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Robert A. Wilson

Publications and source records attributed to Robert A. Wilson.

At least 19 recordsLinked to original sources
arXiv2025

Embeddings of the Standard Model in $E_8$

I present a modified version of the Manogue-Dray-Wilson `octions' model of elementary particles, that overcomes some of the objections to that model that have been raised. In particular, I restore the compactness of the Standard Model gauge group, and show how the symmetry-breaking of the weak $SU(2)$ relates to the symmetry-breaking between the three generations of elementary fermions. In the process of attempting to implement a Dirac equation for three generations of fermions simultaneously, it turns out that some parts of the $E_8$ model are not required for the Standard Model, which is entirely contained in the subalgebra $\mathfrak{so}(7,3)$. In particular a re-interpretation of the Dirac spinors allows us to interpret part of the model as quantum gravity, which is then compared to General Relativity. The general structure of the model shows that the mixing angles depend on masses, and that the masses emerge from quantum interactions with the dynamic background spacetime (vacuum). Some sample calculations are given to support these predictions.

hep-ph↗
arXiv2024

Verification of the conjugacy classes and ordinary character table of the Monster

As part of the programme to re-compute the character tables of all the groups in the Atlas we re-compute the character table of $\mathbb M$, the Monster simple group. We operate under the uniqueness hypotheses of $\mathbb M$ and the existence of an ordinary faithful representation of degree $196883 = 47.59.71$ and determine the conjugacy classes and centralizer orders of the elements of $\mathbb M$. Along the way we re-compute the character tables of centralizers of $p$-elements for $p < 11$ as well as fusions of conjugacy classes of these centralizers in $\mathbb M$.

math.GR↗
arXiv2024

Uniqueness of an $E_8$ model of elementary particles

There are many ways to embed the Lie groups of the Standard Model of Particle Physics in a Lie group of type $E_8$, but so far there is no convincing demonstration that the finite symmetries (and asymmetries) of weak hypercharge, three generations of electrons, three quarks in a proton, and photon polarisation can also be embedded correctly. I show that there is a unique way to embed these finite symmetries consistently, and that the gauge groups of the Standard Model are then uniquely determined. The model is automatically chiral, and the generation symmetry acts as a rotation in a real 2-space, so that the spinors for three generations have only twice as many degrees of freedom in total as the spinors for a single generation. In fact, two distinct generation symmetries arise from the restriction to the Standard Model, related by the CKM and/or PMNS matrices. It therefore appears that these two matrices are not independent. I further speculate on the implications for quantum gravity.

physics.gen-ph↗
arXiv2024

On possible embeddings of the standard model of particle physics and gravity in $E_8$

I investigate the structure of $E_8$ under the action of the subalgebra/subgroup $A_1+G_2+C_3$, as a potential route to unification of the fundamental forces of nature into a single algebraic structure. The particular real form $E_{8(-24)}$ supports a decomposition into compact $G_2$ plus split $A_1+C_3$, which allows a restriction from $G_2$ to $SU(3)$ for the strong force, together with split $SL_2(\mathbb R)$ to break the symmetry of the weak interaction and give mass to the intermediate vector bosons. The factor $C_3$ contains various copies of the Lorentz group $SL_2(\mathbb C)$ and extends the `spacetime' symmetries to the full group of symplectic symmetries of real $3+3$-dimensional phase space. Restricting $G_2$ to the Standard Model $SU(3)$ extends $C_3$ to $A_5$, in the real form $SU(3,3)$, acting on a complex phase space that includes both momentum and current. There is then a natural restriction from $SU(3,3)$ to $SO(3,3)$, describing the action of $SL_4(\mathbb R)$ on phase space. The resulting action of $SL_4(\mathbb R)$ on $E_8$ includes tensors that are equivalent to the stress-energy tensor, the Ricci tensor and the Riemann tensor, and therefore permits the formalism of general relativity to be developed inside $E_{8(-24)}$. The model then suggests unexpected and perhaps subtle ways in which general relativity and particle physics may be forced to modify each other, in order to produce a unified theory.

physics.gen-ph↗
arXiv2024

Octonions, Albert vectors and the group $\mathrm{E}_6(F)$

We present a uniform approach to the construction of the groups of type $\mathrm{E}_6$ over arbitrary fields without using Lie theory. This gives a simple description of the group generators and some of the subgroup structure. In the finite case our approach also permits relatively straightforward computation of the group order.

