N. FUREY

# Humboldt-Universität zu Berlin

Freigeist Fellow in mathematical physics

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## ALGEBRA, PARTICLES, AND QUANTUM THEORY

Welcome to Round 2 of the Algebra, Particles, and Quantum Theory international online seminar series.

Talks are not guaranteed to be recorded/posted, therefore it's always best to attend live. For efficiency reasons, recordings will only be uploaded in large batches. Most talks will begin at 18:00 Berlin time, but not all, due to the accommodation of time zones. This time zone calculator may be helpful.

To be put on the mailing list for future talks, email me at nichol at aims dot ac dot za . You may alternatively find access through the research seminars site.

### 5th December 2022,

18:00 Berlin time

### 16th January 2023,

11:00am Berlin time

Note the earlier time

### Thursday, 2nd March 2023

18:00 Berlin time

Note unusual day of the week

### 28th March 2023

17:00 Berlin time

Note the earlier time

### 3rd April, 2023

18:00 Berlin time

### 1st May 2023

18:00 Berlin time

University of York

Quantization, Dequantization, and Distinguished States on causal sets and beyond

### 15th May 2023

19:00 Berlin time

Note the later time

### 22nd May, 2023

18:00 Berlin time

Tel Aviv University

The many-worlds interpretation of quantum mechanics and the Born rule

## ALGEBRA, PARTICLES, AND QUANTUM THEORY

Round 1 of the Algebra, Particles, and Quantum Theory international online seminar series

For slides, please see the speakers' webpages, or contact them directly.

To be put on the mailing list for future talks, email me at nichol at aims dot ac dot za .

### 1st November 2021,

18:00 Berlin time

### 29th November 2021,

18:00 Berlin time

### 31st January 2022,

18:00 Berlin time

### 14th February 2022,

18:00 Berlin time

### 28th February 2022,

18:00 Berlin time

### 11th April 2022,

17:00 Berlin time

Note the earlier time

Tevian Dray and

Oregon State University

A Division Algebra Description of the Magic Square, including E_8

### 25th April 2022,

17:00 Berlin time

Note the earlier time

### 23rd May 2022,

18:00 Berlin time

Markus Müller

Inst. for quantum optics and quantum information

Quantum theory and Jordan algebras from simple principles

### 13th June 2022,

18:00 Berlin time

## INTRODUCTION

Welcome to my homepage.

I am a VW Stiftung Research Fellow

in mathematical physics

at Humboldt-Universität zu Berlin.

nichol at aims dot ac dot za

My main interests are

division algebras, Clifford algebras, Jordan algebras,

and their relation to particle physics.

## SELECTED PUBLICATIONS

* Furey, An algebraic roadmap of particle theories: Part I: General construction

It turns out that six well-known particle models are each interconnected algebraically. That is,

the Spin(10) model,

the Pati-Salam model,

the Georgi-Glashow model,

the Left-Right Symmetric model,

the Standard Model, and the

Standard Model post-Higgs mechanism.

* Furey, An algebraic roadmap of particle theories: Part II: Theoretical checkpoints

This second paper identifies five obstacles that have recurred in algebraic particle models over the years. That is, it has often proven difficult to:

comply with the Coleman-Mandula theorem,

evade familiar fermion doubling problems,

naturally explain the Standard Model’s chirality,

exclude B-L gauge symmetry at low energy,

explain the existence of three generations.

This second article demonstrates that [Part I] overcomes the first four of these obstacles, but has yet to address the fifth. Perhaps more importantly, it also explains an unexpected appearance of SL(2,C) spacetime symmetries.

* Furey, An algebraic roadmap of particle theories: Part III: Intersections

This third paper points out a direct algebraic route from the Spin(10) GUT to the Standard Model.

Interestingly, it is possible to reconfigure this algebraic constraint so that it may be viewed as the requirement that three inequivalent spin(10) actions be simultaneously satisfied. This appears to mirror certain aspects of the way that division algebraic automorphisms embed inside their triality symmetries.

Finally, this paper shows that the Standard Model’s post-Higgs symmetries can be identified as the intersection of five inequivalent spin(10) actions.

* Furey and Hughes, Division algebraic symmetry breaking, Phys.Lett.B (2022)

This is work together with Mia Hughes of Imperial College London. We first presented these ideas in 2020 for Rutgers' mathematics department. A subsequent video presentation at the University of Edinburgh may be found here. For a popular account, see the cover story written by Michael Brooks at New Scientist.

