Lab Notes for a Scientific Revolution (Physics)

May 7, 2009

Inferring Electrodynamic Gauge Theory from General Coordinate Invariance

I am presently working on a paper to show how electrodynamic gauge theory can be directly connected to generally-covariant gravitational theory.  In essence, we show how there is a naturally occurring gauge parameter in gravitational gemometrodynamics which can be directly connected with the gauge parameter used in electrodynamics, while at the same time local gauge transformations acting on fermion wavefunctions may be synonymously described as general coordinate transformations acting on those same fermion wavefunctions.

This is linked below, and I will link updates as they are developed.

Inferring Electrodynamic Gauge Theory from General Coordinate Invariance

If you check out sci.physics.foundations and sci.physics.research, you will see the rather busy path which I have taken over the last month to go from baryons and confinement to studying the Heisenberg equation of motion and Ehrenfest’s theorem, to realizing that there was an issue of interest in the way that Fourier kernels behave under general coordinate transformations given that a general coordinate x^u is not itself a generally-covariant four vector.  Each step was a “drilling down” to get at underlying foundational issues, and this paper arrives at the most basic, fundamental underlying level.

Looking forward to your feedback.

Jay

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April 4, 2009

Starting a new paper on Baryons and Confinement

Today, I began work on a new paper dealing with the Yang-Mills foundations of baryons and QCD confinement.  The first draft is linked below, and I will provide updates as they develop.

Yang-Mills Foundations of Baryons and Confinement Phenomena

I may get diverted a bit by my US tax filing the next few days, and I am quite busy at work right now so this will mostly be a weekend and after-midnight project, but I do hope to get this paper, which I hope will synthesize many individual insights I have had and subjects I have studied over the past several years, into a something of value for others.

Constructive comments are always appreciated.

Thanks to the Princess and Peter and Ken and Igor and Ben for feedback and insights posted on the various newsgroups.

Jay.

April 24, 2008

Heisenberg Uncertainty and Schwinger Anomaly: Two Sides of the Same Coin?

 In section 3 of Heisenberg Uncertainty and Schwinger Anomaly: Two Sides of the Same Coin?, I have posted a calculation which shows why the Schwinger magnetic anomaly may in fact be very tightly tied to the Heisenberg inequality \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar .  The bottom line result, in (3.11) and (3.12), is that the gyromagnetic “g-factor” for a charged fermion wave field with only intrinsic spin (no angular momentum) is given by:

\left|g\right|=2\frac{\left(\Delta x\Delta p\right)}{\hbar /2} \ge 2   (3.11)

It is also helpful to look at this from the standpoint of the Heisenberg principle as:

\Delta x\Delta p=\frac{\left|g\right|}{2} \frac{\hbar }{2} \ge \frac{\hbar }{2}   (3.12)

The draft paper linked above has details of how I got here.  Below, are some ways to think about this:

 First, if (3.11) is true, then the greater than or equal to inequality of Heisenberg says, in this context, that the magnitude of the intrinsic g-factor of a charged wavefunction is always greater than or equal to 2.  That is, the inequality \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar becomes another way of stating a parallel inequality \left|g\right|\ge 2.  We know this to be true for the charged leptons, which have g_{e} /2=1.0011596521859, g_{\mu } /2=1.0011659203, and g_{\tau } /2=1.0011773 respectively. [The foregoing data is extracted from W.-M. Yao et al., J. Phys. G 33, 1 (2006)]

 Secondly, the fact that the charged leptons have g-factors only slightly above 2, suggests that these a) differ from perfect Gaussian wavefunctions by only a very tiny amount, b) the electron is slightly more Gaussian than the muon, and the muon slightly more-so than the tauon.  The three-quark proton, with g_{P} /2=2.7928473565, is definitively less-Gaussian the charged leptons.  But, it is intriguing that the g-factor is now seen as a precise measure of the degree to which a wavefunction differs from a perfect Gaussian.

 Third, (3.11) states that the magnetic moment anomaly via the g-factor is a precise measure of the degree to which \Delta x\Delta p exceeds \hbar /2.  This is best seen by writing (3.11) as (3.12).

