Physics, and in particular theoretical physics, advances perforce from the bottom up. At the beginning
of the 20th century the base was Euclidean 3-space, nearly axiomatic in being unavoidable
and unignorable. And there was time. Then there was space-time, a unified geometry, but still not derived:
assumed - taken as given because it worked.
Electromagnetism forces us to think of fields. Fermions, which dominate our experiential material universe,
required the discovery of spinor spaces, and the recognition that Fermions could be elegantly described as
The inclusion of Lie group theory into the mix allowed us to group Dirac spinor fields into multiplets in
meaningful ways, and Emmy Noether required there be associated spin-1 Bosonic fields giving rise to interactions
between the Fermions.
The Nobel prize winning successes in this area led to the hope the idea was extendable, and many educated
guesses were made in the effort to find a Grand Unified Theory based on an all encompassing big Lie group
containing all the smaller bits. Many of these ideas fell from popularity as the muscle of quantum field
theory led to predictions that were at odds with experiment.
The accepted core from which the majority of our attempts to advance this science are hoping to branch
is 1,3-dimensional Minkowski space-time, complex Dirac spinors, and the groups of the Standard Model
( U(1) × SU(2) × SU(3) ).
The point of this page is to explain how ideas developed over decades applying division algebras theory
to theoretical physics can be viewed as a top down approach, consisting of hypotheses, conjectures, and
firm mathematics from which much of this widely accepted structure cascades out as consequences of one core idea.
- A spinor field is the source from which everything else cascades as consequences.
- There exist analytic/field-theoretic reasons that require the spinor field to be
built on spaces in which all spheres surrounding any point (or any manifold diffeomorphic to
a sphere) be parallelizable.
- Hence, R2, R4, R8.
Background Conjecture Implying
- Algebraic representations of these spaces are the division algebras C, H, and O.
- Hence the core of the spinor space is
T = C ⊗ H ⊗ O.
- T is not a division algebra, and it admits a resolution of its identity into nontrivial primitive
orthogonal projection operators (idempotents). This fact is used repeatedly in reductions of mathematical
bits into smaller bits.
Connect to Space Geometry.
- Conventional spinors are associated with pseudo-orthogonal geometries via a Clifford algebra.
- T is the spinor space of a 9-dimensional space, much as
P = C ⊗ H
is associated with a 3-dimensional space.
- The Clifford algebra associated with P is PL, isomorphic to the Pauli algebra.
- The Clifford algebra associated with T is TL, a Pauli algebra for a 9-D space.
The spinor spaces P and T are double what what they should be, the extra degree of freedom associated
with the right action of the quaternion algebra,
on both P and T. The subalgebra of HR consisting of unit elements is isospin SU(2).
Borrowing from Dirac
- Although conjectures have been made as to why this is so,
a Dirac spinor is a double Pauli spinor, so the core spinor space used here is
Clifford Algebra and Space-Time Geometry
- The Clifford algebra associated with T2 is
which is the complexification of the Clifford algebra CL1,9, as the
conventional Dirac algebra is the complexification of CL1,3.
Starting from the spinor space T2, with a smidgeon of conjectures and
historical motivation, we now have a spinor space, an associated 1,9-spacetime,
and an internal SU(2) group action.
Unseeable Octonion Conjecture (UOC)
The nonassociativity of the octonions implies (it is conjectured) that only after projecting away
six octonion units, wherever they may appear, are we left with "seeable" physics
Applied to CL1,9, the UOC causes a 1-vector reduction from ten 1-vectors for a 1,9-spacetime,
to four 1-vectors for a 1,3-spacetime.
The space of 2-vectors gives us a representation of the Lie algebra of the Lorentz group: so(1,9). This gets
so(1,3) × so(6).
- NOTE: there are two ways to do this, one to be associated with matter, one with antimatter.
- Each of the two ways to reduce CL1,9 leads to half of T2 being projected away
(leaving us with matter, or antimatter (this is not a guess)).
- Each of these halves still contain the 6 octonion dimensions the UOC requires us to project away. A final reduction
on T2 projects away 3/4 of what was left in both cases (quarks and antiquarks, and again not a guess).
Final Group Reduction
CL1,9 inherits from this final spinor reduction a final reduction of its own. There is no further
reduction of the 1-vector space (still a set of 1,3-spacetime 1-vectors), but the 2-vectors, viewed as a Lie algebra
get further reduced:
so(1,3) × so(6) → so(1,3) × u(1) × su(3).
Including the su(2) arising from righthanded quaternion actions, that leaves us with:
so(1,3) × u(1) × su(2) × su(3).
Final Spinor Reduction
With respect to that reduced group the spinor space T2
transforms exactly like a family plus
antifamily of leptons, quarks, anitleptons, and antiquarks, all represented by Dirac spinors, and all having
the correct charges (color, hypercharge and isospin). This is not a guess.
As recently pointed out
, if you exist in this universe
you will be able to "see" either the matter part, or the antimatter part, but one of those will be hidden.
Lagrangian Density and Interactions
As has been pointed out
, a Dirac-like Lagrangian density constructed from the T2
and including gauge fields is an algebraically complicated entity, parts of which are real (linear over R
with no imaginary elements from C
, or O
). These real parts can be shown to correspond
to allowable interactions (the bits of T2
that correspond to the neutrino, electron, a flavor of quark,
or any of the corresponding antiparticles, are easily identifiable in T2
Cascading from our initial hyper-spinor space, T2
, we have:
- Spacetime geometries, ending with a seeable 1,3-dimensional geometry;
- Internal symmetry groups, with respect to which T2 consists
of multiplets that are consistent with observation;
- The parts of these multiplets are all 1,3-Dirac spinors;
- A hidden antimatter spacetime;
- A hidden extra 6-dimensional geometry (carrying color charges) connecting the matter 1,3-geometry
to the antimatter 1,3-geometry;
- A collection of field interactions consistent with observation that arises before any attempts to produce
a 1st or 2nd quantized version of this model.
- The existence of natural mechanisms that can give rise to chirality and correct symmetry breaking.
On a more speculative level, suggestions have been made
how to extend the model to include 3 families and antifamilies of leptons, quarks, and their antiparticles.
How does a matter carrying geometry, and antimatter carrying geometry, and the color carrying geometry, mesh with
conventional general relativity? How best to quantize the lot, especially in light of the suggestion that we don't
yet have all the mathematics needed for a complete description? Will this lead to a theory in which QM and GR are
naturally unified, even shown to be consequences of something more fundamental than either? Is there even any point
in attempting to explain phenomena like dark matter without this firmer theoretical foundation? Will string theory
survive in some form yet to be determined? Should I go outside and play now?