# Lab Notes for a Scientific Revolution (Physics)

## March 30, 2008

### Revised Paper on Kaluza-Klein and Intrinsic Spin, Based on Spatial Isotropy

I have now prepared an updated revision of a paper demonstrating how the compact fifth dimension of Kaluza-Klein is responsible for the observed intrinsic spin of the charged leptons and their antiparticles.  The more global, underlying view, is that all intrinsic spins originate from motion through the curled up, compact $x^5$ dimension.  This latest draft is linked at:

Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0

Thanks to some very helpful critique from Daryl M. on a thread at sci.physics.relativity, particularly post #2, I have entirely revamped section 5, which is the heart of the paper wherein we establish the existence of intrinsic spin in $x^5$, on the basis of “fitting” oscillations around a $4\pi$ loop of the compact fifth dimension (this is to maintain not only orientation but entanglement version).  From this approach, quantization of angular momentum in $x^5$ naturally emerges, it also emerges that the intrinsic $x^5$ angular momentum in the ground state is given by $(1/2) \hbar$.

In contrast to my earlier papers where I conjectured that the intrinsic spin in $x^5$ projects out into the three ordinary space dimensions by virtue of its orthogonal orientation relative to the $x^5$ plane, which was critiqued in several Usenet posts, I now have a much more direct explanation of how the intrinsic spin projects out of $x^5$ to where we observe it.

In particular, we recognize that one of the objections sometimes voiced with regards to a compactified fifth dimension is the question: how does one “bias” the vacuum toward one of four space dimensions, over the other three, by making that dimension compact?  This was at the root of some Usenet objections also raised earlier by DRL.

In this draft — and I think this will overcome many issues — we require that at least as regards intrinsic angular momentum, the square of the $J^5 = (1/2) \hbar$ obtained for the intrinsic angular momentum in $x^5$, must be isotropically shared by all four space dimensions.  That is, we require that there is to be no “bias” toward any of the four space dimensions insofar as squared intrinsic spin is concerned.  Because $J^5 = (1/2) \hbar$ emanates naturally from the five dimensional geometry, we know immediately that $\left(J^{5} \right)^{2} ={\textstyle\frac{1}{4}} \hbar ^{2}$, and then, by the isotropic requirement, that $\left(J^{1} \right)^{2} =\left(J^{2} \right)^{2} =\left(J^{3} \right)^{2} ={\textstyle\frac{1}{4}} \hbar ^{2}$ as well.  We then arrive directly at the Casimir operator $J^{2} =\left(J^{1} \right)^{2} +\left(J^{2} \right)^{2} +\left(J^{3} \right)^{2} ={\textstyle\frac{3}{4}} \hbar ^{2}$ in the usual three space dimensions, and from there, continue forward deductively.

For those who have followed this development right along, this means in the simplest terms, that rather than use “orthogonality” to get the intrinsic spin out of $x^5$ and into ordinary space, I am instead using “isotropy.”

There is also a new section 8 on positrons and Dirac’s equations which has not been posted before, and I have made other editorial changes throughout the rest of the paper.

## March 29, 2008

### Stepping Back from Kaluza-Klein: Planned Revisions

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” $\Gamma^u_{55}=0$ from $F^{{\rm M} {\rm N} } =-F^{{\rm N} {\rm M} }$ 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 $G_{(5)}$ 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 $J^{\mu } =0$ special case of a broader tensor which includes a $J^\mu A^\nu + J^\nu A^\mu$ term and which applies, e.g., to energy flux densities (Poynting components) $T^{0k}$, 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 $x^5$, and by developing the positron further, we can move from the Pauli spin matrices toward the Dirac $\gamma^\mu$ 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 $n \lambda = 2 \pi R$.  (I actually think that for fermions, one has to consider orientation / entanglement issues, and so to secure the correct “version,” one should use $n \lambda = 4 \pi R$ 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 $x^5 = R\phi$ to define the compact fifth dimension.  However, in obtaining $dx^5$, I have taken $R$ to be a fixed, constant radius.  In light of considering a wavelength $n \lambda = 4 \pi R$ per above, I believe it important to consider variations in $R$ rather than fixed $R$, and so, to employ $dx^5 = Rd\phi + \phi dR$.

