The Larson Research Center

In the Trouble with Physics blog, we are discussing the struggle Lee Smolin is having in taking on the LST community’s propensity to favor string theory as the approach to unifying the LST theory of gravity, a continuum based theory, and the LST theory of matter, a discrete based theory.  In the latest post, entitled The Big Bet, we focus on the UK reaction to the release of his book over there and the learned debates that have been organized and published on the Internet that are driving book sales to record levels for a physics book. 

The ironic thing about all this, however, is that in spite of all the publicity, the brouhaha created by the book is obscuring its main message, which is that string theory is no longer “the only game in town.”  Sure string theory is soaking up the academic/intellectual/economic resources of the world disproportionately, and all the while it is clear to most that it is untestable and non-predictive, which is so exasperating to those who are anxious to get on with “real” science, but to scientists dedicated to its research in perpetuity, those are fighting words, automatically garnering the attention of the media.

Consequently, the more positive message of the book, that now there are viable alternatives to string theory, is almost completely ignored in the publicity, much to the chagrin, I’m sure, of the book’s author.  Unfortunately, what people are missing in the book is a genuine articulation of the motivation for expanding theoretical physics research, in light of recent advances of loop quantum gravity, and similar background-independent quantum gravity theories, which transform them into what is known as preon theories. 

Historically, this class of physical theory has been regarded as a “particle” theory, in which the supposed elementary entities of the standard model (SM) are theorized to actually consist of fundamental entities called “preons,” which is short for pre-fermions. However, though this approach has been around longer than string theory, it’s never been widely persued in the LST community due to some serious problems, which string theory, rather dramatically, promised to solve long ago. 

Nevertheless, “the big bet” of string theory, as Smolin characterizes it, is that there are more than three dimensions of physical magnitude, and, for every particle of the SM, there is a corresponding “sparticle” due to supersymmetry.  After many decades of reseach and billions of dollars spent, no sparticle has ever been detected, and the only reason to suspect that the compactified “extra” dimensions actually exist is that string theory needs them to exist.  Smolin’s point is that, if you are foolish enough to take the bet, don’t quit your day job.

Obviously, making such a point so emphatically, in a publication to the general public, is going to raise the hackles of the string theory establishment, and it has. However, Smolin’s point is that the “core” idea of string theory, which he identifies as the dual nature of fundamental one-dimensional objects, understood as the zero-dimensional entities of gauge theory, wherein these 1D objects theoretically provide for both gravity and the particles of matter, as described in the SM, is the gold in string theory that is hidden in the ore of extra dimensions and supersymmetry. 

Yet, Smolin exclaims that trying to extract the gold from extra dimensions and supersymmetry may be terribly misguided, since, in all likelihood, these may not even exist, but in the meantime, a new gold-bearing ore has been discovered in something thought of as twisted “ribbons” of spacetime, braided to form 3D spacetime triplets, with no need of supersymmetric sparticles, or extra dimensions.  Of course, in the RST science of the LRC, the universe consists of nothing but motion and there is no fabric of spacetime from which to extract ribbons of the fabric and braid them into the entities of the SM, so this approach is just as untenable as string theory’s string objects approach is.

However, this doesn’t mean that the work of Smolin, and that of his collaborators, especially Sundance Bilson-Thompson, can’t be helpful to us in our efforts to combine the discrete units of the RST’s space/time progression, or the universal unit motion, which we assume constitutes the foundation of the theoretical universe of motion, into the observed entities of the SM.  As readers of this blog know, it turns out that the initial motion of the universe of motion forms point-like entities on either side of unity, when periodic “direction” reversals occur in the space, or the time, aspect of the universal space/time progression.  These entities may therefore be regarded as a new concept of preons, and the RST theory that incorporates them may be considered a new preon theory.

