Lab Notes for a Scientific Revolution (Physics)

April 22, 2012

Back to Blogging, Uploaded a paper I wrote in 1986 about Preonic Grand Unification

It has been almost 3 years since my last Blog post.  Much of my time has been diverted into a condo hotel project in Longboat Key Florida, and the focus I need to do good physics has been impossible to come by.  Then, the other day, Ken Tucker, a frequent participant at sci.physics.foundations, emailed me about some new research showing that electrons have constituent substructure.  That brought me back immediately to the half a year I spent back in 1986 developing a 200-page paper about a preonic substructure for quarks and leptons, which culminated six years of study from 1980 to 1986.  I finished that paper in August 1986, and then took an 18 year hiatus from physics, resuming again in late-2004.

Ken’s email motivated me to dig out this 1986 paper which I manually typed out on an old-fashioned typewriter, scan it into electronic form, and post it here.  Links to the various sections of this paper are below.  This is the first time I have ever posted this.

Keep in mind that I wrote this in 1986.  I tend to study best by writing while I study, and in this case, what I wrote below was my “study document” for Halzen and Martin’s book “Quarks and Leptons” which had just come out in 1984 and was the first book to pull together what we now think of as modern particle physics and the (then, still fairly new) electroweak unification of Weinberg-Salam.

What is in this paper that I still to this day believe is fundamentally important, and has not been given the attention it warrants, is the isospin redundancy between (left-chiral) quarks and leptons.  This to me is an absolute indication that these particles have a substructure, so that a neutrino and an up quark both have contain the same “isospin up” preon, and an electron and a down quark both contain the same “isospin down” preon.  Section 2.11 below is the key section, if you want to cut to the chase with what I was studying some 26 years ago.  I did post about this in February 2008 at https://jayryablon.wordpress.com/2008/02/02/lab-note-4-an-interesting-left-chiral-muliplet-perhaps-indicative-of-preonic-structure-for-fermions/, though that post merely showed a 1988 summary I had assembled based on my work in 1986, at the behest of the late Nimay Mukhopadhyay, who at the time was teaching at RPI and had become a good friend and one of my early sources of encouragement.  This is the first time I am posting all of that early up-to-1986 work, in complete detail.

Lest you think me crazy, note that seventeen years later, G. Volovik, in his 2003 book “The Universe in a Helium Droplet,” took a very similar tack, see Figure 12.2 in this excerpt: Volovik Excerpt on Quark and Lepton Preonic Structure.

The other aspect of this 1986 paper that I still feel very strongly about, is taking the Dirac gamma-5 as a fifth-dimension indicator.  I know I have been critiqued by technical arguments as to why this should not be taken as a sign of a fifth dimension, but this fits seamlessly with Kaluza Klein which geometrizes the entirely of Maxwell’s theory and is still the best formal unification of classical electromagnetism and gravitation ever developed.  For those who maintain skepticism of Kaluza-Klein and ask “show me the fifth dimension,” just look to chirality which is well-established experimentally.  Why do we have to assume that this fifth dimension will directly manifest in the same way as space and time, if its effects are definitively observable in the chiral structure of fermions?  Beyond this, I remain a very strong proponent of the 5-D Space-Time-Matter Consortium, see http://astro.uwaterloo.ca/~wesson/, which regards matter itself as the most direct manifestation of a fifth physical dimension.  Right now, most folks think about 4-D spacetime plus matter.  These folks correctly think about 5-D space-time-matter, no separation.  And Kaluza-Klein, which historically predated Dirac’s gamma-5, is the underpinning of this.

After my hiatus of the past couple of years, I am going to try in the coming months to write some big-picture materials about physics, which will pull together all I have studied so far in my life.  I am thinking of doing a “Physics Time Capsule for 2100” which will try to explore in broad strokes, how I believe physics will be understood at the end of this century, about 88 years from now.

