Lab Notes for a Scientific Revolution (Physics)

November 30, 2013

The Yang-Mills Mass Gap Solution

Friends,

I have been a bit behind with this blog, but wanted to let you know that I have pulled together many of the various threads I have posted over the past several years into a complete solution to the Yang-Mills and Mass Gap Problem, which paper is here:

The Mass Gap problem was specified back in 2000 by Arthur Jaffe and Edward Witten at

http://www.claymath.org/millennium/Yang-Mills_Theory

This problem really has four aspects, which are as follows: 1) the mass gap itself, 2) QCD confinement, 3) chiral symmetry breaking and 4)  proof of the existence of a relativistic quantum Yang-Mills field theory in four-dimensional spacetime.  Each of these is respectively presented in sections 10, 11, 12 and 13 of this paper.

You can read the paper abstract, so I will not repeat it here.  But I will also be delivering an oral presentation of this work at the April 2014 APS meeting in Savannah, Georgia.  Yesterday, I submitted the abstract for that presentation, which is below:

APS Abstract: The Yang-Mills Mass Gap problem is solved by deriving SU(3)C Chromodynamics as a corollary theory from Yang-Mills gauge theory.  The mass gap is filled from the finite non-zero eigenvalues of a configuration space inverse perturbative tensor containing vacuum excitations.  This results from carefully developing six equivalent views of Yang-Mills gauge theory as having: 1) non-commuting (non-Abelian) gauge fields; 2) gauge fields with non-linear self-interactions; 3) a “steroidal” minimal coupling; 4) perturbations; 5) curvature in the gauge space of connections; and 6) gauge fields related to their source currents through an infinite recursive nesting.  Based on combining the Yang-Mills electric and magnetic source field equations into a single equation, confinement results from showing how the magnetic monopoles of Yang-Mills gauge theory exhibit color confinement and meson flow and have all the required color symmetries of baryons, from which we conclude that they are one and the same as baryons.  Chiral symmetry breaking results from the recursive behavior of these monopoles coupled with a view of the Dirac gamma matrices as Hamiltonian quaternions extended into spacetime.  Finally, with the aid of the “steroidal” view, the recursive view of Yang-Mills enables polynomial gauge field terms in the Yang-Mills action to be stripped out and replaced by polynomial source current terms prior to path integration.  This enables an exact analytical calculation of a non-linear path integral using a closed recursive kernel and yields a non-linear quantum amplitude also with a closed recursive kernel, thus proving the existence of a non-trivial relativistic quantum Yang–Mills field theory on R4 for any simple gauge group G.

I am of course interested in any comments you may have.

Jay

May 24, 2012

Baryons and Confinement; Exact Quantum Yang Mills Propagators; Mass Gap

To all:

I have started work on physics again this last month after two years “sabbatical.”  I am also again working with my friend Andrej Inopin.

In particular, I am touching up a paper that I was working on in 2008 which shows that baryons are simply magnetic charges in a non-Abelian (Yang Mills) gauge theory, and shows how confinement phenomena are a natural outgrowth of the properties of these “magnetic charge baryons.”  This paper is linked at:

https://jayryablon.files.wordpress.com/2012/05/baryon-paper-3-1.pdf

Now, in returning to this paper after several years, I have always known that my equation (3.5) in the above was a “shortcut” to get to the results  afterwards, because it relies upon an analogy from QED and does not fully develop propagators / inverses for Yang-Mills theory.

This is because back in 2008, I did not know how to quantize Yang-Mill theory and obtain exact propagators that embody all of the non-linearity that comes from Yang-Mills.  Nobody knew / knows how to do this.  That is why people still use perturbation theory even though it breaks up the gauge invariance of Yang-Mills, or use lattice gauge theory even though it breaks up Lorentz symmetry and they have to calculate numerically on computers rather than analytically.  These are “compromises” that everybody uses because exact Yang-Mills quantization solutions simply are not known to date.