When instances of these two entities combine, which the nature of the progression makes possible, the combination forms a one-dimensional entity corresponding to the magnitude interval between -1 and +1, a two unit interval numerically expressed, in the RSM, as

1|2, 1|1, 2|1,

which corresponds to the three integers on the number line, 

-1, 0, +1,

a one-dimensional sequence of zero-dimensional numbers, where the ‘|’ symbol indicates the operational interpretation of the rational number, rather than the quantitative interpretation, indicated by the usual division symbol ‘/’. However, when the two physical units of this displaced unit motion combine (the SUDR & TUDR), the combination (SUDR|TUDR) contains four total units of space and time progression, since both “directions” of the reversing aspects in each oscillating entity must be accounted for.  Thus, the actual equation of the combination is

1|2 + 1|1 + 2|1 = 4|4,

where the space (time) displacement is one unit in the two opposite outward “directions,” contained in the two outside terms, and also where the space (time) displacement is zero units in the single inward “direction” contained in the middle term of the equation. Graphically, this initial combo, the S|T combo, can be illustrated as in figure 1 below.

Figure 1.  The SUDR|TUDR (S|T) Combo as a Space|Time “Ribbon”

While the graphic illustration of the S|T combo necessarily draws it in space, the physical magnitudes that constitute it are created by two scalar expansions/contractions of space and time, which have no spatial separation between them, but only numerical separation corresponding to an ordered sequence of space|time magnitude.  Thus, the “ribbon” of space|time in figure 1 is a representation of an oscillating point, which alternately expands to a unit sphere and contracts to a point, at the speed of light in both space and time, from the perspective of the unit datum.  The colors indicate the “direction” of the space|time displacements with respect to the spatial location: Hence, the green dot in the middle is “higher” than the red dot, just as the frequency of the color green is “higher” than the frequency of the color red, but “lower” than the frequency of the color blue of the right-hand side.

If the magnitude of the space displacement is equal to the magnitude of the time displacement, the color green also represents this balance, but, if one is greater than the other, we can represent the resulting imbalance qualitatively by changing the color of the middle dot accordingly.  This also can be expressed quantitatively with the S|T equation.  For example, the equation of an S|T with two SUDRs and one TUDR would be

S|T = 2|4 + 2|1 + 2|1 = 6|6,

and its graphic “ribbon” would be drawn as

indicating that the unbalance is in the red “direction” of more space displacement than time displacement.  Of course, the equation of the inverse of this imbalance is

S|T = 1|2 + 1|2 + 4|2 = 6|6,

and its corresponding graphic “ribbon” would be drawn as

indicating that the unbalance is in the blue “direction” of more time displacement than space displacement. It is important to note that, as can be seen from the respective equations, a given imbalance in space|time displacement of the two outward terms is always reflected in a corresponding inward space, or inward time, displacement in the middle term of the equation.

Now that we have space|time “ribbons,” the next question is, “Can we “braid” them, in a way that corresponds to Bilson-Thompson’s toy model, to come up with the entites of the SM in our RST theory?”  Of course, our “ribbons” of space|time are not physical lengths, in the sense of one-dimensional spatial objects, so the idea of interweaving them in a physical coordinate space is out of the question.  However, they do represent definite points in coordiante space and time, and, just as the most stable configuration for three adjacent points in coordinate space is a two-dimensional triangle, so too the representation of three S|T combos would be a two-dimensional triangle, where the space displacements of each S|T are combined with the time displacements of an adjacent S|T, forming a very stable combination.  The graphic illustration of this combination, where all three entities are instances of the green (balanced) S|T, is shown below in figure 2.

Figure 2.  Combination of Three Green S|T Combos.

Again, while it’s necessary to draw these space|time relationships in space, to illustrate them, the graphic is only representing numerical magnitudes.  The actual physical configuration would be that of three adjacent points, periodically expanding to unit spheres, and contracting to points.  Questions regarding why, how, and if the inherent space|time expansions and contractions of these oscillations (six of them all together) might be in or out of phase to some degree, will be deferred for now.  The important thing at the moment is the recognition that the positive/negative attraction at the vertices is the most stable combination involving the minimum number of elements forming a stable combination.