math.GR↗
arXiv2024

A Clifford algebra model in phase space

I show how the isomorphism between the Lie groups of types $B_2$ and $C_2$ leads to a faithful action of the Clifford algebra $\mathcal C\ell(3,2)$ on the phase space of 2-dimensional dynamics, and hence to a mapping from Dirac spinors modulo scalars into this same phase space. Extending to the phase space of 3-dimensional dynamics allows one to embed all the gauge groups of the Standard Model as well, and hence unify the electro-weak and strong forces into a single algebraic structure, identified as the gauge group of Hamiltonian dynamics. The gauge group transforms between phase space coordinates appropriate for arbitrary observers, and therefore shows how the apparently arbitrary parameters of the Standard Model transform between mutually accelerating observers. In particular, it is possible to calculate the transformation between an inertial frame and the laboratory frame, in order to explain how macroscopic laboratory mechanics emerges from quantum mechanics, and to show how to write down a quantum theory of gravity that is consistent with quantum mechanics, but is not consistent with General Relativity.

physics.gen-ph↗
arXiv2024

A discrete model for Gell-Mann matrices

I propose a discrete model for the Gell-Mann matrices, which allows them to participate in discrete symmetries of three generations of four types of elementary fermions, in addition to their usual role in describing a continuous group $SU(3)$ of colour symmetries. This model sheds new light on the mathematical (rather than physical) necessity for `mixing' between the various gauge groups $SU(3)$, $SU(2)$ and $U(1)$ of the Standard Model. In particular it shows how the anti-Hermitian version of Pauli matrices can act non-trivially on a unitary version of the Gell-Mann matrices, which leads to a non-trivial mixing between the weak and strong nuclear forces. The unitary version of the Gell-Mann matrices can in turn act non-trivially on a quaternionic version of Dirac matrices, which leads to a non-trivial mixing between the strong force and the shape of spacetime defined by the Dirac matrices. Hence this model implies a mixing between the electro-weak-strong forces on the one hand and gravity, as described by General Relativity, on the other. This mixing in turn implies the necessity for both general relativistic corrections to the Standard Model of Particle Physics, and quantum corrections to General Relativity. Contrary to general expectation, both types of corrections seem to be large enough to be tested experimentally.

math.GR↗
arXiv2024

A New Division Algebra Representation of $E_7$

We decompose the Lie algebra $\mathfrak{e}_{8(-24)}$ into representations of $\mathfrak{e}_{7(-25)}\oplus\mathfrak{sl}(2,\mathbb{R})$ using our recent description of $\mathfrak{e}_8$ in terms of (generalized) $3\times3$ matrices over pairs of division algebras. Freudenthal's description of both $\mathfrak{e}_7$ and its minimal representation are therefore realized explicitly within $\mathfrak{e}_8$, with the action given by the (generalized) matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Along the way, we show how to implement standard operations on the Albert algebra such as trace of the Jordan product, the Freudenthal product, and the determinant, all using commutators in $\mathfrak{e}_8$.

math.GR↗
arXiv2023

On subgroups of the Monster isomorphic to $PSL_2(8)$

We describe computer calculations that were used in 2016 to classify subgroups of the Monster isomorphic to $PSL_2(8)$, containing $7B$-elements. It turns out that there is no such $PSL_2(8)$ in the Monster. These calculations confirm earlier unpublished calculations by P. E. Holmes that obtained the same result. The result has also been confirmed in independent calculations by H. Dietrich, M. Lee and T. Popiel, using different software by M. Seysen. Thus this experimental result is shown to be reproducible.