Anyway, here is my entire 1986 paper:

Preonic Grand Unification and Quantum Gravitation: Capsule Outline and Summary

Abstract and Contents

Section 1.1: Introduction

Section 1.2: Outline and Summary

Section 2.1: A Classical Spacetime Introduction to the Dirac Equation, and the Structure of Five-Dimensional Spacetime with a Chiral Dimension

Section 2.2: Particle/Antiparticle and Spin-Up/Spin-Down Degrees of Quantum Mechanical Freedom in Spacetime and Chirality, Gauge Invariance and the Dirac Wavefunction

Section 2.3: Determination and Labeling of the Spinor Eigensolutions to the Five-Dimensional Dirac Equation, and the High and Low Energy Approximations

Section 2.4: The Fifth-Dimensional Origin of Left and Right Handed Chiral Projections and the Continuity equation in Five Dimensions: Hermitian Conjugacy, Adjoint Spinors, and the Finite Operators Parity (P) and Axiality (A)

Section 2.5: Conjugate and Transposition Symmetries of the Dirac Equation in Five Dimensions, the Finite Operators for Conjugation (C) and Time Reversal (T), and Abelian Relationships Among C, P, T and A

Section 2.6: Charge Conjugation, and the Definitions and Feynman Diagrams for “Electron” and “Positron” Spinors

Section 2.7: Simple Unpolarized s,t,u Scattering Channels with a Covariant Propagator, and the Covariant (Real and Virtual) Polarization States of Massive and Massless Vector Bosons

Section 2.8: Prelude to Preons: The Spinor Decomposition of Four Real Spacetime Dimensions ct,x,y,z into Two Complex Spinor Dimensions Using the Covariant Polarization States of Vector Bosons

Section 2.9: Introduction to Isospin Preons in Electroweak Theory: The Preonic Decomposition of Four Real Electroweak Bosons A, W+, W-, Z into Two Complex Preons Denoting “Isospin Up” and “Isospin Down”

Section 2.10: Summarization of Prior Discussion, and on the Fundamental Importance of Preons in Particle Physics

Section 2.11: The Four-Preon Flavor SU(4) Unification of the Electromagnetic, Weak and Colorless Strong Interactions Excluding Quantum Gravitation; and the Colorless Flavor Classification of Left Handed Real Fermion and Boson Chiral Projections, for a Single Fermion Generation

Section 2.12: The Four-Preon Flavor SU(4)xU(1) Unification of Electromagnetic, Weak, Colorless Strong and Quantum Gravitational Interactions; and the Colorless Flavor Classification of Left and Right Handed Real Fermion and Boson Chiral Projections, for a Single Fermion Generation

Addendum to Section 2.12

Section 2.13: The Six-Preon Unification of Flavor SU(4)xU(1) with High Energy Color SU(4)xU(1) and Two Overlapping Degrees of Freedom; the Flavor and Color Classification of Real Fermions and Vector Bosons for a Single Generation; and the Derivation of Electroweak and Strong/Hyperweak Massless and Massive Neutral Current Vector Bosons

Section 2.14: On the Replication of Fermion Generations: Four Generational Grand Unification with Eighteen Preons and Nine Independent Flavor/Color/Generation Degrees of Freedom, and a Preonic Discussion of Mesons and Meson Decay

References and Bibliography

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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

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.

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.

(more…)

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.

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.

   I welcome your comments and feedback.

   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…)

February 27, 2008

Lab Note 2: Derivation of the Maxwell Stress-Energy Tensor from Five-Dimensional Geometry, using a Four-Dimensional Variation

FOR SOME REASON, THESE EQUATIONS APPEAR CORRUPTED.  I AM CHECKING WITH WORDPRESS TECHNICAL SUPPORT AND HOPE TO HAVE THIS FIXED IN THE NEAR FUTURE. FOR NOW, PLEASE USE THE LINK IN THE FIRST PARAGRAPH, AND GO TO SECTION 10 — JAY.  