However, the most relevant observation for our purpose is that all the possible space|time perturbations of this combination of three S|T units form a group under addition.  The members of this group, following the same group of triplets in Bilson-Thompson’s braids, correspond to the fermions of the SM.  Figure 3 below depicts the members of the group graphically:

Figure 3.  The Triplet Group of S|T Combinations 

The number three in the parentheses of the quarks indicates the three so-called color charges, representing the three possible configurations of the constituent S|T units of the combo; that is, if we label the S|T units so that the left member is A, the right is B, and the bottom is C, then the three possible perturbations of the down quark, for example, with two green and one red S|T, are

Just as in Bilson-Thompson’s braids, we have two symmetries here.  The chiral symmetry of left and right handedness, and the polar symmetry of positive and negative poles, for all entities other than the neutrino and antineutrino, which are the “identity” entity of the group. Again, however, unlike in the case of the braids, we can’t properly employ the concept of spatial orientation to depict these symmetries through rotation, because the symmetry of space|time magnitudes is not the same.  Hence, the left, right, handedness is depicted by the red-blue order of the vertices, indicating a change of perspective from the two sides of unity. 

On this basis, red on the left of blue indicates the material sector, with s/t dimensions of velocity, where we reside, and the blue on the left of red indicates the inverse, or cosmic, sector, with t/s dimensions of inverse velocity, or energy, from our perspective, although the usual convention of signs is reversed here, to permit easier comparison with Bilson-Thompson’s braids. However, it should be stressed that this is a preliminary version of the parallels and is subject to later revision, but the fact that this sense of symmetry just happens to coincide with a 180 degree rotation of the triangle is just that, coincidental, I think.

The braids that constitute the bosons of the SM can also be depicted in the S|T combinations, not as the so-called “trivial braids” of ribbons, of course, but as something we might call “trivial combinations” of S|T units. These would be combinations where each SUDR and TUDR of the combined S|T units are connected to every other in the combo, as shown in figure 4 below:
 

Figure 4. S|T Combos Corresponding to the Bosons of the Standard Model 

As explained previously above, the color of the center dots represents the space|time “direction” of the inward scalar motion of a given S|T unit. Thus, blue in the center indicates motion magnitude in the inward time “direction,” while red in the center indicates magnitude in the inward space “direction.”  In the case of the Z⁰ boson, the magenta color in the center represents a combination of the red and blue units, which is a weird combination of both inward time and inward space scalar motion , but this is all so new that we will have to worry about that later on.

There is much more work to be done on these combinations, and I’m sure changes will be necessary, but nevertheless it’s an exciting breakthrough for us, showing significant contact with the LST theories at last.   

As I pointed out, in the previous post below, Lee Smolin explains, in his newest book The Trouble with Physics, that there are four distinct advantages of the string theory approach over the particle theory approach to physics, which make it extremely compelling to many physicists. These are its ability to “automatically” provide solutions that the particle approach cannot provide:

1. String theory gives us an automatic unification of all the elementary particles, and it also unified the forces with one another. All come from vibrations of one fundamental kind of object.
2. String theory automatically gives us gauge fields, which are responsible for electromagnetism and the nuclear forces. These naturally arise from the vibrations of open strings.
3. String theory automatically gives us gravitons, which come from vibrations of closed strings. As a consequence, we get, for free, an automatic unification of gravity with the other forces.
4. Supersymmetric string theory gives us a unification of bosons and fermions, which are both just oscillations of strings, thus unifying all the forces with all the particles.

However, the key to all this “magic” is that the string concept of a one-dimensional entity has two ends (points) and a middle, which also happens to be the distinguishing feature of the LRC’s SUDR|TUDR combo. In the case of the LST community’s string theory version, it all started with Grebiele Veneziano’s discovery of the remarkable formula that explains a pattern in the data of strongly interacting particles, which, when interpreted physically, has properties analogous to rubber bands. When the entities of the formula gain energy, they stretch, and, when they release energy, they contract, just like rubber bands do. Smolin explains:

Veneziano’s formula thus was a doorway to a world in which the strongly interacting particles were all rubber bands, vibrating as they traveled, colliding with one another and exchanging energy. The various states of vibration would correspond to the various kinds of particles produced in the proton smashing experiments.

Of course, the interest in them at the time is to be understood in the context of the leading edge attempts to explain the strong force of the atomic nucleus, and several theoretical physicists, including Leonard Susskind, came up with the same idea about the same time. However, while the standard model’s QCD theory eventually met the challenges of explaining the strong nuclear force more successfully, it soon became apparent that the idea of using this one-dimensional entity, as a fundamental element, replacing the concept of a point particle altogether, was an even bigger and more exciting prospect than explaining the strong interactions. Hence, string theory, as the fundamental theory of physics, increasingly gained momentum, after a somewhat shaky beginning, as a failed theory of strong interactions.

