Jay R. Yablon (jyablon@nycap.rr.com)
This is at: Derivation
of the Maxwell Stress-Energy Tensor from Five-Dimensional Geometry, using a
Four-Dimensional Variation. This is section
10 from my now-complete paper unifying classical electrodynamics and
gravitation via 5-dimensional Kaluza-Klein at Kaluza-Klein
Theory, Lorentz Force Geodesics and the Maxwell Tensor.
This section 10 derivation is the most important proof
in the paper, which to my knowledge, has not been found before in any of the
zillions of studies that have already been done of Kaluza-Klein. I'd be
interested in your comments specifically on this proof. This paper contains an effort in a brand new
section 11 to begin connecting the 5-D geometry to quantum theory. There
is also, in the complete paper linked above, a full summary at the end, which
can help you to navigate the paper if your are
so-inclined. Email above with comments,
or converse on the weblog Lab Notes
for a Scientific Revolution and / or over at sci.physics.foundations.
The unexpected twist in all this: the fifth-dimensional components of the Riemann
tensor in five-dimension are non-symmetric, and that is what knocks the trace
out of Maxwell's tensor and turns it into luminous energy. Sections 8 to
10 tell that part of the story.
is directly
proportional to the axial (5th-dimensional) component of the gravitational
metric tensor 
. Section 9 then uses
this result to yield a seamlessly-integrated electro-gravitational Lagrangian
density, in which the QED Lagrangian emerges directly and naturally out of the
5-dimensional gravitational Lagrangian density in vacuo. Look at equation (9.11) to cut to the
chase. Section 10 points out how the
connection in section 9 might lead to a possible path to better-understanding
quantum gravitation. Section 11 places
the Maxwell tensor onto a geometrodynamic footing. These new results are also now posted on my weblog Lab Notes for a Scientific Revolution
as “Lab Note 2, Part 3.” In case you
have always wondered, this is what
gravitation and electrodynamics look like, when they are seamlessly unified
into a single geometrodynamic theory.
Enjoy! ;-)
FEBRUARY 6, 2008: I have just today posted an article Gravitational
and Inertial Mass, and Electromagnetism as Geometry, in 5-Dimensional Spacetime
which demonstrates how the Lorentz force motion of charged material bodies in
an electromagnetic field can be seen as geodesic motion through a
five-dimensional Riemannian geometry.
The gravitational and inertial mass of a
material body, as well as its electrostatic charge, is given a
totally-geometric interpretation based on worldline motion through a
two-dimensional time plane consisting of ordinary time, as well as an axial
time dimension motivated by the Dirac/Weyl axial 
matrix. Additionally, it is shown how Maxwell’s
equations are simply the fifth-dimensional components of Einstein’s
gravitational field equation, as well as the known mathematical identities of
Riemannian geometry. This paper, which
is also posted over at my Weblog Lab
Notes for a Scientific Revolution as “Lab Note 2, Part 2”, in effect,
unifies Maxwell’s electrodynamics with Einstein’s gravitation such that
electrodynamics is placed on a completely geometric footing. I hope you will take a look at this and give me
your feedback.
I also recently, on JANUARY 20, 2008, posted an updated article on Rest
Mass as Geometry, which you can read either on the Weblog at Lab
Note 2, Part 1: Rest Mass as Geometry, or, in a PDF file, at Rest Mass
as Geometry.
Baryons and Confinement: Several draft
papers about the possibility that baryons (which include the protons and
neutrons which comprise the bulk of the material world), formally, are best
regarded as third-rank antisymmetric tensors.
Of course, if one should ever come to properly explain a baryon, then
one will have implicitly solved the problem of confinement, because a baryon is
known to contain exactly three quarks in a bound, confined state. Recent drafts on this topic include:
A Possible
Connection Between Baryons and Magnetic Monopoles
(
On The Natural
Origin of Baryons, Short-Range Mesons, and QCD Confinement, from Maxwell’s
Magnetic Equations for a Yang-Mills Field (
Yang-Mills
magnetic sources as the foundation of baryons, mesons, and QCD confinement
(
A Fifth Spacetime Mass Dimension: The Dirac
Gamma matrices can be regarded as the structure matrices, or the generator matrices,
of spacetime itself. There are five of
these. Although any four can be
multiplied together vi 
and so only four of
these are formally independent, the fact is that one can readily form a
five-dimensional metric tensor using the connection 
between the Dirac gamma and the metric tensor and can
then proceed to develop spacetime geometry in five dimensions. This would be a trivial observation, but for the
fact that the fifth dimension has a timelike signature (which suggests that
time is a two-dimensional plane, and on further analysis, for a macroscopic
observer, is a single, complex-valued dimension), and but for the fact that
rest mass can be described on a totally geometric footing in relation to the
angle of movement through this fifth dimension.
