Summer homework
Hello:
I know maybe six professors in physics at MIT. I am the sort to approach them after they have given a talk and pitch my ideas. So far this approach has never worked. I try to be polite and not too much of a pest. The only way an elite physicist will buy into the program is to do the work themselves. Therefore I wrote up the core idea as a problem they could solve and sent an email. Here is what I wrote:
[quote]
Derive the Maxwell equations using quaternions, as Maxwell predicted someone would do someday in the introduction of his "Treatise on Electricity and Magnetism". Proceed by writing the action using the tensor formalism with differentials and 4-potentials you are familiar with. Erase the Greek letters. Add enough conjugate operators, and reverse the order of the differential and potential so the scalar term is B^2 - E^2. Note that the extra 3-vector that tags along is already significant in EM theory. Use the Euler-Lagrange equations to derive the Maxwell equations.
Metrics are symmetric. Gravity can be characterized using changes in symmetric metrics which are also symmetric. Quaternions cannot represent this change since they have the antisymmetric cross product and curl. Rewrite the Maxwell action with hypercomplex numbers, or what I prefer to call the (non-isomorphic) California representation of quaternions where everything is positive: i^2 = j^2 = k^2 = +1, ij=ji=k. You will need to use a few more conjugate operators so that the result is invariant under a Lorentz transformation (easiest to start with the JA term). By applying the Euler-Lagrange equation to the hypercomplex Maxwell action, instead of getting Gauss' law, you should get a form of Newton's law of gravity in the potential form. This equation does have a time-dependent term, removing the need for general relativity as described in MTW, chapter 7. Show how the Rosen metric is a solution to that equation and thereby consistent with all current tests of weak field gravity. For a bonus, show that the J A* term has the phase needed for a spin 2 particle, but it is not a projection operator (Lie algebra), but instead uses a Jordan algebra.
Have a good Summer. I hope you get a chance to work on this
wonderful, specific problem.
Doug, '84 (25th reunion)
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Anyone reading this post can try it themselves, or pass it along to professionals you know.
Doug what pages of ch 7 in MTW are you referring to?
<em>This equation does have a time-dependent term, removing the need for general relativity as described in MTW, chapter 7</em>
__________________
Lowell
Hello Lowell:
I was referring to the first section, second paragraph, in a statement about the field equations, equation 7.2.
"Likewise, the field equation (7.2) is not Lorentz-invariant, since the appearance of a three-dimensional Laplacian operator instead of a four-dimensional d'Almbertian oprator means that the potential
responds instantaneously to changes in the density 
at arbitrarily large distances away. In brief, Newtonian gravitational field propagate with inifinte velocity."
The section goes on to tell how people tried to modify the field equations of Newton to be consistent with special relativity, and end up at Einstein's field equations. I do not understand the details of those papers, but the result is neat. This is not how one should look for field equations. Instead one should start with the action, then derive the field equations using Euler-Lagrange. The reason this is the better road is that different derivatives of the action result in other quantities of interest such as the stress-energy tensor. Quantizing a proposal starts from the action.
In exercise 7.2, they do give a vector gravitational field a try, but it starts out on the wrong foot. We know that the effects of gravity can be accurately described by changes in a metric tensor. A metric tensor is symmectric, so changes in such a metric are symmetric. Yet the use the 4-vector to form an antisymmetric field strength tensor. That approach is destine to fail, which is what their calculation does.
I have had a tense discussion with one or two differential geometry jocks, and they simply will not accept the notion of a symmetric rank 2 field strength tensor. I cannot tell you why they have such an aversion, just that the two folks I interacted with would not look at it.
Doug