In a nutshell:

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 division algebraic reflections that induces a cascade of breaking symmetries: Spin(10) ↦ Pati-Salam ↦ Left-Right symmetric ↦ Standard model + B-L (both pre- and post-Higgs-mechanism). These reflections 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. Upon the breaking of SU(2)_R, our Higgs reduces to the familiar standard model Higgs.

Given that this is the first explicit realization (that I am aware of) of quaternionic triality in the Higgs sector, I have been calling this finding Hughes' Higgs.

* Furey and Hughes, One generation of standard model Weyl representations as a single copy of RCHO, Phys.Lett.B (2022)

In this model, the standard model's unbroken gauge symmetries come out as the Lie subalgebra invariant under the complex conjugate.

* Furey, Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra, Phys.Lett.B (2018)

The addendum of this paper outlines a new project I am currently working on together with Beth Romano at King's College London. The basic idea is to define a universal multiplication rule, known as a multi-action, which splits the Clifford algebra Cl(8,C) into Lie, Jordan, and spinor algebras. This multi-action induces a Z_2 graded structure, reminiscent of a supersymmetry algebra, to a limited extent. With this said, one would not anticipate the introduction of the usual superpartners in this framework. We presented our ideas for the Dublin Institute for Advanced Studies' theoretical physics seminar series.

* Furey, SU(3)xSU(2)xU(1) (xU(1)) as a symmetry of division algebraic ladder operators, Eur. Phys. J. C (2018)

By building from the early work of [3], it is shown how the division algebras R, C, H, and O can combine to yield the basic structure of Georgi and Glashow’s SU(5) grand unified theory. However, there is one significant difference. That is, the extra structure provided by the division algebras may enable an escape from SU(5)’s problematic prediction of proton decay.

For a popular account of this work, and the work below, see Natalie Wolchover's article in Quanta Magazine.

* Furey, Standard model physics from an algebra?, PhD thesis, 2016

In chapter 3, it is shown that each of the Lorentz representations of the standard model (scalars, left- and right-handed Weyl spinors, Dirac spinors, Majorana spinors, four-vectors, and the field strength tensor) can be identified as special invariant subspaces of the complex quaternions. This work extends beyond the early 1937 paper of A. Conway [8]. Specifically, the new finding is that each of the standard model's Lorentz representations generalize, in their own way, the notion of a left ideal.

* Furey, Charge quantization from a number operator, Phys.Lett.B, 742 (2015) 195-199

Gunaydin and Gursey’s early model is extended to show not only quarks and anti-quarks using the octonions. Instead, we find a set of states behaving like the eight quarks and leptons of one full generation. Here we show that unitary symmetries of octonionic ladder operators lead uniquely to the two unbroken gauge symmetries, su(3)_c and u(1)_em.

We also find a straightforward way to understand the quantization of electric charge. In this model, electric charge is proportional to the number operator for the system. Hence, we see that electric charge is quantized because number operators can only take on integer values.

It should be noted that [4] came quite close to this result in the late 1970s; differences between these two models are described in [2], p.39.

* Furey, Generations: three prints, in colour, JHEP 10 (2014) 046

Using only the complex octonions (an 8C dimensional algebra) we show how to build the SU(3)c representations corresponding to not one, but three generations of quarks and leptons. The result is worth noting because natural three-generation models are few and far between.

In each of the three papers above, particles and anti-particles are related simply by the complex conjugate, i to -i.

* DeBenedictis, Furey, Wormhole throats in R^m gravity, Class. Quant. Grav., 2005

### VIDEO SERIES

Filmed by Vincent Lavigne, VIVIDCAM

Division algebras and the standard model

1. Division algebras and physics

2. Overview

3. Introduction to the complex quaternions

4. Biquaternions and the Clifford algebra CL(2)

5. Spinors, ideals, and algebraic black holes

6. Weyl spinors as ideals of the complex quaternions

7. How the complex quaternions give each of the Lorentz representations of the SM

8. Introduction to the complex octonions

9. How to get around the non-associativity of the octonions

10. Octonions, SU(3), and the number operator

11. Quarks and leptons as ideals of the Clifford algebra CL(6)

12. Towards SU(2) weak isospin

13. Summary

14. Three generations under SU(3)_c from the complex octonions

## BACKGROUND

### MOTIVATION FOR MY RESEARCH

The standard model of particle physics is the result of decades of collaboration, which began roughly in the 1930s, and converged finally on its current state in 1979, [1]. It is a perfected set of rules that tells us how the known fundamental particles behave. In the decades since 1979, the standard model has seen little in the way of alterations, and yet has survived rigorous experimental testing, nearly completely unscathed.