Papers which elaborate this are below:
Foundations:
Axial Time as a Possible Fifth Spacetime Mass Dimension (
Why
Quantum Field Theory?:
A Possible Connection Between Complex Time and Path Integrals (
The Size of Elementary Particles: It is my view
that the Dirac relationship is in many ways,
perhaps the most under-utilized, known relationship in all of physics. This of course comes into play with regards
to axial time, see above, but this also means that, starting with a
gravitational metric tensor 
, one can always derive an associated set of 
which include the
effects of gravitation. And, if one uses
these gravitation-containing in the usual way to
calculate magnetic moments, one can actually use the observed Schwinger anomaly in the magnetic moment to determine the
intrinsic size – near the Planck length – of the elementary Fermions. There are actually two directions one can
take with this analysis. One can start
with known and derive a metric tensor , or one can start with a gravitational metric tensor and
derive a set of . The latter approach
is illustrated in:
What
the Magnetic Moment Anomaly May Tell Us About
Planck-Scale Physics (
The former approach, which
leads one to consider the metric tensor in momentum space, is in an arXiv paper at:
Yang Mills Theory: I have been focusing my most
recent research on Yang-Mills Theory, and Quantum Field Theory. My work in this area – very much “in
progress” – can be reviewed here and here.
Some Thoughts and a Gedanken
(and maybe a real experiment) to test for “Hidden Variables.” As an
unrepentant die-hard who agrees with Einstein that God does not play dice and believe
that quantum theory is a transitional, albeit amazingly-successful,
understanding of how nature works, I have developed some thoughts about this
long-standing source of debate and discussion among physicists. A first note, Might Quantum
Probability be Classically-Explainable Based on Motion Through the Planck
Vacuum (March 9, 2007), lays the foundation for a later note which lays out
A proposed double
slit experiment to test for “hidden” variables (March 15, 2007) which was
discussed at some length in an SPF thread here. (I have filed for a patent on any devices
that would perform such an experiment.)
It is well-known that
following summing Feynman graphs, the fermion-boson coupling vertex is modified
according to 
, with 
representing
non-divergent perturbative
corrections. Here, we calculate the anticommutators specified by 
, and then explore some consequences of employing these as a
metric tensor 
in momentum
space. The challenge is that 
and 
must then be
introduced in place of and 
throughout the
Lagrangian density, denoted L ’,
resulting in what appears, superficially, to be different physics from
what is known and observed. However,
with a suitable reparameterization of fermion rest
masses 
, interaction charges 
and momentum vectors 
into their observed
counterparts 
, 
and 
, it turns out that L ’ can
be made to describe physics identical to that of the customary QED Dirac
Lagrangian density L at low photon momentum 
, including the observed magnetic anomaly. That is, we prove that one is able to obtain L 
= L ’
for . We find through the
Ward-Takahashi identity, as summarized in Figure 2, that interaction vertexes
are proportional to the difference between the ordinary and covariant
momentum-space derivatives of the metric tensor, and thus an indicator of
curvature.
SEPTEMBER 10, 2006: Please give me your comments on my new draft
paper Magnetic
Moment Anomalies of the Charged Leptons
(This paper is superseded by the new paper at hep-ph/0610377,
but some of this material may still be relevant to future explorations of hep-ph/0610377.)
The anomalous magnetic
moments of the charged leptons are presently understood in terms of
perturbative corrections of the form 
introduced via the
Dirac gamma matrices . This creates a
gravitational field, because the metric tensor 
is related to these
matrices by , so a change in either of or is necessarily a
change in the other. We characterize the
known experimental data for electron, muon and tau lepton rest masses and magnetic moments 
by converting a metric
such as that of
Schwarzschild into an associated set of and then obtaining
particle solutions to the Dirac equation for a charged electron in an
electromagnetic field. Quite unexpectedly,
the results support the idea that these charged leptons may be “strings”
characterized by three intrinsic parameters which must be specified in relation
to the probe energy 
from which they are
observed: a frequency 
, a length scale 
, and an angle parameter 
. The particle data
lead to being about 430 times
the Planck length, as observed at low .
NOVEMBER 29, 2005: Draft Paper: Is
Quantum Mechanics a Consequence of Requiring The Laws of Nature in Integral
Form to be Invariant Under Special and General Coordinate Transformations?
In this paper, we show how the primary features of quantum mechanics appear to emerge from carefully enforcing invariance under general coordinate transformation when one performs volume integrations in spacetime. In particular, this leads to 1) an inseparable fusion of momentum and displacement into angular momentum which may form the foundation of the similar fusion that takes place from the uncertainty principle, 2) a requirement for a fundamental (Planck’s) constant of angular momentum, and 3) a connection between uncertainty and superluminosity. As regards the third point, it has long been known that he speed of light establishes an upper boundary, and Planck’s constant a lower boundary, on natural phenomena. These two boundaries appear here, to be inseparably linked as flip sides of the same coin. I hope to get this into shape for posting on ArXiV in the near future, and appreciate any feedback you can provide.