But in the last several weeks, I returned to this problem that had been a roadblock for me in 2008, and have now solved it!   The link below is the current version of a paper I have written in the last two weeks which contains this solution.

https://jayryablon.files.wordpress.com/2012/05/quantum-yang-mills-3-for-spf.pdf

Sections 2 and 3 in the above just link replace the “shortcut” of (3.5) in the previous link further up this page.  Section 4 shows that the perturbation which is an important object in this theory actually transforms just like a GRAVITATIONAL field.  I write this with the view that this is a possible path to non-Abelian quantum gravity, but am reserving judgment on this and would like to hear other views.  But what I think is unmistakable is that this shows that gauge transformations in the perturbation — which might be reason to doubt using this perturbation to calculate invariant numbers — are equivalent to no more and no less that plain old general coordinate transformations.  In essence, the perturbation combines several dot products which alone are not invariant, but which together, are.

The work in this paper lays the foundation and provides the calculating machinery for solving the “mass gap” problem.  I will continue developing this in the week ahead, but I have enough already that I wanted to share.

December 26, 2008

S=2, mu=0 Meson Mass Spectrum, and some interesting possible ties to experimental meson data

Before I head out on holiday, I also wanted to post one more item:

In equation (11.8) of my earlier post at:

https://jayryablon.files.wordpress.com/2008/12/su-3-paper-20.pdf

I showed the matrix inverse for mesons based on the values of S=2 and mu=0 using the parameters of the theory developed in that work (which is based on the post I made a few hours ago).

I finished a detailed calculation of the predicted meson masses as a fraction of “.5vg” and put them in ascending order, in the following one-page listing:

https://jayryablon.files.wordpress.com/2008/12/s2-mu0-mass-spectrum.pdf

This is the type of theoretical result that we need to try to fit to experimental meson masses.  That is, this is where the “rubber meets the road.”

In this regard, I point that there are good reasons from the underlying theory to compare and take the ratios of numbers in the above with the 1+/-i factors, and to consider the SU(3) vector to be (uds) from the old quark flavor models (as opposed to the (RGB) of color).

One of these ratios is that of what is the 4,5 mass matrix element to the 1,2 element:

.625727090299/.169470755895=3.69220577135

and this should be related to the ratio of the meson K^0=d s-bar to pi^0=d d-bar.  That experimental ratio is, in fact:

K^0/pi^0 = 497.614 MeV / 134.9766 MeV = 3.6867

This is *very* close (they differ by 1.5 parts per thousand!), and could be an experimental validation of the whole theory, since the only thing not accounted for theoretically are QED corrections!

Another ratio of interest is:

.169470755895/.163577444819=1.03602765089

This is because the experimental pi^+/- to pi^0 ratio is:

pi^+/- / pi^0 = 139.5701 MeV / 134.9766 MeV = 1.0340

This also is rather tantalizing, and is off by just under 2 parts per thousand!

Still trying to figure out the whole fit, but I’ll leave you all with that for now.

Happy new year!

Jay.

Finite Amplitudes Without +i\epsilon

To all,

I have now completed a paper at the link below, which summarizes the work I have been doing for the past two months (and in a deeper sense, for much of my adult life) to lay a foundation for understanding and calculating particle masses:

finite-amplitudes-without-i-epsilon

I have also taken the plunge and submitted this for peer review. ;-)?

The abstract is as follows:

By carefully reviewing how the invariant amplitude M is arrived at in the simplest Yang-Mills gauge group SU(2), we show how to arrive at a finite, pole-free amplitudes without having to resort to the “+i$\epsilon$ prescription.”  We first review how gauge boson mass is generated in the SU(2) action via spontaneous symmetry breaking in the standard model, and then carefully consider the formation of finite, on-shell amplitudes, without +i$\epsilon$.

Comments are welcome, and I wish everyone a happy holiday and New Year!

Jay.

December 11, 2008

Understanding the QCD Meson Mass Spectrum

Dear Friends:

It has been awhile since I last posted and it is good to be back.

Almost two years ago in the course of my work on Yang Mills, I came across what I believe is an approach by which mass spectrum of the massive mesons of QCD might be understood.  I had what I still believe is the right concept, and many of the pieces, but I could not figure out the right execution of the concept in complete detail.  Over the past year and a half I walked away from this to let the dust settle and to also arrive at a place where the basic principles of quantum field theory were no longer “new” to me but had become somewhat ingrained.  Now, I believe I have found the right way to execute this concept, and the results are intriguing.