As mentioned previously, I have been able to rigorously derive the Maxwell tensor from a five dimensional Kaluza-Klein geometry based on Lorentz force geodesics, using a variational principle over the four spacetime dimensions of our common experience.  At the link: Kaluza-Klein Theory, Lorentz Force Geodesics and the Maxwell Tensor with QED, I have attached a complete version of this paper, which includes connections to quantum theory as well as an extensive summary not included in the version of the paper now being refereed at one of the leading journals.  This is a strategic decision not to overload the referee, but to focus on the mathematical results, the most important of which is this derivation of the Maxwell tensor.

Because this paper is rather large, I have decided on this weblog, to post section 10, where this central derivation occurs.  Mind you, there are nine sections which lay the foundation for this, but with the material below, plus the above link, those who are interested can see how this all fits together.  The key result emerges in equation (10.15) below.  Enjoy!

Excerpt: Section 10 — Derivation of the Maxwell Stress-Energy Tensor, using a Four-Dimensional Variation

 In section 8, we derived the energy tensor based on the variational calculation (8.4), in five dimensions, i.e., by the variation delta g^{{rm M} {rm N} } .  Let us repeat this same calculation, but in a slightly different way. 

 In section 8, we used (8.3) in the form of {rm L}_{Matter} =-{textstylefrac{1}{8kappa }} boverline{kappa }g^{5{rm B} } J_{{rm B} } =-{textstylefrac{1}{8kappa }} boverline{kappa }g^{{rm M} {rm N} } delta ^{5} _{{rm M} } J_{{rm N} } , because that gave us a contravariant g^{{rm M} {rm N} } against which to obtain the five-dimensional variation delta {rm L}_{Matter} /delta g^{{rm M} {rm N} } .  Let us instead, here, use the very last term in (8.3) as {rm L}_{Matter} , writing this as:

{rm L}_{Matter} equiv {textstylefrac{1}{2kappa }} R^{5} _{5} =-{textstylefrac{1}{8kappa }} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right)=-{textstylefrac{1}{8kappa }} boverline{kappa }left(g^{mu nu } delta ^{5} _{nu } J_{mu } +{textstylefrac{1}{4}} g^{55} g^{mu nu } boverline{kappa }F_{mu } ^{tau } F_{nu tau } right). (10.1)

It is important to observe that the term g^{5beta } J_{beta } is only summed over four spacetime indexes.  The fifth term, g^{55} J_{5} ={textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } , see, e.g., (6.8).  For consistency with the non-symmetric (9.5), we employ g^{5beta } J_{beta } =g^{mu nu } delta ^{5} _{nu } J_{mu } rather than g^{5beta } J_{beta } =g^{mu nu } delta ^{5} _{mu } J_{nu } .  By virtue of this separation, in which we can only introduce g^{mu nu } and not g^{{rm M} {rm N} } as in section 8, we can only take a four-dimensional variation delta {rm L}_{Matter} /delta g^{mu nu } , which, in contrast to (8.4), is now given by:

T_{mu nu } equiv -frac{2}{sqrt{-g} } frac{partial left(sqrt{-g} {rm L}_{Matter} right)}{delta g^{mu nu } } =-2frac{delta {rm L}_{Matter} }{delta g^{mu nu } } +g_{mu nu } {rm L}_{Matter} . (10.2)

Substituting from (10.1) then yields:

T_{mu nu } ={textstylefrac{1}{4kappa }} boverline{kappa }left(delta ^{5} _{nu } J_{mu } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F_{mu } ^{tau } F_{nu tau } right)-{textstylefrac{1}{2}} g_{mu nu } {textstylefrac{1}{4kappa }} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.3)

Now, the non-symmetry of sections 8 and 9 comes into play, and this will yield the Maxwell tensor.  Because delta ^{5} _{nu } =0, the first term drops out and the above reduces to:

kappa T_{mu nu } ={textstylefrac{1}{4}} boverline{kappa }left({textstylefrac{1}{4}} g^{55} boverline{kappa }F_{mu } ^{tau } F_{nu tau } right)-{textstylefrac{1}{2}} g_{mu nu } {textstylefrac{1}{4}} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.4)

Note that this four-dimensional tensor is symmetric, and that we would arrive at an energy tensor which is identical if (10.3) contained a delta ^{5} _{mu } J_{nu } rather than delta ^{5} _{nu } J_{mu } .  One again, the screen factor delta ^{5} _{nu } =0 is at work. 