The “warmup” exercise I did to develop the mathematics for invariant volume integration, in a short draft piece called Volume Integration and Stokes’ / Gauss’ Theorem in Curved Spacetime, is also posted for your perusal. I point out that the analysis in Is Quantum Mechanics a Consequence of Requiring The Laws of Nature in Integral Form to be Invariant Under Special and General Coordinate Transformations? ultimately need to be applied to the integral expression of Maxwell’s equations. Ordinarily, the flux of electromagnetic field out of a closed surface is equal to the charge enclosed by that surface. I will preview that when the requirement of invariance under general coordinate transformation for volume integrations changes the quantum mechanical expression to the following: the total flux of angular momentum through a close surface is equal to the momentum enclosed by that surface. It may be a few weeks before I have the time to write up that result.
because of the special
property that 
. I was able show how
the Lorentz force, in differential form, is the equation for geodesic motion in
an adiabatic fluid. But, when I wanted
to integrate back up to a finite region of spacetime (last page) to reconstruct
the usual expression for Lorentz force (7), (24), I started to mull what
happens when one integrates in spacetime, picked up the first sense that the
uncertainty principle might be lurking nearby, and started on the present
course.
General Relativity, Maxwell’s
Electrodynamics, and the Foundations of the Quantum Theory of Gravitation and
Matter (gr-qc/0511050)
Abstract: The formalism of electric – magnetic duality,
first pioneered by Reinich and Wheeler, extends General Relativity to encompass
non-Abelian fields. Several energy
tensors 
with non-vanishing trace matter are developed as a function
solely of the field strength tensor 
, including the Euler tensor, and tensors for matter in flux,
pressure in flux, and stationary pressure.
The spacetime metric 
is not only a solution
to the second-order Einstein equation based on , but is also constrained by a third-order equation involving
the Bianchi identity together with the gravitational energy components 
for each . The common
appearance of in all of the and makes it possible to
obtain quantum solutions for the spacetime metric, thereby geometrizing quantum
physics as a non-linear theory.
Magnetic Monopoles, Chiral
Symmetries, and the NuTeV Anomaly (hep-ph/0509223)
Magnetic Monopoles and Duality
Symmetry Breaking in Maxwell's Electrodynamics (hep-ph/0508257)
(1)
shown in the footnote on page 28 of
hep-ph/0508257. If one draws a Feynman diagram for each of
the fermions labeled with σ,τ,ν, and
recognizes that, for example, the non-Abelian dualon
field Cσν contains a vertex
coupling of the 
to vector bosons Gσ and a Gν,
and if 1, 2, 3 designate a given q in “initial,” “intermediate,” and “final”
states, and if the dotted arrow “recycles” from the final to the initial state,
then the above tensor can be represented in the Feynman and scattering diagrams
shown below:
There is no presupposition about what this is; this is
simply the drawing one gets when carefully applying Feynman rules to the
equation above. This looks very much
like a bound system of three fermions, and for this reason, I think we will
eventually show that baryons are represented by third rank antisymmetric
tensors which are the first / third rank duals of individual fermions, as also
discussed in hep-ph/0508257,
section 4. This is in the same sense
that fermions are represented in first rank tensors, a.k.a. vectors, in the form 
.
I have come in light of the drawing above which emerges from equation (1), to view the baryon as a “finite state machine.” Suppose we start everything in state 1: three quarks. Then, each quark emits a gluon and enters state 2: three quarks and three gluons. Then, the gluons are reabsorbed, and we get to state 3: three quarks again, which we reiterate to state 1. And, we do this reiteration trillions and trillions of times (> 1020) per second. As each gluon is emitted or absorbed, there is a Dirac delta at the vertex, which brings in some probabilistic uncertainty. Suppose we start out with 1/3 of the total baryon energy-momentum in each quark. After a number of iterations, and using the Dirac delta to bring in some uncertainty, we will quickly arrive at a probability distribution of the energy for each quark, which in the 1 and 3 states, will have a peak at 1/3 but will have a probabilistic distribution about 1/3 which should relate to the “valence” structure functions of quarks inside baryons. In the 2 state, some of the energy will also be carried by the gluons, and so the structure function for each quark will pick up some “sea” enhancements at <1/3. What we observe, since again, this finite state machine we call a baryon is iterating over 1020 times per second, is a juxtaposition of the pure 1 and 3 valence states, and the mixed 2 sea states. This is the path along which I would foresee development of the above, to establish the baryons as third rank antisymmetric tensors.