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!

Jay.

## March 22, 2008

### A Possible Kaluza-Klein Experiment

It has been suggested — appropriately so — that I consider whether there might be one or more experiments which can be designed to validate or falsify some of the Klauza-Klein results which I have been posting of late. I believe that one possible experiment resides in the non-symmetric energy tensor of trace matter derived in (11.6) of my latest posted paper. Thus, I have added a new section 15 to this paper, and reposted the entire paper, with this new section 15, at Kaluza-Klein Theory and Lorentz Force Geodesics Rev. 6.0 Because this is of particular interest as it may open some new experimental windows, I have posted section 15 below as well. Please note: the specific discussion of the connection between the compactified fifth dimension, and intrinsic spin, is not updated in this paper, and the latest discusssion I have written up on this topic, is at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.

Section 15: At this juncture, we have enough information to propose an experiment to validate or falsify some of the results derived thus far.  We turn for this purpose to the stress energy tensor of matter (11.6), which we raise into contravariant notation as follows:

$\kappa T^{\nu \mu } =-\kappa \left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)+{\textstyle\frac{\sqrt{2} }{2}} \overline{\kappa }g^{5\mu } J^{\nu } =\kappa T^{\mu \nu } _{Maxwell} +{\textstyle\frac{\sqrt{2} }{2}} \overline{\kappa }g^{5\mu } J^{\nu }$. (15.1)

The Maxwell tensor $T^{\mu \nu } _{Maxwell} =T^{\nu \mu } _{Maxwell}$ is, of course, a symmetric tensor.  But the added trace matter term $g^{5\mu } J^{\nu }$ is not necessarily symmetric, that is, there is no a priori reason why $g^{5\mu } J^{\nu }$ must be equal to $g^{5\nu } J^{\mu }$.  The origin of this non-symmetry was discussed earlier in Section 9.

With an eye toward conducting an experiment, let us now consider (15.1) in the linear approximation of (13.6) where ${\rm L}_{QCD} \approx -A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau }$.  In the linear approximation, as used to reach (13.3), (12.11) reduces to $g^{5\mu } \approx \overline{\kappa }\left(\frac{\phi ^{5\mu } -{\textstyle\frac{1}{2}} bA^{\mu } }{1+{\textstyle\frac{1}{2}} \overline{\kappa }\phi } \right)\approx -{\textstyle\frac{1}{2}} \overline{\kappa }bA^{\mu }$, and (15.1) becomes:

$T^{\nu \mu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-2J^{\nu } A^{\mu } =T^{\mu \nu } _{Maxwell} -2J^{\nu } A^{\mu }$, (15.2)

where we have also used $b^{2} =8$ and $2\kappa =\overline{\kappa }^{2}$, and divided out $\kappa$.  The transpose of this non-symmetric energy tensor is:

$T^{\mu \nu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-2J^{\mu } A^{\nu } =T^{\mu \nu } _{Maxwell} -2J^{\mu } A^{\nu }$, (15.3)

Now, it is known that a non-symmetric energy tensor, physically, is indicative of a non-zero spin density.  In particular, using (15.2) and (15.3), the non-symmetry of the energy tensor is related to a non-zero spin density tensor $S^{\mu \nu \alpha }$ according to: [A good, basic discussion of the spin tensor is at http://en.wikipedia.org/wiki/Spin_tensor.]

$S^{\mu \nu \alpha } _{;\alpha } =T^{\mu \nu } -T^{\nu \mu } =-2J^{\mu } A^{\nu } +2J^{\nu } A^{\mu }$. (15.4)

For such a non-symmetric tensor, the “energy flux” is not identical to the “momentum density, as these differ by (15.4), for $\mu =0$, $\nu =k=1,2,3$ and vice versa.  If the spin density $S^{\mu \nu \alpha } =0$, then  in this special case, (15.4) yields:

$J^{\mu } A^{\nu } =J^{\nu } A^{\mu }$. (15.5)

So, for $S^{\mu \nu \alpha } =0$, (15.3) may be written using (15.5) as the explicitly-symmetric tensor:

$T^{\mu \nu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-J^{\mu } A^{\nu } -J^{\nu } A^{\mu } =T^{\mu \nu } _{Maxwell} -J^{\mu } A^{\nu } -J^{\nu } A^{\mu }$. (15.6)

Now, let’s consider a experiment which is entirely classical.  The $T^{0k}$ “Poynting” components of (15.4), (15.6) represent the energy flux across a two-dimensional area, for a flux of matter which we will take to be a stream of electrons, while the $T^{k0}$ components represent the momentum density.  The proposed experiment, then, will be to fire a stream of a very large number of electrons thereby constituting an electron “wave,” and to detect the aggregate flux of energy across a two-dimensional surface under various spin preparations, in precisely the same manner that one might test the flow of luminous energy across a surface when using light waves rather than electron waves.  Specifically, we propose in test I to fire electrons without doing anything to orient their spins, so that, statistically, the number of electrons flowing through the flux surface with positive helicity is equal to the number with negative helicity and so the spin density is zero, and (15.6) applies.  In test II, we fire electrons, but apply a magnetic field before detecting the flux, to ensure that all of the electrons are aligned to positive helicity.  In this event, the spin density, by design, is non-zero, and one of (15.2) or (15.3) will apply.  In test III, we do the same, but now apply the magnetic field to ensure that all of the electrons have negative helicity, before detecting the flux.

## March 20, 2008

### Derivation of Heisenberg Uncertainty from Kaluza Klein Geometry

For those who have followed my Kaluza-Klein (KK) work, I believe that it is now possible to derive not only intrinsic spin, but Heisenberg uncertainty directly from a fifth, compactified dimension in Kaluza Klein.  This would put canonical quantum mechanics on a strictly Riemannian geometric foundation which — as a side benefit — unites gravitation and electromagnetism.

I need to consolidate over the next few days and will of course make a more expanded post when I am ready, but here is the basic outline.  First, take a look at:

Intrinsic Spin and the Kaluza-Klein Fifth Dimension

where I show how intrinsic spin is a consequence of the compactified fifth dimension.  This paper, at present, goes so far as to show how the Pauli spin matrices emerge from KK.

Next, go to the two page file:

Spin to Uncertainty

This shows how one can pop Heisenberg out of the spin matrices.

Finally, go to the latest draft paper on KK generally, at:

Kaluza-Klein Theory and Lorentz Force Geodesics with Non-linear QED

This lays out the full context in which I am developing this work.  Please note that the discussion on intrinsic spin in the third link is superseded by the discussion thereof in the first link.

More to follow . . .

Jay.

## March 3, 2008

### Intrinsic Spin and the Kaluza-Klein Fifth Dimension: Journal Submission

I mentioned several days ago that I had submitted a Kaluza Klein paper to one of the leading journals.  That lengthy paper was not accepted, and you can read the referee report and some of my comments here at sci.physics.foundations or here, with some other folks’ comments, at sci.physics.relativity.  The report actually was not too bad, concluding that “the author must have worked a considerable amount to learn quite a few thing in gravitation theory, and a number of the equations are correctly written and they do make sense, however those eqs. do not contain anything original.”  I would much rather hear this sort of objection, than be told — as I have been in the past — that I don’t know anything about the subject I am writing about.

In fact, there is one finding in the above-linked paper which, as I thought about it more and more, is quite original, yet I believe it was lost in the mass of this larger paper.  And, frankly, it took me a few days to catch on to the full import of this finding, and so I downplayed it in the earlier paper.  Namely:  that the compactified fifth dimension of Kaluza-Klein theories is the mainspring of the intrinsic spins which permeate particle physics.

I have now written and submitted for publication, a new paper which only includes that Kaluza-Klein material which is necessary to fully support this particular original finding.  You may read the submitted paper at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.  I will, of course, let you know what comes from the review of this paper.

Jay.

## February 29, 2008

### Lab Note 2: Why the Compactified, Cylindrical Fifth Dimension in Kaluza Klein Theory may be the “Intrinsic Spin” Dimension

FOR SOME REASON, THESE EQUATIONS APPEAR CORRUPTED. I AM CHECKING WITH WORDPRESS TECHNICAL SUPPORT AND HOPE TO HAVE THIS FIXED IN THE NEAR FUTURE — JAY.

