Those who have followed my Weblog are aware that I have been putting in a lot of work on Kaluza-Klein theory. This post is to step back from the canvas, lay out the overall picture of what I am pursuing, and summarize what I plan at present to change or correct in the coming days and weeks. This is in keeping with the concept of this Weblog as “Lab Notes,” or as a public “scientific diary.”
There are really two main aspects to this Kaluza-Klein work:
First, generally, I have found that 5-D Kaluza-Klein theory is most simply approached by starting with (classical) Lorentz force motion, and requiring the Lorentz force motion to be along the geodesics of the five dimensional geometry. I am far from the first person who recognizes that the Lorentz force can be represented as geodesic motion in a 5-D model. But I have found, by starting with the Lorentz force, and by requiring the 5-D electromagnetic field strength tensor to be fully antisymmetric, that all of the many “special assumptions” which are often employed in Kaluza-Klein theory energy very naturally on a completely deductive basis, with no further assumptions required. I also believe that this approach leads to what are perhaps some new results, especially insofar as the Maxwell tensor is concerned, and insofar as QED may be considered in a non-linear context. The latest draft of this global work on Kaluza-Klein may be seen at Kaluza-Klein Theory and Lorentz Force Geodesics.
Second, specifically, within this broader context, is the hypothesis that the fifth-dimensional “curled” motion is the direct mainspring of intrinsic spin. More than anything else, the resistance by many physicists to Kaluza-Klein and higher-dimensional theories, rests on the simple fact that this fifth dimension — and any other higher dimensions — are thought to not be directly observable. In simplest form, “too small” is the usual reason given for this. Thus, if it should become possible to sustain the hypothesis that intrinsic spin is a directly-observable and universally-pervasive outgrowth of the fifth dimension, this would revitalize Kaluza-Klein as a legitimate and not accidental union of gravitation and electrodynamics, and at the same time lend credence to the higher-dimensional efforts also being undertaken by many researchers. The latest draft paper developing with this specific line of inquiry is at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.
Now, the general paper at Kaluza-Klein Theory and Lorentz Force Geodesics is very much a work in progress and there are things in this that I know need to be fixed or changed. If you should review this, please keep in mind the following caveats:
First, sections 1-4 are superseded by the work at Intrinsic Spin and the Kaluza-Klein Fifth Dimension and have not been updated recently.
Second, sections 5-7 are still largely OK, with some minor changes envisioned. Especially, I intend to derive the “restriction” from rather than impose it as an ad hoc condition.
Third, sections 8-11 needs some reworking, and specifically: a) I want to start with an integration over the five-dimensional volume with a gravitational constant suited thereto, and relate this to the four dimensional integrals that are there at present; and b) I have serious misgivings about using a non-symmetric (torsion) energy tensor and am inclined to redevelop this to impose symmetry on the energy tensor — or at least to explore torsion versus no torsion in a way that might lead to an experimental test. If we impose symmetry on the energy tensor, then the Maxwell tensor will be the special case of a broader tensor which includes a term and which applies, e.g., to energy flux densities (Poynting components) , k=1,2,3 for “waves” of large numbers of electrons.
Fourth, I am content with section 12, and expect it will survive the next draft largely intact. Especially important is the covariant derivatives of the electrodynamic potentials being related to the ordinary derivatives of the gravitational potentials, which means that the way in which people often relate electrodynamic potentials to gravitational potentials in Kaluza-Klein theory is valid only in the linear approximation. Importantly, this gives us a lever in the opposite direction, into non-linear electrodynamics.
Fifth, I expect the development of non-linear QED in section 13 to survive the next draft, but for the fact that the R=0 starting point will be removed as a consequence of my enforcing a symmetric energy tensor in sections 8-11. Just take out all the “R=0” terms and leave the rest of the equation alone, and everything else is more or less intact.
Finally, the experiment in Section 15, if it stays, would be an experiment to test a symmetric, torsionless energy tensor against a non-symmetric energy tensor with torsion. (Basically, metric theory versus Cartan theory.) This is more of a “back of the envelope” section at present, but I do want to pursue specifying an experiment that will test the possible energy tensors which are available from variational principles via this Kaluza-Klein theory.
The paper at Intrinsic Spin and the Kaluza-Klein Fifth Dimension dealing specifically with the intrinsic spin hypothesis is also a work in progress, and at this time, I envision the following:
First, I will in a forthcoming draft explore positrons as well as electrons. In compactified Kaluza-Klein, these exhibit opposite motions through , and by developing the positron further, we can move from the Pauli spin matrices toward the Dirac and Dirac’s equation.
Second, I have been engaged in some good discussion with my friend Daryl M. on a thread at sci.physics.relativity. Though he believes I am “barking up the wrong tree,” he has provided a number of helpful comments, and especially at the bottom of post #2 where he discusses quantization in the fifth dimension using a wavelength . (I actually think that for fermions, one has to consider orientation / entanglement issues, and so to secure the correct “version,” one should use which introduces a factor of 2 which then can be turned into a half-integer spin.) I am presently playing with some calculations based on this approach, which you will recognize as a throwback to the old Bohr models of the atom.
Third, this work of course uses to define the compact fifth dimension. However, in obtaining , I have taken to be a fixed, constant radius. In light of considering a wavelength per above, I believe it important to consider variations in rather than fixed , and so, to employ .
There will likely be other changes along the way, but these are the ones which are most apparent to me at present. I hope this gives you some perspective on where this “work in flux” is at, and where it may be headed.
Thanks for tuning in!