 In mixed form, starting from (10.3), there are two energy tensors to be found.  If we raise the mu index in (10.3), the first term becomes delta ^{5} _{nu } J^{mu } =0 and we obtain:

-kappa T^{mu } _{nu } =-{textstylefrac{1}{4}} boverline{kappa }left({textstylefrac{1}{4}} g^{55} boverline{kappa }F^{mu tau } F_{nu tau } right)+{textstylefrac{1}{2}} delta ^{mu } _{nu } {textstylefrac{1}{4}} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.5)

with this first term still screened out.  However, if we transpose (10.3) and then raise the mu index, the first term becomes g^{5mu } J_{nu } and this term does not drop out, i.e.,

-kappa T_{nu } ^{mu } =-{textstylefrac{1}{4}} boverline{kappa }left(g^{5mu } J_{nu } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{mu tau } F_{nu tau } right)+{textstylefrac{1}{2}} delta ^{mu } _{nu } {textstylefrac{1}{4}} boverline{kappa }left(g^{5beta } J_{beta } +{textstylefrac{1}{4}} g^{55} boverline{kappa }F^{sigma tau } F_{sigma tau } right). (10.6)

So, there are two mixed tensors to consider, and this time, unlike in section 8, these each yield different four-dimensional energy tensors.  Contrasting (10.5) and (10.6), we see that delta ^{5} _{nu } =0 has effectively “broken” a symmetry that is apparent in (10.6), but “hidden” in (10.5).  At this time, we focus on (10.5), because, as we shall now see, this is the Maxwell stress-energy tensor T^{mu } _{nu } =-left(F^{mu tau } F_{nu tau } -{textstylefrac{1}{4}} delta ^{mu } _{nu } F^{sigma tau } F_{sigma tau } right), before reduction into this more-recognizable form.
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February 25, 2008

Lab Note 2 Term Paper: Kaluza-Klein Theory and Lorentz Force Geodesics . . . and the Maxwell Tensor

Dear Friends: 

I have just today completed a paper titled “Kaluza-Klein Theory and Lorentz Force Geodesics,” which I have linked below:

Kaluza-Klein Theory and Lorentz Force Geodesics.

I have also submitted the draft linked above, to one of the leading physics journals for consideration for publication.

One of the things I have been beating my head against the wall over these past few weeks, is to deduce the Maxwell stress-energy tensor from the 5-dimensional geometry using Einstein’s equation including its scalar trace.  I finally got the proof nailed down this morning, and that is section 10 of the paper linked above.

I respectfully submit that the formal derivation of the Maxwell stress-energy tensor in section 10, provides firm support for the Spacetime-Matter (STM) viewpoint that our physical universe is a five-dimensional Kaluza-Klein geometry in which the phenomenon we observe in four dimensions are “induced” out of the fifth dimension, and that it supports the correctness of the complete line of development in this paper.  Section 10 — as the saying goes — is the “clincher.”

As is apparent to those who have followed the development of this particular “Lab Note,” my approach is to postulate the Lorentz force, and require that this be geodesic motion in 5-dimensions.  Everything else follows from there.  The final push to the Maxwell tensor in section 10, rests on adopting and implementing the STM viewpoint, and applying a 4-dimensional variational principle in a five-dimensional geometry.  If you have a serious interest in this subject, in addition to my paper, please take a look at The 5D Space-Time-Matter Consortium.

Best to all,

Jay.

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