arXiv subjects

N. Furey

Publications and source records attributed to N. Furey.

10 recordsLinked to original sources

arXiv2026

Standard Model Symmetries and the Nested Embeddings of $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}$

Where does the Standard Model's internal structure come from? Treating $\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}$ as a module for its own multiplication algebra enables a particular origin story for the Standard Model's pre-Higgs, $\mathfrak{g}_{SM}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus \mathfrak{u}(1)_{Y},$ and post-Higgs, $\mathfrak{g}_{LE}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q},$ symmetries. We recognize the endomorphisms and their modules alike as $\mathbb{Z}_2^n$-graded algebras. Then, annihilating certain highest grade (volume) elements, and enacting an equal-trace condition on anti-hermitian operators leads precisely to $\mathfrak{g}_{SM}$ and $\mathfrak{g}_{LE}$. Weak hypercharge and electric charge operators, $Y$ and $Q,$ take on a remarkably simple form: $\sum \frac{1}{n}\mathbb{I}_{n\times n}$. Upon the introduction of Cayley-Dickson imaginary units, this 15 $\mathbb{R}$ dimensional $\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}$ embeds naturally as a vector space into several well-studied 16 $\mathbb{R}$ dimensional algebras, which we generically refer to as $\mathbb{V}.$ With this embedding, the Standard Model's internal symmetries may then be seen to arise in part from the sequence of nested inclusions: $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset \mathbb{O}\subset\mathbb{V}.$ We define the notion of endomorphic models of particle physics, and connect $End_\mathbb{R}(\mathbb{V})\simeq Cl(0,8)$ to the earlier ideas of Bott Periodic Particle Physics. We comment on a possible connection between the existence of multiple complex structures and the baryon asymmetry problem. In closing, we identify an appearance in this model of the fully connected tree of division algebraic Hopf fibrations that starts at $S^{15},$ and simultaneously involves the four parallelizable spheres $S^7, S^3, S^1, S^0.$

hep-ph↗

arXiv2025

A Superalgebra Within: representations of lightest standard model particles form a $\mathbb{Z}_2^5$-graded algebra

It is demonstrated how a set of particle representations, familiar from the Standard Model, collectively form a superalgebra. Those representations mirroring the internal behaviour of the Standard Model's gauge bosons, and three generations of fermions, are each included in this algebra, with exception only to those irreps involving the top quark. This superalgebra is isomorphic to the Euclidean Jordan algebra of $16\times 16$ hermitian matrices, $H_{16}(\mathbb{C}),$ and is generated by division algebras. The division algebraic substructure enables a natural factorization between internal and spacetime symmetries. It also allows for the definition of a $\mathbb{Z}_2^5$ grading on the algebra. Those internal symmetries respecting this substructure are found to be $\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y,$ in addition to four iterations of $\mathfrak{u}(1)$. For spatial symmetries, one finds multiple copies of $\mathfrak{so}(3)$. Given its Jordan algebraic foundation, and its apparent non-relativistic character, the model may supply a bridge between particle physics and quantum computing. We close by describing current research directions. These include (1) detailing how this construction fits into the larger picture of Bott Periodic Particle Physics, first introduced in [1], [2], [3], (2) investigating how the origin of these particle representations might be grounded in the unsung algebra $\mathbb{O} \oplus \mathbb{H} \oplus \mathbb{C} \oplus \mathbb{R}$, and (3) proposing how these two directions may merge by reframing the model in terms of the 16$\hspace{.5mm}\mathbb{R}$ dimensional sedenion algebra, $\mathbb{S}.$

hep-ph↗

arXiv2024

Three Generations and a Trio of Trialities

We identify the Standard Model's $\mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathfrak{u}(1)$ internal symmetries within the triality symmetries $\mathfrak{tri}(\mathbb{C}) \oplus \mathfrak{tri}(\mathbb{H}) \oplus \mathfrak{tri}(\mathbb{O})$. From here, the corresponding Standard Model group action is applied to the triality triple $\left( \Psi_+, \Psi_-,V\right)$ for $\Psi_+, \Psi_-, V \in \mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Together, $\Psi_+$ and $\Psi_-$ provide the correct irreducible representations for two generations. Owing to a certain Cartan Factorization, which we define, $V$ provides the irreducible representations for a third generation. Said more explicitly in another way, division algebraic multiplication merges a third generation of spinor representations into a set of scalar bosons. This set of scalar bosons includes the familiar Standard Model Higgs representation.

hep-ph↗

arXiv2023

An Algebraic Roadmap of Particle Theories, Part III: Intersections

In this article, we bypass the detailed symmetry breaking pathways established in [1]. Instead, a direct route from the Spin(10) model to the Standard Model is enabled via a single algebraic constraint. This single constraint, however, may be reconfigured as a requirement that three $\mathfrak{so}(10)$ actions coincide on a fixed space of multi-vector fermions. This $\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus{u}(1)_{Y}$ breaking (from a three-way intersection) mirrors, in certain ways, the $\mathfrak{so}(8) \mapsto \mathfrak{g}_2$ breaking (from a three-way intersection) in the context of octonionic triality. By extending this result to include quaternions and complex numbers, we find that a five-way intersection breaks $\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q}$. These are the Standard Model's unbroken gauge symmetries, post-Higgs-mechanism.