The most important development in the early history of string theory, the one that really prepared it as a candidate for a fundamental theory, was Pierre Ramond’s modification that made it a discovery of supersymmetry. In this form, bosons and fermions were mixed together, and the twenty-six dimensions, which the theory required originally, were reduced to ten dimensions. This was huge, and when it was discovered that these “strings,” as they came to be called, could interact in ways consistent with quantum mechanics and special relativity, and that the bosons of the theory not only contained photons, but also the theoretical quantum of gravity as well, the so-called gravitons, things really started to heat up. Smolin writes:

The fact that string theory contained gauge bosons and gravitons changed everything. Scherk and Schwarz proposed immediately that string theory, rather than being a theory of the strong interactions, was instead the fundamental theory - the theory that unifies gravity with other forces.

The way this works is engaging, to say the least. The gravitons arise from the vibrations of closed strings; that is, the photon bosons are the vibrations of open strings, but when the two ends of a vibrating string are joined together, to form a closed loop, the vibrations correspond to graviton bosons. In the case of open strings, the boson vibrations are connecting the two opposite ends of the string, just as the photon “connects” the electron and positron. Smolin explains:

The ends of the open string can be seen as charged particles. For example, one end could be a negatively charged particle, such as an electron; the other would then be its antiparticle, the positron, which is positively charged. The massless vibration of the string between them describes the photon that carries the electrical force between the particle and the antiparticle. Thus you get particles and forces alike from the open strings, and if the theory is designed cleverly enough, it can produce all the forces and all the particles of the standard model.

Whoa, no wonder this gets people excited, but, then, again, on top of all this, when the ends of the string are joined together, the theoretical graviton boson is born. Again, Smolin explains:

If there are only open strings, there is no graviton, so it seems as though gravity is left out. But it turns out that you must include the closed strings. The reason is that nature produces collisions between particles and antiparticles. They annihilate, creating a photon. From the string point of view, this is described by the two ends of the string coming together and joining. The ends go away and you’re left with a closed loop.

Here, Smolin concentrates on the inflexibility of the theory’s consequences, something that is most desirable in a physical theory, but he doesn’t explain very well why “gravitons come only from vibrations of closed strings.” This is unfortunate, but probably due to the fact that to Smolin, and to many, many, others, the most significant part of the development of string theory is that it must include gravitons, while particle theory cannot be made to include gravity, in spite of all efforts to find a way to do so. He writes:

…the difference between gravity and the other forces is naturally explained, in terms of the difference between open and closed strings. For the first time, gravity plays a central role in the unification of the forces.
Isn’t this beautiful? The conclusion of gravity is so compelling that a reasonable and intelligent person might easily come to believe in the theory based on this alone, whether or not there was any detailed experimental evidence for it. Especially if that person has been searching for years for a way to unify the forces, and everything else has failed.

But the beauty and strength of the “unification of the forces” in this concept are not to be understood simply in the straightforward logic of open strings becoming closed strings, by joining the two ends together, a mechanical action that all of us are familiar with, in one way or another, but rather in the deeper significance of the implication that there exists an underlying law of nature that requires the process to take place. “But what gives rise to it?” Smolin asks, “Is there a law that requires the ends of strings to meet and join?” His answer is positive. It describes a deeper meaning, or a deeper unification of fundamental concepts, if you will.

He writes, “Herein lies one of the most beautiful features of the theory, a kind of unification of motion and forces.” Now, readers of this blog know that one of the primary tenets of Larson’s RST is that force cannot exist as an autonomous entity, apart from motion. Force is necessarily a property of motion, by definition, as Larson explains at length in his Neglected Facts of Science, but the LST community, following Newton’s program of research, interpreted as a dictum to focus on the forces of interaction, has treated them as something fundamental, existing autonomously and independently of motion, but here, Smolin is delighted that string theory has now lead to “a kind of unification of motion and forces.” Perhaps we would prefer to characterize it as a “reunification of motion and forces.” Unfortunately, however, that point would be lost on those not familiar with Larson’s works. Smolin continues:

In most theories [i.e. LST theories], particle motion and the fundamental forces are two separate things. The law of motion tells how the particle moves in the absence of external forces. Logically, there is no connection between that law and the laws that govern the forces.
In string theory, the situation is very different. The law of motion dictates the laws of the forces. This is because all forces in string theory have the same simple origin - they come from the breaking and joining of strings. Once you describe how strings move freely, all you have to do to add forces is add the possibility that a string can break into two strings. By reversing the process in time, you can rejoin two strings into a single string. The law for breaking and joining turns out to be strongly prescribed, to be consistent with special relativity and quantum theory. Force and motion are unified in a way that would have been impossible in a theory of particles as points.

