Non-associative quaternion multiplication
Hello Lawrence:
This is the definition for a Hamilton quaternion product I am using:
<br />
i^2~=~j^2~=~k^2~=~ijk~=~-1.<br />
While this is true:
The nonassociative product.
<br />
(e_ie_j)e_k~-~e_i(e_je_k)~=~\upsilon_{ijkl}e_l<br />
can be defined where \upsilon_{ijkl} is defined without reference to octonions.
it is not relevant because I am using a conjugate operator, or more technically an anti-involutive automorphism. The conjugate flips the sign of the 3-vector.
What I call a Euclidean product has a closed multiplication table:
i^{*} i= j^{*} j = k^{*} k= (i^{*} j)^{*} k = 1
The standard definition of a non-associative product you provide does not use a conjugate operator as I do. Because the automorphism keeps the quaternions within the quaternion manifold, the multiplication table is closed: i^{*} i = 1. Let's see this in action, how always taking the conjugate of the first quaternion in the binary quaternion multiplication makes the triple product non-associative.
a (b c) = a^{*} (b^{*} c) = a^{*} b^{*} c
(a b) c = (a^{*} b)^{*} c = a b^{*} c
These are not the same. The norm of these two are same, but they point in different directions.
Our current gauge theories for gravity and EM are both spectacular and inadequate. Both match experimental data extraordinarily well. Yet we cannot quantize EM without choosing a gauge. Even if a gauge is chosen, gravity cannot be quantized.
Let me reestablish how I am working from the "domain of experience". Gravity and EM as currently formulated work in 4 dimensions, as does GEM. It is odd that simple statement is at odds with the majority of work on gravity today. The group Diff(M) was used by Einstein to make his equations covariant, and it plays a role in GEM. EM symmetry must be broken by a scalar field, as it is in GEM by the trace of the field strength tensor. The 4D EM wave equation can be quantized by the Gupta-Bleuler method, but two of the modes of emission must be eliminated by a supplementary condition. GEM uses the same creation and annihilation operators, but it also has a spin 2 field that will not have the same technical issues the spin 1 scalar mode has. And in keeping with the most important of all scientific traditions, should we refine the level of accuracy of measuring light around the Sun, GEM predicts 0.7 microacrsecond more than GR, so unlike all the work in string theory, I am willing to put every chip I own on the line and let the data decide.
How things get written matters. You write the field strength tensor as F = dA. Nothing can be simpler than that. Sure, it has all the differential forms stuff behind it, but since that is the starting point, I can appreciate that there appears nothing simpler. I write it differently, also in an established way, as a F=\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu}. In my notation, I can explain even to my girlfriend that the F thingie is not a complete story because of the act of subtraction: there are 4 parts in the triangle, 4 parts in the A, sixteen parts total, but the study of light needs only 6 of the 16. What I cannot explain to her is why the brightest folks on the planet do not share my feelings of obligation to work with the other 10 terms.
dougHistorical note: Maxwell was familiar with quaternions because Hamilton trumpeted their cause (too loudly). Scalar, vector, div, grad, curl were inventions of Hamilton in his quest to understand quaternions. In Maxwell's first edition of his treatise, he used "pure quaternions" which is one with a zero for the scalar, effectively as a stand-in for 3-vectors. He did nothing at all fancy with quaternion algebra. By the third edition, he only hoped that quaternions would play a role in describing EM. Many folks have figured out how to toss in an extra imaginary number and get the Maxwell equations. Peter Jack was the first person to do so with real quaternion (1996?) and I did the same independently a year later. There is a rumor on the Internet that some 200 quaternion equations were deleted from the first to the third edition. From my viewing of an on-line version of the first treatise, that is an inaccurate description: the quaternions were used like any 3-vector, and he did not write the homogeneous or source equations as an exercise in quaternion algebra.