hep-ph↗

arXiv2023

An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints

An optimal algebraic model of particle physics has a number of checkpoints to pass. As a minimum, models should $\langle 1 \rangle$ conform to the Coleman-Mandula theorem (or establish a loophole), $\langle 2 \rangle$ evade familiar fermion doubling problems, $\langle 3 \rangle$ naturally explain the Standard Model's chirality, $\langle 4 \rangle$ exclude B-L gauge symmetry at low energy, and $\langle 5 \rangle$ explain the existence of three generations. We demonstrate how the model introduced in [1] passes checkpoints $\langle 1 \rangle, \langle 2 \rangle, \langle 3 \rangle, \langle 4 \rangle$, and has yet to cross $\langle 5 \rangle$. We close by elucidating an unexpected appearance of spacetime symmetries.

hep-ph↗

arXiv2023

An Algebraic Roadmap of Particle Theories, Part I: General construction

Expanding the results of [1], [2], [3], we demonstrate a network of algebraic connections between six well-known particle theories. These are the Spin(10) model, the Georgi-Glashow model, the Pati-Salam model, the Left-Right Symmetric model, the Standard Model both pre- and post-Higgs mechanism. A new inclusion of a quaternionic reflection within the network further differentiates $W^{\pm}$ bosons from the $Z^0$ boson in comparison to the Standard Model. It may introduce subtle new considerations for the phenomenology of electroweak symmetry breaking.

hep-ph↗

arXiv2022

Division algebraic symmetry breaking

Reframing certain well-known particle models in terms of normed division algebras leads to two new results for BSM physics. (1) We identify a sequence of complex structures which induces a cascade of breaking symmetries: Spin(10) $\mapsto$ Pati-Salam $\mapsto$ Left-Right symmetric $\mapsto$ Standard model + B-L (both pre- and post-Higgs-mechanism). These complex structures derive from the octonions, then from the quaternions, then from the complex numbers. (2) We provide, also for the first time we believe, an explicit demonstration of left-right symmetric Higgs representations stemming from quaternionic triality, tri($\mathbb{H}$). Upon the breaking of $su(2)_R$, our Higgs reduces to the familiar standard model Higgs.

hep-ph↗

arXiv2022

One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$

Peering in from the outside, $\mathbb{A} := \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ looks to be an ideal mathematical structure for particle physics. It is 32 $\mathbb{C}$-dimensional: exactly the size of one full generation of fermions. Furthermore, as alluded to earlier in arXiv:1806.00612, it supplies a richer algebraic structure, which can be used, for example, to replace SU(5) with the SU(3)$\times$SU(2)$\times$U(1) / $\mathbb{Z}_6$ symmetry of the standard model. However, this line of research has been weighted down by a difficulty known as the fermion doubling problem. That is, a satisfactory description of SL(2,$\mathbb{C}$) symmetries has so far only been achieved by taking two copies of the algebra, instead of one. Arguably, this doubling of states betrays much of $\mathbb{A}$'s original appeal. In this article, we solve the fermion doubling problem in the context of $\mathbb{A}$. Furthermore, we give an explicit description of the standard model symmetries, $g_{sm}$, its gauge bosons, Higgs, and a generation of fermions, each in the compact language of this 32 $\mathbb{C}$-dimensional algebra. Most importantly, we seek out the subalgebra of $g_{sm}$ that is invariant under the complex conjugate - and find that it is given by $su(3)_C \oplus u(1)_{EM}$. Could this new result provide a clue as to why the standard model symmetries break in the way that they do?

hep-ph↗

arXiv2019

Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra

A considerable amount of the standard model's three-generation structure can be realised from just the $8\hspace{.3mm}\mathbb{C}$-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a $64\hspace{.3mm}\mathbb{C}$-dimensional space. Here we identify an $su(3)\oplus u(1)$ action which splits this $64\hspace{.3mm}\mathbb{C}$-dimensional space into complexified generators of $SU(3)$, together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge. Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ (the Clifford algebra $\mathbb{C} l(8)$). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.

hep-th↗

arXiv2004

Wormhole throats in $R^{m}$ gravity

We consider wormhole geometries subject to a gravitational action consisting of non-linear powers of the Ricci scalar. Specifically, wormhole throats are studied in the case where Einstein gravity is supplemented with a Ricci-squared and inverse Ricci term. In this modified theory it is found that static wormhole throats respecting the weak energy condition can exist. The analysis is done locally in the vicinity of the throat, which eliminates certain restrictions on the models introduced by considering the global topology.

gr-qc↗