On
I have posted this DRAFT paper because several of the
sections of this paper are now planned as individual topical papers over the
upcoming months. By looking at these
sections, one can get an idea of the direction I am headed. The
introduction is outdated. The discussion
below should be taken as a more current guide to reading this DRAFT paper.
Section 2, will be the topic of a distinct paper. In this section, we show how duality invariance goes hand-in-hand with the geometrodynamic vacuum. In particular, we show that duality symmetry between electric and magnetic sources exists if and only if the energy tensor is that of the vacuum, Tμν=0. This means that by breaking duality symmetry, one is inherently going from an energy state characterized by Tμν=0 to one characterized by Tμν≠0. As such, we expect that duality symmetry (or lack thereof) is very closely aligned with the creation of gravitating, inertial matter.
Section 3 is still a work in progress. This began as an effort to understand how the strong interaction might be mediated by massless gluons yet still be short range. An important clue, in my estimation, is obtained by the contrast set up in equation (3.7). The next equation, (3.8), which uses an “=” sign, is both intriguing yet problematic. It is not clear to me whether the degrees of freedom (i.e., the spin 1 polarizations) match up, or can be made to match up. But, it is very important to recognize that the extra term in the wave equation for a massive vector boson on the left side of (3.8) seems to bear a very close relationship to the extra term in the wave equation for non-Abelian gauge bosons on the right hand side of (3.8). If one can develop and understand a method of symmetry breaking that explains the similarities highlighted in (3.7) and (3.8), we may be able to move beyond the Higgs-Goldstone mechanism for generating mass which I believe is an interim not a final understanding of how mass arises, and which many recognize as something of a “weak link” in electroweak theory. This section also shows, from an historical context, how I first discovered how to integrate the Dirac Quantization Condition with Reinich-Wheeler duality, as is developed more cleanly in hep-ph/0508257.
Sections 4 and 5 are earlier, QCD versions of what is developed in hep-ph/0508257 for QED. All of what is in these sections still applies to QCD. But, hep-ph/0508257 takes the very important additional step of developing duality as a local symmetry. I also call your attention particularly to equations (5.8) though (5.14) and (5.61) through (5.66). This was the context in which I first came upon the notion of a connection between electric and magnetic charges, and chiral symmetry. This connection was developed more formally in my second e-print at hep-ph/0509223, and led to the cross section calculations and the connection with the NuTeV anomaly. But, equations (5.8) though (5.14) and (5.61) through (5.66) lay the foundation for section 8 of this draft paper, which deals with the unification of the electroweak and strong interactions. All of this is also envisioned as a distinct paper.
Section 6 is an interesting read if one wishes to understand the complexion angle of hep-ph/0508257 in a broader context that includes weak and strong interactions as well as chiral symmetry.
Section 7 is alluded to in the * footnote on the bottom of page 21 in hep-ph/0508257, and will be the subject of a distinct paper. These section presents a possible exact relationship between probe energy and running couplings, even for couplings which are very large. It does so by making use of the complexion angle in the context of QCD, and in so-doing, uses the mass and radius of the proton as the basis for deriving the Particle Data Group’s experimental strong running coupling curve. I have refrained from putting this work on arxiv.org to date, because I want to make sure that these results are possibly “true,” and not “too good to be true.” Because, if these results are “true,” then they do indeed teach us how to move beyond perturbation theory and to deal in an exact manner even with very large interaction couplings. I would very much appreciate specific comments about section 7. If these results are plausible, I want to get them into a stand-alone paper as soon as possible, and post these to arxiv.org. But I want to be sure first, that I am not overlooking anything important.
Section 8, which will also be the topic of a distinct paper in its own right, deals with electroweak and strong unification, together with lepto-quark unification using B-L, and makes use of the connections between electric and magnetic charges and chiral symmetry as developed in the earlier-noted equations (5.8) though (5.14) and (5.61) through (5.66) as well as hep-ph/0509223 where this was used as the basis for deriving cross sections which account for at least part of the NuTeV anomaly. I have referred often to Volovik’s section 12.2. Section 8 here, expounds in detail, the connection which I envision with Volovik’s section 12.2, where the consolidation of weak and strong interactions uses [SU(4)R x SU(4)L] x [SU(2)R x SU(2)L], with the charge generator Q of electromagnetism sitting across weak and strong interactions according to Q = ˝ (B-L) + I3L + I3R. This shows how the extra freedom that arises from having both electric and magnetic charges solves the problem with spin which required Volovik to resort to the inartful use of “holons” and “spinons.” Also included in this section are some comments about superconductivity, which I recently elaborated in several posts on sci.physics, sci.physics.relativity, and sci.physics.particle.
On
Contact Information
E-mail address
jyablon@nycap.rr.com
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