two-qubit commutation picture $GQ(2,2)$ (at index $15$) as well as the combinatorial Grassmannian Gr$(2,8)$ (at index $28$) , (ii) it allows the interpretation of MICs measurements as arising from such exotic (space-time) $R^4$'s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of 'quantum gravity'.

math.GT↗

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↗

arXiv2019

Group geometrical axioms for magic states of quantum computing

Let $H$ be a non trivial subgroup of index $d$ of a free group $G$ and $N$ the normal closure of $H$ in $G$. The coset organization in a subgroup $H$ of $G$ provides a group $P$ of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of MIC states associated to minimal informationally complete measurements. It is shown that, in most cases, the existence of a MIC state entails that the two conditions (i) $N=G$ and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups), or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group $G=π_1(M)$ of some manifolds encountered in our recent papers, e.g. the $3$-manifolds attached to the trefoil knot and the figure-eight knot, and the $4$-manifolds defined by $0$-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

math.GR↗

arXiv2019

Quantum Gravity at the Fifth Root of Unity

We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within $SU(2)$ quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering fermionic cycles through the foam we couple this $SU(2)$ quantum group with the same deformation of $SU(3)$, so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. The generalization to higher-dimensional Lie groups $SU(N)$, $G_2$ and $E_8$ is suggested. On this basis we discuss a unifying framework for quantum gravity. Inside the transition amplitude or partition function for geometries, we have the quantum numbers of particles and fields interacting in the form of a spin foam network $-$ in the framework of state sum models, we have a sum over quantum computations driven by the interplay between aperiodic order and topological order.

hep-th↗