In the file linked below, which I will update on a regular basis in the coming days:

I review how mass is known to be generated in SU(2), as a template for considering SU(3) QCD.  I have tried to explain as simply as possible, what I believe to be the origin of QCD meson masses, as well as to lay the foundation for theoretically predicting these.  Keep in mind, finding out how the vector mesons of QCD obtain their non-zero masses, which make the QCD interaction short range despite supposedly-massless gluons, is one aspect of the so-called “mass gap” problem, see point 1) on page 3 of
http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf at
http://www.claymath.org/millennium/Yang-Mills_Theory/.

Then, I extend this development, in detail, to SU(3).

Several interesting results are already here:

1)  This approach neatly solves the problem of propagator poles (infinities) in a manner which I believe has not heretofore been discovered.  Goodbye to the +i\eta prescription, off mass-shell particles, and other inelegant dodges to achieve a finite propagator.

2) This approach may solve the confinement and the mass gap problems simultaneously.  It is important to understand that electroweak SU(2)xU(1) is a special case in which the gauge bosons are synonymous with the observed vector mesons, but that in SU(3) and higher order theories they are not.  The gauge bosons aka gluons, which show up in the Lagrangian, are not observed.  What is observed are the vector mesons which pass through to the denominator of the propagator in the invariant amplitude.

3) There emerges is a quantum number that is restricted to three discrete values, and depending on which value of chosen, all the meson masses are scaled up or down on a wholesale basis.  I believe that this may resolve the problem of generation replication.

I expect to be churning out mass calculations in the next day or two.  You may wish to check out the meson mass tables at http://pdg.lbl.gov/2008/tables/rpp2008-qtab-mesons.pdf, because that table contains the data which I am going to try to fit to equation (6.1), via (6.5).

Hope you enjoy!

Jay.

June 19, 2008

A New Lab Note: Commutation of Linear Rest Mass with Canonical Position

It has been awhile since my last blog entry, but if you want to check out some my recent wanderings through physicsland, check out sci.physics.foundations, relativity, and research.

Here, I would like to show a rather simple calculation, which may cast a different light on how one needs to think about the canonical commutation relationship $\left[x_{j} ,p_{k} \right]=i\eta _{jk} ;\; j,k=1,2,3$.  I would very much like your comments in helping me sort this through.  You may download this in pdf form at https://jayryablon.files.wordpress.com/2008/06/linear-mass-commutator-calculation.pdf.

I.  A Known Square Mass Commutation Calculation

Consider a particle of mass m as a single particle system.  Consider canonical coordinates $x_{\mu }$, and that at least the space coordinates $x_{j} ;\; j=1,2,3$ are operators.  If we require that the mass m must commute with all operators, then we must have $\left[x_{\mu } ,m\right]=0$, and by easy extension, $\left[x_{\mu } ,m^{2} \right]=0$.  It is well known that the commutation condition $\left[x_{\mu } ,m^{2} \right]=0$, taken together with the on-shell mass relationship$m^{2} =p^{\sigma } p_{\sigma }$ and the single-particle canonical commutation relationship $\left[x_{j} ,p_{k} \right]=i\eta _{jk} ;\; j,k=1,2,3$, where $diag\left(\eta _{\mu \nu } \right)=\left(-1,+1,+1,+1\right)$ is the Minkowski tensor, leads inexorably to the commutation relationship:

$\left[x_{k} ,p_{0} \right]=-ip_{k} /p^{0} =-iv_{k}$   (1.1)

where $v_{k}$ is the particle velocity (in c=1 units) along the kth coordinate.  I leave the detailed calculation as an exercise for the reader not familiar with this calculation, and refer also to the sci.physics.research thread at http://www.physicsforums.com/archive/index.php/t-142092.html or http://groups.google.com/group/sci.physics.research/browse_frm/thread/d78cbfecf703ff6a.