Thus, for the electron, \left(\Delta x\Delta p\right)_{e} =1.0011596521859\cdot \left(\hbar /2\right), to give an exact numerical example.  For a different example, for the proton, \left(\Delta x\Delta p\right)_{P} =2.7928473565\cdot \left(\hbar /2\right).

 Fourth, as a philosophical and historical matter, one can achieve a new, deeper perspective about uncertainty.  Classically, it was long thought that one can specify position and momentum simultaneously, with precision.  To the initial consternation of many and the lasting consternation of some, it was found that even in principle, one could at best determine the standard deviations in position and momentum according to \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar .  There are two aspects of this consternation:  First, that one can never have\Delta x\Delta p=0 as in classical theory.  Second, that this is merely an inequality, not an exact expression, so that even for a particle with \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar , we do not know for sure what is its exact value of \Delta x\Delta p.  This latter issue is not an in-principle limitation on position and momentum measurements; it is a limitation on the present state of human knowledge.

 Now, while {\tfrac{1}{2}} \hbar is a lower bound in principle, the question remains open to the present day, whether there is a way, for a given particle, to specify the precise degree to which its \Delta x\Delta p exceeds {\tfrac{1}{2}} \hbar , and how this would be measured.  For example, one might ask, is there any particle in the real world that is a perfect Gaussian, and therefore can be located in spacetime and conjugate momentum space, down to exactly {\tfrac{1}{2}} \hbar .  Equation (3.12) above suggests that if such a particle exists, it must be a perfect Gaussian, and, that we would know it was a perfect Gaussian, if its g-factor was experimentally determined to be exactly equal to the Dirac value of 2.  Conversely, (3.12) tells us that it is the g-factor itself, which is the direct experimental indicator of the magnitude of \Delta x\Delta p for any given particle wavefunction.  The classical precision of \Delta x\Delta p=0 comes full circle, and while it will never return, there is the satisfaction of being able to replace this with the quantum  mechanical precision of (3.12), \Delta x\Delta p=\left|g\right|\hbar /4, rather than the weaker inequality of \Delta x\Delta p\ge {\tfrac{1}{2}} \hbar .

 Fifth, if (3.12) is correct, then since it is independently known from Schwinger that \frac{g}{2} =1+\frac{a}{2\pi } +\ldots , this would mean that we would have to have:

\Delta x\Delta p=\frac{\left|g\right|}{2} \frac{\hbar }{2} =\left(1+\frac{a}{2\pi } +\ldots \right)\frac{\hbar }{2}   (3.13)

Thus, from the perturbative viewpoint, the degree to which \Delta x\Delta p exceeds {\tfrac{1}{2}} \hbar would have to be a function of the running coupling strength \alpha =e^{2} /4\pi in Heaviside-Lorentz units.  As Carl Brannnen has explicitly pointed out to me, this means that a Gaussian wavepacket is by definition non-interacting; as soon as there is an interaction, one concurrently loses the exact Gaussian.

 Sixth, since deviation of the g-factor above 2 would arise from a non-Gaussian wavefunction such as \psi (x)=N\exp \left(-{\tfrac{1}{2}} Ax^{2} +Bx\right), the rise of the g-factor above 2 would have to stem from the Bx term in this non-Gaussian wavefuction.  In this regard, we note to start, that N\int \exp \left(-{\tfrac{1}{2}} Ax^{2} +Bx\right)dx= \sqrt{2\pi /A} \exp \left(B^{2} /2A\right), for a non-Gaussian wavefunction, versus N\int \exp \left(-{\tfrac{1}{2}} Ax^{2} \right)dx= \sqrt{2\pi /A} for a perfect Gaussian.

