Split-Octonion Physics

In 1843 Sir William Rowan Hamilton invented the quaternions, carving their defining equations into the stones of Brougham Bridge in Dublin Ireland. Simultaneously the cross product and dot product were born, in one unified package.

By the 1880 Josiah Gibbs and Oliver Heaviside had developed a competing vector notation, with the added benefit that the dot product could be generalized to arbitrary dimensions.

This utility won the day, and lead to tensor arithmetic that biases the mathematical language in favor of higher order dimensions and the possibility of curved metrics. The deal was sealed with general relativity’s use of curved space-time to derive the precession of Mercury.

Unfortunately, this also leads to difficulties. We must eventually learn to distinguish between polar and axial vectors, leading to the notion of tensors and pseudo-tensors. At the quantum mechanical level this leads to non-associative behavior that can not be expressed in the normally associative tensor arithmetic.

Within one year of publication, the quaternion notation was generalized to octonions. This algebra is a non-associative normed division algebra, and a Moufang loop (similar to associativity, but weaker). Thus the properties of nature requiring of bra-ket notation are naturally embedded in the octonions.

Further, the split-complex numbers and split-quaternions (historically also called coquaternions) were known by 1849.  Had the quaternion notation won the day, these algebras would have been familiar when the Lorentz transformations were derived. In one dimension, the algebra of the Lorentz transformations are identical to to the split-complex numbers.

Adding additional split-roots to this algebra, similar to how the octonions were constructed, leads to the split-octonions. This algebra represents a scalar, an axial vector, a polar vector and a pseudo-scalar in one algebraic format. Aside for its zero divisors, with norm 0, this algebra has many of the same algebraic properties of the octonions. This algebra, however, is Lorentzian. In fact, the zero divisors can be seen to have the same singular properties as light cones.

Further, this algebra can be closed over a lattice with natural half-integer values.

So I ask, Were Gibbs and Heaviside Wrong?

To make this argument, we need a theory of gravity that agrees with general relativity to the parametrized post-Newtonian formalism, but in the flat space-time of the split-octonions. The postulates of the refractive field theory of gravity are:

1) Space-time is Minkowskian (flat and relativistic).
2) Gravitational mass IS energy, including the energy of the gravitational field.
3) The gravitational field has an index of refraction proportional to its energy density, impacting all processes.

This theory predicts higher order variations from general relativity that could become measurable in our lifetime, in particular with the proposed LATOR mission.

Chapter 1: The Split-Octonions
Chapter 2: Classical Physics
Chapter 3: The Refractive Field Theory of Gravity

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