I am posting here a further excerpt from my paper at Kaluza-Klein Theory, Lorentz Force Geodesics and the Maxwell Tensor with QED. Notwithstanding some good discussion at sci.physics.relativity, I am coming to believe that the intrinsic spin interpreation of the compacified, hypercylindrical fifth dimension presented in section 4 of this paper may be compelling. The math isn’t too hard, and you can follow it below: The starting point for discussion equation is (3.2) below,

$frac{dx^{5} }{dtau } equiv -frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =-frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m}$. (3.2)

which is used to connect the $q/m$ ratio from the Lorentz law to geodesic motion in five dimensions, and $b$ is a numeric constant of proportionality. Section 4 below picks up from this.

Excerpt from Section 4:

Transforming into an “at rest” frame, $dx^{1} =dx^{2} =dx^{3} =0$, the spacetime metric equation $d/tau ^{2} =g_{/mu /nu } dx^{/mu } dx^{/nu }$ reduces to $dtau =pm sqrt{g_{00} } dx^{0}$, and (3.2) becomes:

$frac{dx^{5} }{dx^{0} } =pm frac{1}{b} sqrt{frac{g_{00} }{4pi G} } frac{q}{m}$. (4.1)

For a timelike fifth dimension, $x^{5}$ may be drawn as a second axis orthogonal to $x^{0}$, and the physics ratio $q/m$ (which, by the way, results in the $q/m$ material body in an electromagnetic field actually “feeling” a Newtonian force in the sense of $F=ma$ due to the inequivalence of electrical and inertial mass) measures the “angle” at which the material body moves through the $x^{5} ,x^{0}$ “time plane.”

For a spacelikefifth dimension, where one may wish to employ a compactified, hyper-cylindrical $x^{5} equiv Rphi$ (see [Sundrum, R., TASI 2004 Lectures: To the Fifth Dimension and Back, http://arxiv.org/abs/hep-th/0508134 (2005).], Figure 1) and $R$ is a constant radius (distinguish from the Ricci scalar by context), $dx^{5} equiv Rdphi$. Substituting this into (3.2), leaving in the $pm$ ratio obtained in (4.1), and inserting $c$ into the first term to maintain a dimensionless equation, then yields:

$frac{Rdphi }{cdtau } =pm frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =pm frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m}$. (4.2)

We see that here, the physics ratio $q/m$ measures an “angular frequency” of fifth-dimensional rotation. Interestingly, this frequency runs inversely to the mass, and by classical principles, this means that the angular momentum with fixed radius is independent of the mass, i.e., constant. If one doubles the mass, one halves the tangential velocity, and if the radius stays constant, then so too does the angular momentum. Together with the $pm$ factor, one might suspect that this constant angular momentum is, by virtue of its constancy independently of mass, related to intrinsic spin. In fact, following this line of thought, one can arrive at an exact expression for the compactification radius $R$, in the following manner:

Assume that $x^{5}$ is spacelike, casting one’s lot with the preponderance of those who study Kaluza-Klein theory. In (4.2), move the $c$ away from the first term and move the $m$ over to the first term. Then, multiply all terms by another $R$. Everything is now dimensioned as an angular momentum $mcdot vcdot R$, which we have just ascertained is constant irrespective of mass. So, set this all to $pm {textstylefrac{1}{2}} nhbar$, which for $n=1$, represents intrinsic spin. The result is as follows:

$mfrac{Rdphi }{dtau } R=pm frac{1}{b} frac{sqrt{hbar c^{3} alpha } }{sqrt{G} } R=pm frac{1}{b} frac{c}{sqrt{4pi G} } qR=pm frac{1}{2} nhbar$. (4.3)

Now, take the second and fourth terms, and solve for $R$ with $n=1$, to yield:

$R=frac{b}{2sqrt{alpha } } sqrt{frac{Ghbar }{c^{3} } } =frac{b}{2sqrt{alpha } } L_{P}$, (4.4)

where $L_{P} =sqrt{Ghbar /c^{3} }$ is the Planck length. This gives a definitive size for the compactification radius, and it is very close to the Planck length. (more…)

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