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 Yang-Mills Mass Gap Solution.

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

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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

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  and currents , this non-linear result is:

, (13.6) 

where the approximation shows the connection to the linear approximation.  Re-expressed solely in terms of the fifth-dimensional gravitational metric tensor components and energy tensor source components , this result is: 

. (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.

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 .  Let us repeat this same calculation, but in a slightly different way. 

 In section 8, we used (8.3) in the form of , because that gave us a contravariant against which to obtain the five-dimensional variation .  Let us instead, here, use the very last term in (8.3) as , writing this as:

. (10.1)

It is important to observe that the term is only summed over four spacetime indexes.  The fifth term, , see, e.g., (6.8).  For consistency with the non-symmetric (9.5), we employ rather than .  By virtue of this separation, in which we can only introduce and not as in section 8, we can only take a four-dimensional variation , which, in contrast to (8.4), is now given by:

. (10.2)

Substituting from (10.1) then yields:

. (10.3)

Now, the non-symmetry of sections 8 and 9 comes into play, and this will yield the Maxwell tensor.  Because , the first term drops out and the above reduces to:

. (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 rather than .  One again, the screen factor is at work. 

 In mixed form, starting from (10.3), there are two energy tensors to be found.  If we raise the index in (10.3), the first term becomes and we obtain:

. (10.5)

with this first term still screened out.  However, if we transpose (10.3) and then raise the index, the first term becomes and this term does not drop out, i.e.,

. (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 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 , before reduction into this more-recognizable form.

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.

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. 

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 which generalizes (5.4) to five dimensions, and recognizing that the field strength tensor is related to the four-vector potential according to , let us now examine the relationship between and the metric tensor .  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 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 , expanding the Christoffel connections , making use of which as shown in (6.5) is equivalent to , and using the symmetry of the metric tensor, we may write:

.  (8.1)

It is helpful to lower the indexes in field strength tensor and connect this to the covariant potentials , generalized into 5-dimensions as , using , as such:

. (8.2)

The relationship expresses clearly, the antisymmetry of in terms of the remaining connection terms involving the gravitational potential.  Of particular interest, is that we may deduce from (8.2), the proportionality

. (8.3)

(If one forms from (8.3) and then renames indexes and uses , one arrives back at (8.2).)  Further, we well know that , i.e., that the covariant derivatives of the potentials cancel out so as to become ordinary derivatives when specifying , i.e., that is invariant under the transformation .  Additionally, the Maxwell components (7.10) of the Einstein equation, are also invariant under , because (7.10) also employs only the field strength .  Therefore, let is transform in the above, then perform an ordinary integration and index renaming, to write:

. (8.4)

In the four spacetime dimensions, this means that the axial portion of the metric tensor is proportional to the vector potential, , and that the field strength tensor and the gravitational field equations are invariant under the transformation used to arrive at (8.4).  We choose to set , 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 , where g is the metric tensor determinant and is the Ricci tensor.  Let us now examine a Lagrangian based upon the 5-dimensional Ricci scalar, which we specify by:

. (9.1)

Lab Note 2, Part 2: Gravitational and Inertial Mass, and Electrodynamics as Geometry, in 5-Dimensional Spacetime

(You may obtain a PDF version of this lab note at Electrodynamic Geodesics) Note: See also Part 3 of this Lab Note, Gravitational and Electrodynamic Potentials, the Electro-Gravitational Lagrangian, and a Possible Approach to Quantum Gravitation, which contains further development.

1.  Introduction  It has been understood at least since Galileo’s refutation of Aristotle which legend situates at the Leaning Tower of Pisa, that heavier masses and lighter masses similarly-disposed in a gravitational field will accelerate at the same rate and reach the ground after identical times have elapsed.  Physicists have come to describe this with the principle that the “gravitational mass” and the “inertial mass” of any material body are “equivalent.”  As a material body becomes more massive and so more-susceptible to the pull of a gravitational field (back when gravitation was viewed as action at a distance), so too this increase in massiveness causes the material body in equal measure to resist the gravitational pull.  By this equivalence, the result is a “wash,” and so with the neglect of any air resistance, all the bodies accelerate and fall at the same rate.  (The other consequence of Galileo’s escapade, is that it strengthened the role of experimental testing, in relation to the “pure thought” upon which Aristotle had relied to make the “obvious” but untested and in fact false argument that heavy objects should fall faster.  In this way, it spawned the essence of what we today know as the scientific method which remains a dynamic blend of thought and creativity, with experience and cold, hard numbers derived from measurement of masses, lengths, and times.)

  Along his path to developing the General Theory of Relativity (GTR), Albert Einstein made a brief stop in 1911 in an imaginary elevator, to conduct a gedanken in which he concluded that the physical experience of an observer falling freely in a gravitational field before terminally hitting the ground is no different from what was commonly thought of as Newton’s inertial motion in which a body in motion remained in motion unless acted upon by a “force.”  (GTR later showed that this was not quite true, the “asterisk” to this insight arising from the so-called tidal forces.)  And, he concluded that the force one feels standing on the floor of an elevator in free fall to which a constant force is then applied, is no different from the force one feels when standing on the surface of the earth.

  The General Theory of Relativity, in the end, captured inertial motion and its close cousin of free-fall motion in a gravitational field, in the most elegant way, as simple geodesic motion in a curved geometry along geodesic paths which coincide precisely with the paths one observes for bodies moving under gravitational influences.  This was a triumph of the highest order, as it placed gravitational theory on the completely-solid footing of Riemannian geometry, and became the “gold standard” against which all other physical theories are invariably measured, even to this day.  (“Marble and wood” is another oft-employed analogy.)

  However, the question of “absolute acceleration,” that is, of an acceleration which is not simply a geodesic phenomenon of unimpeded free fall through a swathe carved out by geometry, but rather one in which an observer actually “feels” a “force” which can be measured by a “weight scale” in physical contact between the observer and that body which applies the force, is in fact not resolved by GTR.  To this day, it is hotly-debated whether or not there is such a thing as “absolute acceleration.”  Surely, the forces we feel on our bodies in elevators and cars and standing on the ground are real enough, but the question is whether there is some way to understand these forces — which are impediments to what would otherwise be our own geodesic free fall motion in spacetime under the influence of gravity and nothing more — as geodesic forces in their own right, simply of a different, supplemental, and perhaps more-subtle character than the geodesics of gravitation.  That is the central question to be examined in this lab note.