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. We explain both these approaches, and how to reconcile
them.
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. It
also sheds light on why the Standard Model gauge group acts on 10d
Euclidean space, or Minkowski spacetime, while preserving a 4+6 splitting.
Dubois-Violette and Todorov noticed that the gauge group of the
Standard Model of particle physics is the intersection of two maximal
subgroups of \(\text{F}_4\), which is the automorphism group of the
exceptional Jordan algebra \(\mathfrak{h}_3(\mathbb{O})\). Here we
show that these can be taken to be any subgroups preserving copies of
\(\mathfrak{h}_2(\mathbb{O})\) and \(\mathfrak{h}_3(\mathbb{C})\)
that intersect in a copy of \(\mathfrak{h}_2(\mathbb{C})\). Thus, the
Standard Model gauge group acts on the octonionic projective plane in
a way that preserves an octonionic projective line and a complex
projective plane intersecting in a complex projective line. This is
joint work with Paul Schwahn.
Part 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under SU(3).
Part 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.
Part 3. How a lepton and a quark fit together into an octonion — at least if we only consider them as representations of SU(3), the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group SU(3).
Part 4. Introducing the exceptional Jordan algebra: the 3×3 self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the 2×2 adjoint octonionic matrices form precisely the Standard Model gauge group.
Part 5. How to think of the 2×2 self-adjoint octonionic matrices as 10-dimensional Minkowski space, and pairs of octonions as left- or right-handed Majorana–Weyl spinors in 10 dimensional spacetime.
Part 6. The linear transformations of the exceptional Jordan algebra that preserve the determinant form the exceptional Lie group E6. How to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and left-handed spinors in 10d Minkowski spacetime.
Part 7. How to describe the Lie group E6 using 10-dimensional spacetime geometry. This group is built from the double cover of the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
Part 8. A geometrical way to see how E6 is connected to 10d spacetime, based on the octonionic projective plane.
Part 9. Duality in projective plane geometry, and how it lets us break the Lie group E6 into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
Part 10.
Jordan algebras, their symmetry groups, their invariant structures — and how they connect quantum mechanics, special relativity and projective geometry.
Part 11. Particle physics on the spacetime given by the exceptional Jordan algebra: a summary of work with Greg Egan and John Huerta.
Part 12. The bioctonionic projective plane and its connections to algebra, geometry and physics.