Well, that might be so, but then, as we have been discussing below, and also in the New Math blog, the whole idea of points in the continuum, and the assumption of a correspondence of these with real numbers, has been confused from the beginning. Given the assumptions of the RST, the concept of a point charge changes, from the enigmatic one of an object with no spatial extent, to one of a spherical vibration, or change, a displacement, in the rates of an eternal space/time progression, at a given location in the progression. One of these, formed by the “direction” reversals in the space aspect of the universal progression, at that location, is the dual of the other, formed by the “direction” reversals in the time aspect of the universal progression.

When these two join together, and the probability that they will do so exists, because of the progression, the combination forms a one-dimensional object, consisting of two “points” separated by a “length” of magnitude. We refer to the space vibration as a SUDR (space unit displacement ratio) and the time vibration as a TUDR (time unit displacement ratio). When the two are joined together, they become the SUDR|TUDR combo (or S|T), which is analogous to the string vibration of string theory, in that it has two end terms and a middle term, but these are space/time magnitudes of motion, not an undulating object. Hence, while there is a useful comparison of the two concepts, they are vastly different.

Nevertheless, it is clear that what the three or four decades of research has revealed about string theory, that it “automatically” provides tremendous theoretical advantages in terms of the standard model and quantum gravity theory, is also an indication of the potential of the new system’s theory. Obviously, we can describe forces in terms of the joining and separating of the S|T entities in a similar manner, as the string theorists have done with their strings. However, we can expect much more than that as well, because, with the new math forming the basis of analysis, in the context of the Chart of Motion, the inductive aspect that string theory so conspicuously lacks becomes a big part of the new theory.

Just as the algebra of the new math is necessarily ordered, distributive, commutative and associative, in all dimensions, the rules of combining the discrete units of motion in the new physics follows the same straightforward, well understood, principles of discrete numbers.  This means that the only thing needed to form the initial units of the system is a universal space/time progression, which, while we must assume it exists, and that too in a suitable form of a workable hypothesis, it is clearly observable for all to see: We know that both time and space are inexorably marching forward, do we not? We also observe no more than three dimensions of magnitude, do we not?  Finally, while some may think that a “smooth” continuum of discrete points is possible, it’s clear that a non-divisible length cannot be imagined anymore than can a non-extendable length. Is this not so, surely?

Therefore, we conclude that string theory is giving us a valuable glimpse of the way forward.  However, if we are to understand what it is telling us, we need to go back and correct the misguided concepts of the past.  We need to recognize that ad hoc inventions, while they are certainly useful in the short run, will always come back to bite us in the long run.

Not only does Lee Smolin write clearly and forcefully about the The Trouble With Physics, which he shows is ultimately about the age-old problem of reconciling the discrete (quantum mechanics) and continuum (general relativity) theories of modern physics, in light of the fact that nature obviously has no problem with the task, but, in the process, he manages to clearly explain why string theory has been, and still is, so compelling to theoretical physicists.

He begins his exposition, on page 103 of his book, by establishing the context in which string theory arose: He asserts that the idea that elementary particles might be vibrations of strings, instead of pointlike particles, is possibly another one of those rare insights, where the discovery of a missing element to a puzzle suddenly clarifies an otherwise perplexing physical mystery. He reminds us that it was the discovery of a hidden, or missing, motion that first led ancient astronomers to understand the motion of the planets. He writes:

The discovery of this third motion - the missing element - must have been one of the earliest triumphs of abstract thinking. We see two objects, the sun and the moon. Each has a period, known from earliest times. It took an act of imagination to see that something else was moving as well: the paths themselves. This was a profound step, because it required realizing that behind the motion you observe there are other motions whose existence can only be deduced. Just a few times since has science progressed by discovery of such a missing element.