math.GR↗
arXiv2023

Finite symmetry groups in physics

Finite symmetries abound in particle physics, from the weak doublets and generation triplets to the baryon octet and many others. These are usually studied by starting from a Lie group, and breaking the symmetry by choosing a particular copy of the Weyl group. I investigate the possibility of instead taking the finite symmetries as fundamental, and building the Lie groups from them by means of a group algebra construction. Finite group algebras are the natural algebraic structures in which finite symmetry groups, such as the symmetry group of three generations of elementary fermions, are embedded in Lie groups, that are necessary for the formalism of quantum field theory, including the gauge groups of the fundamental forces. They are also the natural algebraic structures for describing representations of groups, which are used for describing elementary particles and their quantum properties. It is natural therefore to ask the question whether finite group algebras can provide a formal underpinning for the standard model of particle physics, and if so, whether this foundation can explain any aspects of the model that are otherwise unexplained, such as the curious structure of the combined gauge group, or the mixing angles between the different forces. In this paper I investigate the relationships between finite symmetry groups and the gauge groups of each of the fundamental forces individually and in combination, and show that the geometry of representations of the finite groups can be used to predict accurate values for a number of the mixing angles in the standard model, including the electro-weak mixing angle, one lepton mixing angle, one quark mixing angle and one of the CP-violating phases.

math.GR↗
arXiv2023

A New Division Algebra Representation of $E_6$

We construct the well-known decomposition of the Lie algebra $\mathfrak{e}_8$ into representations of $\mathfrak{e}_6\oplus\mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation of $\mathfrak{e}_6$, namely the Albert algebra, is thus realized explicitly within $\mathfrak{e}_8$, with the action given by the matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in $\mathfrak{e}_8$, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group $E_6\subset E_8$.

math.GR↗
arXiv2023

Is there a universal concept of mass in fundamental physics?

The concept of mass was introduced as a mathematical abstraction and unifying principle in physics by Newton in the 17th century, and calibrated on a Solar System scale by Cavendish at the end of the 18th century. In the 19th century, this concept proved adequate to explain a vast range of physical processes on all scales from the microscopic to the Solar System. But in the 20th century, attempts to extend this range upwards to the galactic scale, and downwards to subatomic particles, have led to increasing difficulties. Modifications to the concept of mass by Einstein and Dirac have not prevented these difficulties. In this paper, I ask the question, can these difficulties be overcome by further modification of the definitions, or is the concept of mass an unavoidably local (Solar System scale) rather than global concept?

physics.hist-ph↗
arXiv2023

Tetrions: a discrete approach to the standard model

I show how the symmetry-breaking of a recently proposed embedding of the standard model of particle physics in $E_8$ can be explained in terms of the representation theory of the binary tetrahedral group. This finite group provides a link between various types of spin and isospin that can be exploited to `explain' the chirality of the weak interaction, and the existence of three generations of fermions. Two apparently small technical differences between the finite group model and the standard model turn out to have profound consequences for the ways in which the weak and strong forces create mass.

physics.gen-ph↗
arXiv2022

Chirality in an $E_8$ model of elementary particles

We show how chirality emerges naturally from an embedding of the standard model of particle physics into $E_{8(-24)}$. The well-known argument that there is no chiral theory of fundamental physics in $E_8$ is avoided by implementing chirality not as a property of the complexified Lorentz group, but as a property of the complex representations of the real Lorentz group, combined with a real scalar. This avoids the problems of complexification, and ensures that the model is completely contained in the real Lie group.

physics.gen-ph↗
arXiv2022

Octions: An $E_8$ description of the Standard Model

We interpret the elements of the exceptional Lie algebra $\mathfrak{e}_{8(-24)}$ as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra $\mathfrak{su}(3)+\mathfrak{su}(2)+\mathfrak{u}(1)$, and the Lorentz Lie algebra $\mathfrak{so}(3,1)$. Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains GUTs based on $SO(10)$ (Georgi--Glashow), $SU(5)$ (Georgi), and $SU(4)\times SU(2)\times SU(2)$ (Pati--Salam). We then briefly speculate on the role of the remaining elements of $\mathfrak{e}_8$, and propose a mechanism leading to exactly three generations of particles.

hep-ph↗
arXiv2022

Remarks on the group-theoretical foundations of particle physics

I propose the group SL(4,R) as a generalisation of the Dirac group SL(2,C) used in quantum mechanics, as a possible basis on which to build a more general theory from which the standard model of particle physics might be derived as an approximation in an appropriate limit.

physics.gen-ph↗
arXiv2022

A toy model for the W/Z mass ratio

The recently reported $7σ$ anomaly in measurements of the W/Z mass ratio, if confirmed, will demand an extension of some kind to the standard model of particle physics. In this paper I consider whether some recently proposed models are capable, in principle, of resolving this anomaly, subject to experimental confirmation.

hep-ph↗