I would ask for your comments on the following calculation, which is totally analogous to the calculation that leads to (1.1), but which is done using the linear mass m rather than the square mass $m^{2}$, and using the Dirac equation written as $m\psi =\gamma ^{\nu } p_{\nu } \psi$, in lieu of what is, in essence, the Klein Gordon equation $m^{2} \phi =p^{\sigma } p_{\sigma } \phi$ that leads to (1.1).

2.  Maybe New?? Linear Mass Commutation Calculation

$m\psi =\gamma ^{\nu } p_{\nu } \psi$ .  (2.1)

Require that:

$\left[x_{\mu } ,m\right]=0$   (2.2)

Continue to use the canonical commutator $\left[x_{j} ,p_{k} \right]=ig_{jk}$.  Multiply (2.1) from the left by $x_{\mu }$ noting that $\left[\gamma ^{\nu } ,x_{\mu } \right]=0$ to write:

$x_{\mu } m\psi =\gamma ^{\nu } x_{\mu } p_{\nu } \psi =\gamma ^{0} x_{\mu } p_{0} \psi +\gamma ^{j} x_{\mu } p_{j} \psi$ .  (2.3)

This separates into:

$\left\{\begin{array}{c} {x_{0} m\psi =\gamma ^{0} x_{0} p_{0} \psi +\gamma ^{j} x_{0} p_{j} \psi } \\ {x_{k} m\psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} x_{k} p_{j} \psi } \end{array}\right.$ .  (2.4)

Now, use the canonical relation $\left[x_{j} ,p_{k} \right]=i\eta _{jk}$ to commute the space (k) equation, thus:

$\begin{array}{l} {x_{k} m\psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} x_{k} p_{j} \psi =\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} \left(p_{j} x_{k} +i\eta _{jk} \right)\, \psi } \\ {=\gamma ^{0} x_{k} p_{0} \psi +\gamma ^{j} p_{j} x_{k} \psi +i\gamma _{k} \psi } \\ {=\gamma ^{0} x_{k} p_{0} \psi +mx_{k} \psi -\gamma ^{0} p_{0} x_{k} \psi +i\gamma _{k} \psi } \end{array}$ .  (2.5)

In the final line, we use Dirac’s equation written as $mx_{\mu } \psi =\gamma ^{\nu } p_{\nu } x_{\mu } \psi =\gamma ^{0} p_{0} x_{\mu } \psi +\gamma ^{j} p_{j} x_{\mu } \psi$, and specifically, the $\mu =k$ component equation $\gamma ^{j} p_{j} x_{k} \psi =mx_{k} \psi -\gamma ^{0} p_{0} x_{k} \psi$.

If we require that $\left[x_{\mu } ,m\right]=0$, which is (2.2), then (2.5) reduces easily to:

$\gamma ^{0} \left[x_{k} ,p_{0} \right]\psi =-i\gamma _{k} \psi$ ,  (2.6)

Finally, multiply from the left by $\gamma ^{0}$, and employ $\gamma ^{0} \gamma _{k} \equiv \alpha _{k}$ and $\gamma ^{0} \gamma ^{0} =1$ to write:

$\left[x_{k} ,p_{0} \right]\, \psi =-i\alpha _{k} \psi$ .  (2.7)

If we contrast (2.7) to (1.1) written as $\left[x_{k} ,p_{0} \right]\phi =-iv_{k} \phi$, we see that the velocity $p_{k} /p^{0} =v_{k}$ has been replaced by the Dirac operator $\alpha _{k}$, that is, $v_{k} \to \alpha _{k}$.

3.  Questions

Here are my first set of questions:

1)  Is the calculation leading to (2.7) correct, and is (2.7) a correct result, or have I missed something along the way?

2)  If (2.7) is correct, has anyone seen this result before?  If so where?