The idea of a point particle in physics is not just a puzzle, but a puzzle wrapped in mystery inside an enigma, as they say. It is the mother of all physical mysteries. Interestingly enough, though, its definition is the physical counterpart to the definition of an irrational number, because, like the definition of an irrational number by mathematicians, the definition of a physical point by physicists is actually an ad hoc definition, based on the so-called Archimedean Axiom, where it is assumed that an infinitesimal quantity cannot be defined, indeed, must not be defined; that is, a quantity is an infinitesimal quantity, if it exists, yet, at the same time, it must be smaller than any division of 1 that can be imagined, and, since this is impossible, such a quantity cannot exist. In other words, for physicists, there is a need to limit how far the number 1, or the continuum, can be subdivided.

Of course, there is no limit to numbers, or limit to the elements of the set of natural numbers, to be a little more precise. Which is to say that there is no limit to n/1, and if there is no limit to n/1, it follows that there also is no limit to 1/n. However, this presents a problem in physics, when elementary particles, such as electrons, are regarded as point particles that cannot be subdivided. In his Lectures on Physics, Richard Feynman explains the problem as follows:

…the limit at r = ∞ gives no difficulty. But for a point charge we are supposed to integrate down to r = 0, which gives an infinite integral. [The equations say] that there is an infinite amount of energy in the field of a point charge, although we began with the idea that there was energy only between points charges…one way out of the difficulty would be to say that elementary charges, such as an electron, are not points, but are really small distributions of charge. Alternatively, we could say that there is something wrong in our theory of electricity at very small distances, or with the idea of the local conservation of energy. There are difficulties with either point of view. These difficulties have never been overcome; they exist to this day.

Thus, in physics, just as in mathematics, either an artificial cutoff of the infinitesimal is devised, or else the theories, whether classical or quantum mechanical, will fail due to fundamental incompatibilities of the momentum of charged particles and the energy of their associated fields. The only way out in physics is a solution that corresponds to the solution of the mathematical problem of the incommensurabilities in the rational and irrational numbers, the invention of an ad hoc concept. In mathematics, the ad hoc invention used is the invention of real numbers (see latest post in New Math).

In physics, the ad hoc invention used is the invention of renormalization. Feynman, one of the inventors of renormalization, who received a share of the Nobel Prize for it, concluded in his 1965 Nobel Lecture:

I don’t think we have a completely satisfactory relativistic quantum-mechanical model, even one that doesn’t agree with nature, but, at least, agrees with the logic that the sum of probability of all alternatives has to be 100%. Therefore, I think that the renormalization theory is simply a way to sweep the difficulties of the divergences of electrodynamics under the rug. I am, of course, not sure of that.

However, since then, LST physicists have become quite accustomed to Feynman’s renormalization theory and simply regard it as a useful technique in getting at the final solution. As Frank Wilczek explained in his 2005 Nobel Lecture:

The central observation that is exploited in renormalization theory is that, although interactions with high-energy virtual particles appear to produce divergent corrections, they do so in a very structured way. That is, the same corrections appear over and over again in the calculations of many different physical processes. For example, in quantum electrodynamics (QED), exactly two independent divergent expressions appear, one of which occurs when we calculate the correction to the mass of the electron, the other of which occurs when we calculate the correction to its charge. To make the calculation mathematically well defined, we must artificially exclude the highest energy modes, or dampen their interactions, a procedure called applying a cutoff, or regularization. In the end we want to remove the cutoff, but at intermediate stages we need to leave it in, so as to have well defined (finite) mathematical expressions. If we are willing to take the mass and charge of the electron from experiment, we can identify the formal expressions for these quantities, including the potentially divergent corrections, with their measured values. Having made this identification, we can remove the cutoff. We thereby obtain well defined answers, in terms of the measured mass and charge, for everything else of interest in QED.

But then that’s just another way of stating that the fundamental theoretical problem of how the energy of a particle’s electric field can be temporarily treated as finite, for the purposes of calculations (since it obviously is finite), until the calculations that depend on it are complete, and then we can safely ignore the theoretical implications of this procedure, because the calculations work exceedingly well. Clearly, Feynman’s characterization of this as “sweeping the difficulties…under the rug,” is more honest.