However, despite its long string of victories, the standard model is in some ways hollow, or incomplete. Roughly speaking, we know how to use the model to make predictions, but we do not know why it is the way it is.

To be more explicit, we do not know why the standard model is based on the gauge group SU(3)xSU(2)xU(1)/Z_6, and not some other gauge group. Even given that gauge group, the standard model does not specify why it uses such a long, apparently arbitrary, list of particles to represent that group. The standard model does not explain why its quarks and leptons are organized into three generations. It does not explain why SU(2) weak isospin acts only on left-handed states. These questions, and others, have gone unanswered now for nearly 40 years.

The main goal of my work is to try to answer questions like these.

### UNDERLYING MATHEMATICAL STRUCTURE

Ultimately, what we are seeking out is an underlying mathematical structure which could explain the behaviour of fundamental particles. But which mathematical structure should this be?

One possibility is to consider a special set of number systems, which generalize the real numbers that we are already accustomed to using in everyday life. They are known as the normed division algebras over the reals. Curiously, it turns out that there exist only four of these algebras. They are the real numbers, R, the complex numbers, C, the quaternions, H, and the octonions, O. They have dimension 1, 2, 4, and 8, respectively.

The real numbers are used nearly everywhere in physics; the complex numbers are central to quantum theory; the quaternions underlie SL(2,C), and hence are tightly entwined with special relativity. In fact, in chapter 3 of [2], it is shown how invariant subspaces of the complex quaternions can concisely describe each of the Lorentz representations of the standard model.

But what is to be made of the octonions, the fourth and final division algebra? Currently, this algebra is not central to any widely accepted theory in physics. However, with R, C, and H undeniably etched into fundamental physics, it is hard not to wonder: is it really the case that O has been omitted in nature?

This becomes a particularly unavoidable question, once one realizes that the number of quarks and leptons within a generation of standard model fermions adds up to eight. (eg. red, green, and blue up quarks, red, green and blue down quarks, electron, and electron-neutrino) Eight, of course, is also the dimension of the octonionic algebra.

One of the earliest breakthroughs along these lines belong to Gunaydin and Gursey, [3], who showed the SU(3)_c structure for quarks and anti-quarks within the octonions. Several authors since then have built directly from Gunaydin and Gursey's model, notably, Casalbuoni et al [4] and Dixon [5]. Much of my own work over the last few years has also made direct use of [3]. Although [4] and [5] were not used in the construction of my model, I would highly recommend that readers explore these earlier texts.

### ONLY AN ALGEBRA ACTING ON ITSELF

My work to date has aimed to describe standard model physics only in terms of division algebras acting on themselves. That is, just one copy of R, one copy of C, one copy of H, and one copy of O. In particular, this means that there is then no liberty to alter the original division algebraic structure by augmenting it to matrices or column vectors. This is a statement worth emphasizing: there are no matrices or column vectors in this formalism. We deal strictly with the division algebras themselves.

Consequently, this means that particles, and the transformations that act on them, both come from the same algebra. Although this model is far from complete, it is anticipated that both bosons and fermions will likewise arise from the same algebra.

### REFERENCES

[1] P. Ramond, Journeys beyond the standard model, Perseus Books, 1999.

[2] C. Furey, Standard model physics from an algebra?, PhD thesis, 2015.

[3] M. Gunaydin, F. Gursey, Quark structure and the octonions, J. Math. Phys., 1973.

M. Gunaydin, F. Gursey, Quark statistics and octonions, PRD, 1974.

[4] A. Barducci, F. Buccella, R. Casalbuoni, L. Lusanna, and E. Sorace, Quantized grassmann variables and unifed theories, Phys. Letters B, 1977.

[5] G. Dixon, Division algebras: octonions, quaternions, complex numbers and the algebraic design of physics, Kluwer Academic Publishers, 1994.

[6] C. Furey, Charge quantization from a number operator, Phys.Lett.B, 742 (2015) 195-199

[7] C. Furey, Generations: three prints, in colour, JHEP 10 (2014) 046

[8] A. Conway, Quaternion treatment of relativistic wave equation, Proceedings of the Royal Society of London, Series A, Mathematical and physical sciences, 162, No 909 (1937).

Division algebras as a unifying structure for elementary particle physics

## MOTIVATION FOR MY RESEARCH

(FOR A GENERAL AUDIENCE)

Note 1: More precisely, four algebras generalize the real numbers *in a particular way*. To see what we mean by *in a particular way*, please see Section 1.1 of https://arxiv.org/pdf/math/0105155.pdf

Note 2: No dividing by zero, of course.