3)  Now use the plane wave $\psi =ue^{ip^{\sigma } x_{\sigma } }$ so that we can work with the Dirac spinors $u\left(p^{\mu } \right)$, and rewrite (2.7) as:

$\left\{\begin{array}{c} {\left(\alpha _{k} -\lambda \right)\, u=0} \\ {\lambda =i\left[x_{k} ,p_{0} \right]} \end{array}\right.$

The upper member of (3.1) is an eigenvalue equation.  Reading out this equation, I would say that the commutators $\lambda =i\left[x_{k} ,p_{0} \right]$ are the eigenvalues of the Dirac $\alpha _{k}$ matrices, which are:

${\bf \alpha }=\left(\begin{array}{cc} {0} & {{\bf \sigma }} \\ {{\bf \sigma }} & {0} \end{array}\right)$ and ${\bf \alpha }=\left(\begin{array}{cc} {-{\bf \sigma }} & {0} \\ {0} & {{\bf \sigma }} \end{array}\right)$ ,  (3.2)

in the respective Pauli/Dirac and Weyl representations, and that the u are the eigenvectors associated with these eigenvalues $\lambda =i\left[x_{k} ,p_{0} \right]$.  Am I wrong?  If not, how would one interpret this result?  Maybe the commutators $\left[x_{j} ,p_{k} \right]=i\eta _{jk}$ can be discussed in the abstract, but it seems to me that the commutators $\lambda =i\left[x_{k} ,p_{0} \right]$ can only be discussed as the eigenvalues of the matrices $\alpha _{k}$ with respect to the eigenstate vectors u.  This, it seems, would put canonical commutation into a somewhat different perspective than is usual.

Just as Dirac’s equation reveals some features that cannot be seen strictly from the Klein Gordon equation, the calculation here seems to reveal some features about the canonical commutators that the usual calculation based on $\left[x_{\mu } ,m^{2} \right]=0$ and $m^{2} =p^{\sigma } p_{\sigma }$ cannot, by itself, reveal.

I’d appreciate your thoughts on this, before I proceed downstream from here.

Thanks,

Jay.

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.

Jay.

February 14, 2008

Lab Note 2, Part 3: Gravitational and Electrodynamic Potentials, the Electro-Gravitational Lagrangian, and a Possible Approach to Quantum Gravitation

Note:  You may obtain a PDF version of Lab Note 2, with parts 2 and 3 combined, at Lab Note 2, with parts 2 and 3.

Also Note: This Lab Note picks up where Lab Note 2, Part 2, left off, following section 7 thereof.  Equation numbers here, reference this earlier Lab Note.

8.  The Electrodynamic Potential as the Axial Component of the Gravitational Potential

Working from the relationship $F^{{\rm M} } _{{\rm T} } \propto 2\Gamma ^{{\rm M} } _{{\rm T} 5}$ which generalizes (5.4) to five dimensions, and recognizing that the field strength tensor $F^{\mu \nu }$ is related to the four-vector potential $A^{\mu } \equiv \left(\phi ,A_{1} ,A_{2} ,A_{3} \right)$ according to $F^{\mu \nu } =A^{\mu ;\nu } -A^{\nu ;\mu }$, let us now examine the relationship between $A^{\mu }$ and the metric tensor $g_{{\rm M} {\rm N} }$.  This is important for several reasons, one of which is that these are both fields and so should be compatible in some manner at the same differential order, and not the least of which is that the vector potential $A^{\mu }$ is necessary to establish the QED Lagrangian, and to thereby treat electromagnetism quantum-mechanically.  (See, e.g., Witten, E., Duality, Spacetime and Quantum Mechanics, Physics Today, May 1997, pg. 28.)

Starting with ${\tfrac{1}{2}} F^{{\rm M} } _{{\rm T} } \propto \Gamma ^{{\rm M} } _{{\rm T} 5}$, expanding the Christoffel connections $\Gamma ^{{\rm A} } _{{\rm B} {\rm N} } ={\tfrac{1}{2}} g^{{\rm A} \Sigma } \left(g_{\Sigma {\rm B} ,{\rm N} } +g_{{\rm N} \Sigma ,{\rm B} } -g_{{\rm B} {\rm N} ,\Sigma } \right)$, making use of $g^{{\rm M} {\rm N} } _{,5} =0$ which as shown in (6.5) is equivalent to $F^{{\rm M} {\rm N} } =-F^{{\rm N} {\rm M} }$, and using the symmetry of the metric tensor, we may write:

${\tfrac{1}{2}} F^{{\rm M} } _{{\rm T} } \propto \Gamma ^{{\rm M} } _{{\rm T} 5} ={\tfrac{1}{2}} g^{{\rm M} \Sigma } \left(g_{\Sigma {\rm T} ,5} +g_{5\Sigma ,{\rm T} } -g_{{\rm T} 5,\Sigma } \right)={\tfrac{1}{2}} g^{{\rm M} \Sigma } \left(g_{5\Sigma ,{\rm T} } -g_{5{\rm T} ,\Sigma } \right)$.  (8.1)

It is helpful to lower the indexes in field strength tensor and connect this to the covariant potentials $A_{\mu }$, generalized into 5-dimensions as $A_{{\rm M} }$, using $F_{\Sigma {\rm T} } \equiv A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma }$, as such:

$A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma } \equiv F_{\Sigma {\rm T} } =g_{\Sigma {\rm M} } F^{{\rm M} } _{{\rm T} } \propto g_{\Sigma {\rm M} } g^{{\rm M} {\rm A} } \left(g_{5{\rm A} ,{\rm T} } -g_{5{\rm T} ,{\rm A} } \right)=\left(g_{5\Sigma ,{\rm T} } -g_{5{\rm T} ,\Sigma } \right)$. (8.2)

The relationship $F_{\Sigma {\rm T} } \propto \left(g_{5\Sigma ,{\rm T} } -g_{5{\rm T} ,\Sigma } \right)$ expresses clearly, the antisymmetry of $F_{\Sigma {\rm T} }$ in terms of the remaining connection terms involving the gravitational potential.  Of particular interest, is that we may deduce from (8.2), the proportionality

$A_{\Sigma ;{\rm T} } \propto g_{5\Sigma ,{\rm T} }$. (8.3)

(If one forms $A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma }$ from (8.3) and then renames indexes and uses $g_{{\rm M} {\rm N} } =g_{{\rm N} {\rm M} }$, one arrives back at (8.2).)  Further, we well know that $F_{\Sigma {\rm T} } =A_{\Sigma ;{\rm T} } -A_{{\rm T} ;\Sigma } =A_{\Sigma ,{\rm T} } -A_{{\rm T} ,\Sigma }$, i.e., that the covariant derivatives of the potentials cancel out so as to become ordinary derivatives when specifying $F_{\Sigma {\rm T} }$, i.e., that $F_{\Sigma {\rm T} }$ is invariant under the transformation $A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} }$.  Additionally, the Maxwell components (7.10) of the Einstein equation, are also invariant under $A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} }$, because (7.10) also employs only the field strength $F^{\sigma \mu }$.  Therefore, let is transform $A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} }$ in the above, then perform an ordinary integration and index renaming, to write:

$A_{{\rm M} } \propto g_{5{\rm M} }$. (8.4)

In the four spacetime dimensions, this means that the axial portion of the metric tensor is proportional to the vector potential, $g_{5\mu } \propto A_{\mu }$, and that the field strength tensor $F_{\Sigma {\rm T} }$ and the gravitational field equations $-\kappa T^{{\rm M} } _{{\rm N} } =R^{{\rm M} } _{{\rm N} } -{\tfrac{1}{2}} \delta ^{{\rm M} } _{{\rm N} } R$ are invariant under the transformation $A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} }$ used to arrive at (8.4).  We choose to set $A_{\Sigma ;{\rm T} } \to A_{\Sigma ,{\rm T} }$, and can thereby employ the integrated relationship (8.4) in lieu of the differential equation (8.3), with no impact at all on the electromagnetic field strength or the gravitational field equations, which are invariant with respect to this choice.

9.  Unification of the Gravitational and QED Lagrangians

The Lagrangian density for a gravitational field in vacuo is ${\rm L}_{gravitation} =\sqrt{-g} R$, where g is the metric tensor determinant and $R=g^{\mu \nu } R_{\mu \nu }$ is the Ricci tensor.  Let us now examine a Lagrangian based upon the 5-dimensional Ricci scalar, which we specify by:

${\rm R} \equiv R^{\Sigma } _{\Sigma } =R^{\sigma } _{\sigma } +R^{5} _{5} =R+R^{5} _{5}$. (9.1)
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