The truth is that the theoretical concept of elementary particles, as point particles, remains problematic. The problems are of the most fundamental nature too, regardless of the much touted success of the standard model of particle physics. The idea that the vibration of strings is a concept that “provides a plausible answer to several big problems of physics,” to put it in Smolin’s words, is a very powerful one indeed, for those who are keen to discover the true physical structure of nature and are not satisfied with just a practical approach to calculations. Some physicists are clearly not willing to let “the ends justify the means.”

Of course, the bigger picture is that while string theory “provides a plausible answer to several big problems of physics,” it also introduces a whole range of new problems at the same time, and, what is worse, these new problems are not as easily “swept under the rug,” as the original problems of infinities are. Nevertheless, and not withstanding this, the genie is now out of the bottle, and there is no going back. It seems that either there has to be an unknown way to solve the problems of string theory, so that we can return to science as we know it, where we can have confidence that the structure of the physical universe can be determined, as a consequence of the fundamental nature of reality, or else we have to give up the idea of science, as we know it, and accept the idea that the physical structure of nature can be anything at all, that there is an infinite number of possibilities, all of which are consistent with nature, but only one of which is consistent with the instance of nature that we observe, with no apparent way of ever discovering which possibility that might be.

Smolin clearly explains how each of these string theory possibilities corresponds to one of the ways in which 9 or 10 spatial dimensions can be compactified to fit in the 3 dimensions of space that we observe. Each way this compactification of “extra” dimensions proceeds determines a unique set of coupling constants (particle masses, charges, etc.) that characterizes the results. This lack of uniqueness is a major problem, but is characteristic of all approaches that incorporate extra dimensions, ever since Kaluza and Klein first tried it. In string theory, these extra dimensions are necessary in order to describe the several ways that strings can move, and this change of position (M₂) motion (see our Chart of Motion), requires extra constants, hundreds of them. Smolin explains:

[The use of extra dimensions] is how string theory solves the basic dilemma facing attempts to unify physics[, but even] if everything comes from a basic principle [like this], you [still] have to explain how the variety of particles and forces arises. In the simplest possibility, where space has nine dimensions, string theory is very simple; all the particles of the same kind are identical. But [in the case] when the strings are allowed to move in the complicated geometry of the six extra dimensions, there arise lots of different kinds of particles, associated with different ways to move and vibrate in each of the extra dimensions.

“So,” Smolin continues, “we get a natural explanation for the apparent differences among the particles…but there is a cost.” The cost is that the theory turns out not to be a single unified theory, but many different theories. So the challenge of unifying the discrete and continuous theories, which string theory promised to meet in the beginning, actually ended up severely exasperated by the concept of strings moving and vibrating in nine dimensions of space. But even then, Smolin admits, the theory would still be useful, if it had led to predictions of the twenty-some odd standard model constants:

…this scheme might have been compelling if it had led to unique predictions of the standard model. If by translating the standard model’s constants into constants denoting the geometry of the extra dimensions. we had found out something new about the standard model’s constants, and if these findings had agreed with nature, that would have constituted strong evidence that string theory must be true.

But, of course, “this is not what happened,” he writes. The given masses, charges, etc, constituting the constants of the standard model particles and forces were just replaced with the different geometries of string theory:

The constants that could be freely valued in the standard model were translated into geometries that could be freely valued in string theory. Nothing was constrained or reduced. And because there were a huge number of choices for the geometry of the extra dimensions, the number of free constants went up, not down.

This was bad enough, but it is not the end of the bad news. String theory equations are not able to produce the exact combinations of particles and forces seen in nature, but just the general features of fermions and gauge fields, and what’s even worse, they predict extra particles and forces of supersymmetry, something that, in spite of years of effort, has never been observed in nature.

In fact, the idea of supersymmetry is inextricably connected with string theory today. It is an extension of the standard model, where the names of sparticles are modified names of particles, etc., doubling the number of entities and greatly expanding the forces in the model. What binds string theory to the supersymmetric version of the standard model is a particular way to compactify the extra dimensions of string theory, called a Calabi-Yau space, which enables one to correlate the constants of the supersymmetric standard model to the geometries of the extra dimensions, compactified as Calabi-Yau spaces. This would have been a good thing, except for the fact that there are practically an infinite number of ways to construct these spaces. Smolin explains:

This was great progress. But there was an equally great problem. Had there been only one Calabi-Yau space, with fixed constants, we would have had the unique unified theory we yearned for. Unfortunately, there turned out to be many Calabi-Yau spaces. No one knew how many, but Yau himself was quoted as saying that there were at least a hundred thousand. Each of these spaces gave rise to a different version of particle physics. And each space came with a list of free constants governing its size and shape. So there was no uniqueness, no new predictions, and nothing was explained.

