Collection Number C21001
Collection Date -
Collection Type Conference/School

Over the years, various researchers have suggested connections between the octonions and the standard model of particle physics. The past few years, in particular, have been marked by an upsurge of activity on this subject, stimulated by the recent observation that the standard model gauge group and fermion representation can be elegantly characterized in terms of the octonions. This workshop, which will be the first ever on this topic, is intended to bring this new community together in an attempt to better understand these ideas, establish a common language, and stimulate further progress.

The workshop will consist of an hour-long talk every Monday at noon (EST), with the first talk on Monday February 8, and the final talk on Monday May 17.


The standard model, left/right symmetry, and the "magic square"

Latham Boyle Perimeter Institute for Theoretical Physics
Recently, an intriguing connection between the exceptional Jordan algebra h_3(O) and the standard model of particle physics was noticed by Dubois-Violette and Todorov (with further interpretation by Baez). How do the standard model fermions fit into this story? I will explain how they may be neatly incorporated by complexifying h_3(O) or, relatedly, by passing from RxO to CxO in the so-called "magic square" of normed division algebras.

Jordan algebras: from QM to 5D supergravity to … Standard Model?

Paul Townsend University of Cambridge
This talk will be about two applications of Jordan algebras. The first, to quantum mechanics, follows on from the talk of John Baez. I will explain how time dependence makes use of the associator, and how this is related to the commutator in the standard density matrix formulation. The associator of a Jordan algebra also determines the curvature of a Riemannian metric on its positive cone, invariant under the symmetry group of the norm (mentioned in the talk of John Baez); the cone is foliated by hypersurfaces of constant norm.

Spin(11,3), particles and octonions

Kirill Krasnov University of Nottingham
The fermionic fields of one generation of the Standard Model, including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S of the group Spin(11,3). I will describe an octonionic model for Spin(11,3) in which the semi-spinor representation gets identified with S=OxO', where O,O' are the usual and split octonions respectively. It is then well-known that choosing a unit imaginary octonion u in Im(O) equips O with a complex structure J.

Can We Understand the Standard Model Using Octonions?

John Baez University of California
Dubois-Violette and Todorov have shown that the Standard Model gauge group can be constructed using the exceptional Jordan algebra, consisting of 3×3 self-adjoint matrices of octonions. After an introduction to the physics of Jordan algebras, we ponder the meaning of their construction. For example, it implies that the Standard Model gauge group consists of the symmetries of an octonionic qutrit that restrict to symmetries of an octonionic qubit and preserve all the structure arising from a choice of unit imaginary octonion.

Can We Understand the Standard Model?

John Baez University of California
40 years trying to go beyond the Standard Model hasn't yet led to any clear success. As an alternative, we could try to understand why the Standard Model is the way it is. In this talk we review some lessons from grand unified theories and also from recent work using the octonions. The gauge group of the Standard Model and its representation on one generation of fermions arises naturally from a process that involves splitting 10d Euclidean space into 4+6 dimensions, but also from a process that involves splitting 10d Minkowski spacetime into 4d Minkowski space and 6 spacelike dimensions.

The Higher Algebra of Supersymmetry

John Huerta University of Lisbon
We have already met the octonionic Fierz identity satisfied by spinors in 10-dimensional spacetime. This identity makes super-Yang-Mills "super" and allows the Green-Schwarz string to be kappa symmetric. But it is also the defining equation of a "higher" algebraic structure: an L-infinity algebra extending the supersymmetry algebra. We introduce this L-infinity algebra in octonionic language, and describe its cousins in various dimensions. We then survey various consequences of its existence, such as the brane bouquet of Fiorenza-Sati-Schreiber.

Clifford algebra of the Standard Model

Ivan Todorov Bulgarian Academy of Sciences - Institute for Nuclear Research and Nuclear Energy
We explore the Z2 graded product C`10 = C`4⊗ˆC`6 (introduced by Furey) as a finite quantum algebra of the Standard Model of particle physics. The gamma matrices generating C`10 are expressed in terms of left multiplication by the imaginary octonion units and the Pauli matrices. The subgroup of Spin(10) that fixes an imaginary unit (and thus allows to write O = C⊗C 3 expressing the quark-lepton splitting) is the Pati-Salam group GP S = Spin(4) × Spin(6)/Z2 ⊂ Spin(10). If we identify the preserved imaginary unit with the C`6 pseudoscalar ω6 = γ1...γ6, ω2 6 = −1 (cf.

Division algebraic symmetry breaking

Can the 32C-dimensional algebra R(x)C(x)H(x)O offer anything new for particle physics? Indeed it can. Here we identify a sequence of complex structures within R(x)C(x)H(x)O which sets in motion a cascade of breaking symmetries: Spin(10) -> Pati-Salam -> Left-Right symmetric -> 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.