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,

. (3.2)

which is used to connect the ratio from the Lorentz law to geodesic motion in five dimensions, and is a numeric constant of proportionality. Section 4 below picks up from this.

Excerpt from Section 4:

Transforming into an “at rest” frame, , the spacetime metric equation reduces to , and (3.2) becomes:

. (4.1)

For a timelike fifth dimension, may be drawn as a second axis orthogonal to , and the physics ratio (which, by the way, results in the material body in an electromagnetic field actually “feeling” a Newtonian force in the sense of due to the inequivalence of electrical and inertial mass) measures the “angle” at which the material body moves through the “time plane.”

For a spacelikefifth dimension, where one may wish to employ a compactified, hyper-cylindrical (see [Sundrum, R., TASI 2004 Lectures: To the Fifth Dimension and Back, http://arxiv.org/abs/hep-th/0508134 (2005).], Figure 1) and is a constant radius (distinguish from the Ricci scalar by context), . Substituting this into (3.2), leaving in the ratio obtained in (4.1), and inserting into the first term to maintain a dimensionless equation, then yields:

. (4.2)

We see that here, the physics ratio 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 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 , in the following manner:

Assume that is spacelike, casting one’s lot with the preponderance of those who study Kaluza-Klein theory. In (4.2), move the away from the first term and move the over to the first term. Then, multiply all terms by another . Everything is now dimensioned as an angular momentum , which we have just ascertained is constant irrespective of mass. So, set this all to , which for , represents intrinsic spin. The result is as follows:

. (4.3)

Now, take the second and fourth terms, and solve for with , to yield:

, (4.4)

where is the Planck length. This gives a definitive size for the compactification radius, and it is very close to the Planck length.

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

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

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

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 3, Part 2: Unification of Particle, Nuclear and Atomic Phenomonology

This lab note will be brief.

On April 28, 2007, I posted a paper which went from baryons and confinement to strings to particle phenomenology to atomic physics and deuterons and a whole range of phenomenology including fermion generation replication which appeared to lend itself to a common, underlying explanation based on the work I have previously discussed with respect to baryons and confinement in particular.  The underlying thread throughout, is to connect spacetime symmetry to internal symmetry using the Pauli fermionic exclusion principle.   I am afraid, however, that this paper may have been buried amidst all of the other postings, so I want to specifically call it to your attention, at the link below:

On The Natural Origin of Baryons, Short-Range Mesons, and QCD Confinement, from Maxwell’s Magnetic Equations for a Yang-Mills Field

In the spirit of “Lab Notes” which are a scientific diary of theoretical explorations, I ask you in particular to look at the second half of this paper, starting at section 6.  In football, there is something known as a “Hail Mary” pass where the quarterback throws the ball all the way down the field hoping for a touchdown.  The second half of the above paper is just that.  While certainly speculative, it seems to me that this ties together a very diverse range of observable phenomenology which has not previously been tied together.   It is probably the most audacious piece of physics writing I have done, and I don’t want it to get lost in the shuffle.

So, if nothing comes of it, so be it.  But, it may well be that someone in the end zone will catch this long pass, and physics will come to rest in a different place from where it rests today.  That is why it is so important to take good lab notes!

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.

Lab Note 4: An Interesting Left-Chiral Muliplet Perhaps Indicative of Preonic Structure for Fermions

I return in this brief Lab Note to the underlying spirit of one of the basic premises of this Weblog, which is that these are a series of “Lab Notes.”  We often tend these days to think and speak about “theories,” rather than “notes,” and certainly, many of the “Lab Notes” which I am presenting here are intended to be thought of as “theories,” or “theories-under-development,” as much as “lab notes.”

But when one talks about “Lab Notes,” what should be the prevailing thought is that sometimes, in the course of research, one uncovers an interesting “data point.”   Perhaps that data point goes nowhere; perhaps, when one looks hard at that data point and follows it up carefully, it leads in a whole new direction and changes many things in physics.

