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Jay.
Lab Notes for a Scientific Revolution (Physics) 
I’ve been reading your essay on Kaluza Klein theory. There’s a serious mistake:
If g_MN is symmetric
and if g^MN is the inverse matrix of g_MN
then g^MN is symmetric
Then, ∂ g^MN/∂ g^NM = 1
So if M ≠ N, ∂ L_matter / ∂ g^MN picks up two terms from the expression
L_matter = – b κ‾ g^MN δ^5_N J_M / 8 κ
with opposite permutations of M and N.
This means symmetrization is automatic with the ordinary assumptions. If you actually want to write a theory with T_5M ≠ T_M5, one of three assumptions has to be modified:
* the symmetry g_MN = g_NM
* the inversion law
* the definition of the Christoffel Symbol
There is also an intuitive problem with the fifth-dimensional velocity being inversely proportional to mass. The electric charge q is the conserved charge of the U1 symmetry. The rate at which a particle rotates around the U1 circle, in cycles per unit time, is mc² / 2πћ. It would seem natural to identify the U1 circle with the fifth dimension, so the proportionality should be direct, not inverse.
By the way, the Uncertainty Principle is a tautology in terms of the position and momentum operators. Its status as a physical law arises from the fact that it applies to position and momentum variables, which are defined differently from the operators. Thus it is equivalent to the mysterious duality between the variables and the operators.
Comment by Collin — May 19, 2008 @ 11:36 pm |
Hi Collin:
Thank you for your comment, and I agree about the symmetry of T^5M. For a brief time I became enamored with antisymmetric field theory, but in the end after reading all about Cartan theory and torsion / spin density tensors and the like, became convinced that one must, in fact, work with a symmetrized energy tensor. I also thought that perhaps I had found a way around modifying any of the three assumptions you lay out, but I had not.
I have not had an opportunity to redraft the material you were reading at:
https://jayryablon.files.wordpress.com/2008/02/kaluza-klein-and-lorentz-force-complete-paper.pdf , (there is a later version at https://jayryablon.wordpress.com/files/2008/03/kaluza-klein-theory-and-lorentz-force-geodesics-60.pdf )specifically eqs. (8.3)and (8.4) therein, but when I do, that is the most important change I will make.
The downstream implication of such a change, in the above essay, is that the Maxwell tensor no longer emerges “as is” from the variational calculation (which is why I considered a non-symmetric tensor so seriously). Rather, one gets a stress energy tensor T^uv (Maxwell) + J^uA^v + J^vA^u so that the Maxwell tensor is a special case where J^u = 0, i.e., absent a current density. Actually, that now seems to me to be desirable, because in this context the Maxwell tensor is the energy tensor for pure EM radiation (no current), but then, if one considers electron radiation (firing “waves” of electrons rather than light at a target), one gets a modified energy tensor with a non-zero trace.
I will comment separately on the spin interpretation of the fifth dimension, though will note that at this time, I have focused on understanding spin totally within the confines of four dimensions, see the Ohanian paper at https://jayryablon.wordpress.com/files/2008/04/ohanian-what-is-spin.pdf which I have periodically discussed.
I would be interested if you could amplify your remarks about Heisenberg Uncertainty Principle.
Thanks again!
Jay.
Comment by Jay R. Yablon — May 20, 2008 @ 11:22 am |
Thanks for the link to the Ohanian paper. It’s similar to my thoughts about spin. (I’ll get back with you later about the Uncertainty Principle.)
One thing I would add, which may have something to do with the magnetic moment anomaly, is the question of how to interpret the spinor itself. In the fully 4-dimensional formalism, a pair of spinors always has γ^0 between them. In the Clifford Algebra interpretation the gamma matrices have directions like vectors. So what does γ^0 point to? It cannot be the laboratory velocity, because it doesn’t change under boosts. My theory is that it is — if I may be so bold to use the word — an aether. A spinor is traditionally considered to be a column, whose components have no simple relationship to spacetime directions. However, this could be considered a representation of incomplete information. The column can be augmented into a complete matrix. With the restriction that the matrix be a real polynomial of gammas, that the transformation is linear, and that the effect of multiplying on the left by gammas is unchanged, the possibilities span the surface of a sphere in the rest frame of γ^0. Furthermore, the sum of the representations on opposite points of the sphere is the same throughout the sphere, and for a direction γ^a, they are formed by multiplying this common value on the right by (1 ± σ^(0a))/2. In other words, they are projection operators. If σ^(0a) is, in some sense, a “polarization” of the laboratory device, then it could be said that the physical field contains twice as much information as can be observed, which is sufficient to determine a spinor representation for any choice of the three spatial axes. The time axis γ^0, however, cannot be fixed either actively by the laboratory device nor passively by a reference frame. However, the augmentation allows operators to be placed on both sides of the spinor, so that matrix products can be decomposed into even and odd Clifford products, which can be converted into coordinate notation. Thus the Dirac equations can be rewritten in a form that depends dynamically on γ^0. I will be preparing a draft of this in actual formulas.
Comment by Collin — May 22, 2008 @ 11:21 pm |
Can I upload a PDF?
Comment by Collin — May 22, 2008 @ 11:26 pm |
Collin, Send it to me at jyablon@nycap.rr.com, and if it looks suitable, I’ll upload it, then send you the link so you can refer to it in a blog post. Jay.
PS: The Dirac gamma^0 is a projection operator that eliminates two of four spinor degree of freedom. See,e.g. Zee, QFT in a Nutshell, at pp. 90-91, so as to leave behind only the two physical degrees of freedom for a fermion, spin up and spin down.
Comment by Jay R. Yablon — May 22, 2008 @ 11:45 pm |