 Finally, to calculate this all out precisely, one would need to employ a calculation similar to that shown at http://en.wikipedia.org/wiki/Uncertainty_principle#Wave_mechanics, but for the non-Gaussian N\int \exp \left(-{\tfrac{1}{2}} Ax^{2} +Bx\right)dx= \sqrt{2\pi /A} \exp \left(B^{2} /2A\right) rather than the GaussianN\int \exp \left(-{\tfrac{1}{2}} Ax^{2} \right)dx= \sqrt{2\pi /A} , to arrive at the modified bottom line equation of this Wiki section.  That is the next calculation I plan, but this is enough, I believe, to post at this time.

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.

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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 6, 2008

Electrodynamic Potentials and Non-Linear QED in Kaluza-Klein

I have now added new sections 12, 13 and 14 to the Kaluza-Klein paper earlier posted.  These sections examine the relationship between the electrodynamic potentials and the gravitational potentials, and the connection to QED.  You may view this all at:

Electrodynamic Potentials and Non-Linear QED

Most significantly, these three new sections not only connect to the QED Lagrangian, but, they show how the familiar QED Lagrangian density

{\rm L}_{QCD} =-A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau }

emerges in the linear approximation of 5-dimensional Kaluza-Klein gravitational theory.

Then, we go in the opposite direction, to show the QED Lagrangian density / action for non-linear theory, based on the full-blown apparatus of gravitational theory. 

Expressed in terms of the electrodynamic field strength F^{\sigma \tau } and currents J_{\beta }, this non-linear result is:

{\rm L}_{QCD} =0={\textstyle\frac{1}{8\kappa }} b\overline{\kappa }g^{5\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau } \approx -A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau } , (13.6) 

where the approximation \approx shows the connection to the linear approximation.  Re-expressed solely in terms of the fifth-dimensional gravitational metric tensor components g_{5\sigma } and energy tensor source components T_{\beta 5}, this result is: 

\kappa {\rm L}_{QCD} =0={\textstyle\frac{1}{2}} g^{5\beta } \kappa T_{\beta 5} +{\textstyle\frac{1}{8}} g^{\sigma \alpha } \partial ^{\beta } g_{5\alpha } \left[\partial _{\sigma } g_{5\beta } -\partial _{\beta } g_{5\sigma } \right]. (14.4)

You may also enjoy the derivations in section 12 which decompose the contravariant metric tensor into gravitons, photons, and the scalar trace of the graviton. 

Again, if you have looked at earlier drafts, please focus on the new sections 12, 13 and 14.  Looking for constructive feedback, as always.

February 19, 2008

Lab Note 2 Progress Report: Draft Paper on Kaluza-Klein Theory and Lorentz Force Geodesics

Hello to all my readers and contributors:

I have been very busy these past several days preparing my research on Kaluza-Klein five-dimensional theory into a formal paper.  I now have a draft paper sufficiently advanced, that I would like to share it with my readers and contributors for their comment.

I am not going to reproduce this directly on the blog as there are dozens of equations and the paper itself is already 25 pages.  However, I have linked a PDF copy of the latest draft below, for your perusal and comment.

Kaluza-Klein Theory and Lorentz Force Geodesics — 2-19-08 Draft

I know that there are literally dozens if not hundreds of Kaluza-Klein papers already out there in the world.  This one, I believe, is the one that actually describes how nature works, and how classical gravitation and electrodynamics actually do become united in nature.

Looking forward to your thoughts.

Jay. 

February 16, 2008

Lab Note 2 Intermezzo: Change of View to a Spacelike Fifth Dimension, as the Geometric Foundation of Intrinsic Spin

(A PDF version of this Lab Note may be downloaded from Lab Note 2 Intermezzo.) 

 Those who have followed the development of this lab note know that I have been working with a Kaluza-Klein theory which regards the fifth dimension as timelike, rather than spacelike.  After reviewing some key literature in the field including a Sundrum Lecture recommended by Martin Bauer and several articles by Paul Wesson linked over at The 5-D Space-Time-Matter Consortium, I have undergone a conversion to the view that the fifth dimension needs to be spacelike – not timelike – and specifically, that it needs to be a compact, spacelike hypercylinder.  In this conversion, I am motivated by the following reasoning, which gives a geometric foundation to intrinsic spin: 
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