It is in this spirit that I present Lab Note 4, “An Interesting Left-Chiral Muliplet Perhaps Indicative of Preonic Structure for Fermions,” which I offer in the spirit of an interesting theoretical “data point” I uncovered back in 1988.  No more, no less.

For some history, I had been auditing some physics courses at nearby Rensselaer Polytechnic Institute (which my daughter Paula is now attending as a Freshman Chemical Engineer), and in the course of my studies there began a dialogue and I later became friends with the tragically-late Professor Nimai Mukhopadhyay, who was a particle physicist. 

At the time, particularly because of the “isospin redundancy” between quarks and leptons — which is my way of saying that both quarks and leptons can exist in both an “isospin up” and an “isospin down” state — I began to realize that there are really two distinct “attributes” which specify the “flavor” of an “elementary” fermion, within each generation.   First: is it a “quark” or a “lepton”?  Second, is its isospin “up” or “down.”  And, this, I began to suspect, was indicative of a preon substructure for the fermions — part of which provided the quark versus lepton aspect of flavor, the other part of which provided the up versus down isospin aspect of flavor. 

Following this thinking, I found that if one were to consider the flavor quantum numbers for only the left-chiral fermions, it turned out that the simple gauge group SU(4) could be used to represent the flavor symmetry of these left-chiral fermions, and that four preons, simply, A, B, C, D so as to avoid any preconceptions at all, could be used in pairs so as to construct the left-chiral fermion flavors.  I wrote this up for the 1988 “Excited Baryons” conference at RPI, and Dr. Mukhopadhyay included the writeup with the conference program.  For your consideration as a “lab note,” i.e., as an interesting piece of research data to keep in one’s mind, I link to a copy of that 1988 writeup below:

Left-Chiral Flavor Muliplet 

At this point in time, 20 years later, it is clear to me that I am not the only person to have thought in this way.  For example, G. Volovik, in his 2003 book “The Universe in a Helium Droplet,” includes an excellent discussion in section 12.2, which I have uploaded to the following link:  

Volovik Excerpt on Quark and Lepton Preonic Structure

Volovik makes a separation very similar to what I was going after in 1988, does so very clearly, and also, nicely handles the right-chiral states which drove me to fits back in 1988.  In fact, this excerpt from Volovik is another very important “lab note,” in and of itself, and I commend it to the reader.   The main problem which I perceive with this excerpt from Volovik, however, is his handling of the spins, which motivates the use of “holons” (spin 0) and “spinons” (spin 1/2) to construct a spin 1/2 fermion out of two preons.  In my view, it would be preferred for each of these preons to have spin 1/2, and when combined, for the resulting particle to also have spin 1/2.  That is, we need to find a way to have 1/2 = 1/2 + 1/2.

How we do this is another story which utilizes the fact that a fermion is a four-component Dirac spinor, and that left- and right-chirality each occupy two of the four components.  Thus, for the left- and right-chiral components of a fermion , each of which has spin 1/2, one can combine those into the whole four-component fermion — which still has spin 1/2.  That is, may be a way to implement 1/2 = 1/2 + 1/2 within the context of Volovik, thereby avoiding the seemingly-artificial (to me) holons and spinons.  But, that is a topic for another lab note, and this approach clearly exploits pre-existing chiral properties of fermions. 

In any event, please take a look at my 1988 publication linked above, look also at the linked Volovik excerpt, think about his and my separation of the fermion flavor attributes into quark/lepton and isospin up/down, think about the inelegance of using holons and spinons (at least if you and I have the same sense of elegance), and think about the chiral properties of fermions.  I do continue to believe the believe that the clearly-established “isospin redundancy” between quarks and leptons is the best evidence we have, of a preonic substructure for fermions which is waiting to be better understood, and cast into a suitable formal pedagogical structure.

That concludes this lab note.