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Dixon, DivisionAlgebras, and Physics:

In July 1994, Geoffrey Dixon's book

Division Algebras: Octonions, Quaternions, Complex Numbers, andthe Algebraic Design of Physics

was published by Kluwer (ISBN: 0-7923-2890-6)

(see clf-alg/dixo9401 on the Clifford Algebra bulletin board):

Abstract: The four real division algebras (reals, complexes,quaternions and octonions) are the most obvious signposts to a richand intricate realm of select and beautiful mathematical structures.Using the new tool of adjoint division algebras, with respect towhich the division algebras themselves appear in the role of spinorspaces, some of these structures are developed, including parallelizablespheres, exceptional Lie groups, and triality. In the case oftriality the use of adjoint octonions greatly simplifies itsinvestigation. Motivating this work, however, is a strong convictionthat the design of our physical reality arises from this selectmathematical realm. A compelling case for that conviction ispresented, a derivation of the standard model of leptons andquarks.


Table of contents:I Underpinnings  1.1  The Argument  1.2  Clifford Algebras  1.3  Conjugations and Spinors  1.4  Algebraic Fundamentals of the Standard ModelII Division Algebras Alone  2.1  Mostly Octonions  2.2  Adjoint Algebras  2.3  Clifford Algebras, Spinors  2.4  Resolving the Identity of OL  2.4  Lie Algebras, Lie Groups, from OL  2.5  From Galois Fields to Division Algebras: An InsightIII Tensor Algebras  3.1 Tensoring Two: Clifford Algebras and Spinors  3.2 Tensoring Two: Spinor Inner Product  3.3 Tensoring Three: Clifford Algebras and Spinors  3.4 Tensoring Three: Spinor Inner Product  3.5 Derivation of the Standard Symmetry  3.6 SU(2)xSU(3) Multiplets and U(1)IV Connecting to Physics  4.1 Connecting to Geometry  4.2 Connecting to Particles  4.3 Parity Nonconservation  4.4 Gauge Fields  4.5 Weak Mixing  4.6 Gauging SU(3)V  Spontaneous Symmetry Breaking  5.1 Scalar Fields  5.2 Scalar Lagrangians  5.3 Fermions and ScalarsVI 10 Dimensions  6.1 Fermion Lagrangian  6.2 More SU(3)  6.3 Freedom from Matter/antimatter Mixing  6.4 (1,9)-Scalar Lagrangian  6.5 Charge Conjugation on TL(2)  6.6 Charge Conjugation on T2  6.7 10 Other DimensionsVII Doorways  7.1 Moufang and Other Identities  7.2 Spheres and Lie Algebras  7.3 Triality  7.4 LG2 and Tri  7.5 LG2 and the X-ProductVIII Corridors  8.1 Magic Square  8.2 The Ten MS(KK')  8.3 Spinor(KK') Outer Products  8.4 LF4 = MS(RO)  8.5 J(3,O) and F4  8.6 More Magic Square


Dixon's papers include:

Division Algebras, Galois Fields, Quadratic Residues hep-th/9302113

Division Algebras, (1,9)-Space-Time, Matter-antimatter Mixinghep-th/9303039

Octonion X-product Orbits hep-th/9410202

Octonion X-product and Octonion E8 Lattices hep-th/9411063

Octonions: E8 Lattice to /\16 hep-th/9501007

Octonion XY-Product hep-th/9503053

Octonions: Invariant Representation of the Leech Lattice hep-th/9504040


Octonion X,Y Product G2 Variants hep-th/9604116

Division Algebras: 26 Dimensions; 3 Families hep-th/9902016

(1,9)-spacetime to (1,3)-spacetime: Reduction to U(1)xSU(2)xSU(3)hep-th/9902050

AlgebraicSpinor Reduction Yields the Standard Symmetry and FamilyStructure

TensoredDivision Algebras: Resolutions of the Identity and ParticlePhysics


Here are some of my thoughts ( as of 5 September 2000 ) onreading TensoredDivision Algebras: Resolutions of the Identity and ParticlePhysics:

As I understand it, the starting point is M(64,C) = Cl(12,1) with2^13 = 2^6 x 2^6 x 2 = 64x64x2 = 8,192 real dimensions, and whosegraded structure is

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

and whose bivector Lie algebra is 78-dimensional Spin(12,1), anon-euclidean version of Cl(13).

Another 78-dimensional algebra is E6, which has a 32-dim HermitianSymmetric Space (I'll call it HSS32) such that

HSS32 = E6 / Spin(10)xU(1)

so that 78-dim E6 breaks down into 32-dim and (45+1)-dim.

HSS32 has complex structure, so it is really a 16-complex-dimspace, and in fact is just the 32-dim spinor space of the Cl(10)Clifford algebra of Spin(10).

Do the 16 complex dimensions of HSS32 correspond to the 16orthogonal subspaces of the form Tmn+Tmn of Geoffrey Dixon'sdecomposition of your Cl(13) spinor space T+T, the space of 2x1column matrices over T = CxQxO ?

From Geoffrey Dixon's point of view, it looks like he is lookingat Cl(12,1) in terms of the Cl(9,1) spinors, and

from my point of view, it looks like you could also (if youchanged the multiplication table somewhat) build E6 from Cl(9,1) andits spinors.

Since the chain E6 -> D5 -> D4 is the basis of the VoDouD4-D5-E6-E7 physics model, it would be nice to write it in termsof

Cl*(13) -> Cl(10) -> Cl(8)

where Cl* is something like (but maybe not exactly) a Cliffordalgebra.

Putting in non-Euclidean signature ought to be straightforward, Ithink.

HERE is my rough attempt tocompare Dixon's approach with my D4-D5-E6-E7physics model,


HERE is my roughattempt to combine Dixon's approach with KenichiHorie's GeometricInterpretation of Electromagnetism in a Gravitational Theory withTorsion and Spinorial Matter.




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