Soon, though, it was shown that there were many more ways to construct Calabi-Yau spaces, as many as 10⁵⁰⁰ ways to do it! Certainly, this fact was very discouraging, but it was later discovered, to the absolute delight of string theorists, that there were only five string theories that needed to be defined in these spaces, and, moreover, that extending the extra space dimensions from 6 to 7, enabled physicists to unify these five theories, through a principle of duality, even though the unified string theory, dubbed M-theory, has not been successfully formulated yet, and, if it exists at all, it is only dimly understood at this point in time.

Nevertheless, the discovery of Calabi-Yau space and the duality of the five string theories, with its promise of a unified string theory, in an eleven-dimensional version of some unspecified theory of strings, represents a great deal of progress, in the minds of many, that seems to clearly indicate that the whole enterprise will eventually lead to a theoretical unification of the discrete and continuous magnitudes observed in nature. However, to many others, including Smolin, replacing the paradoxical notion of point particles, given their inexplicable properties of mass and charge, with vibrating strings, given their inexplicable properties of extra dimensions, only shifts the mystery of the physical structure of nature from one challenge to another, from how a point of space can have the necessary properties of elementary particles, such as mass and charge, to how vibrating strings can have the necessary properties of motion in ten or eleven dimensions.

The reason for this is found in the definition of motion. Without the different definitions of motion in the chart of motion (CM), the only known definition is change of position (M₂) motion, but this type of motion requires a fixed background of point locations to be defined, or a given geometry, which would constitute the one background of M-theory that is suitable for all five string theories that are supposed to be part of it, with all their different geometries that can be compactified in one to ten dimensions of space. As Smolin explains it:

[All these geometries] provide backgrounds for strings and branes to move. But if they are part of one unified theory, that theory cannot be built on any one background, because it must encompass all backgrounds.

Thus, we have come full circle. A concept of magnitudes of motion, in the form of vibrating strings, was invented to overcome the problematic concept of point magnitudes of force, a property of motion. This seems to work much better, in some ways, but the necessary magnitudes of motion can only be defined in terms of different sets of points that satisfy the postulates of a given geometry, requiring one to ten dimensions. So, the fundamental question has evolved from how a single point of space can have the necessary physical properties to explain nature, to how a single set of points can have the necessary properties to explain nature. As Smolin writes:

The key problem in M-theory, then, is to make a formulation of it that is consistent with quantum [or discrete] theory and background independence. This is an important issue, perhaps the most important open question in string theory. Unfortunately, not much progress has been made on it. There have been some fascinating hints, but we still do not know what M-theory is, or whether there is any theory deserving of the name…

Unfortunately, M-theory remains a tantalizing conjecture. It’s tempting to believe it. At the same time, in the absence of a real formulation, it is not really a theory - it is a conjecture about a theory we would love to believe in.

The most fascinating aspect of this modern drama is the light it throws on the subject of fundamental concepts of magnitudes of geometry and magnitudes of motion.  In the CM, we find that magnitudes of space are actually manifestations of past motion, but this motion can take the form of three types of motion, M₂, M₃, and M₄ motion, and only M₂ motion requires a fixed background, or a set of points, satisfying the postulates of Euclidean geometry. The M₄ motion of the CM actually defines these points, and it does so in a completely background free manner, which, of course, it would have to do, because the space of geometry cannot be defined in terms of itself, which, in the final analysis, is the tautological concept that string theory leads to.

Nevertheless, the LST community’s fascination with string theory’s ability to automatically provide

1. Unification of all elementary particles
2. Gauge fields of electromagnitism and nuclear forces
3. Gravitons and unifications of gravity with other forces
4. Unification of particles and forces

will not permit them to abandon it anytime soon, even with all the trouble it’s revealing with the attempt of theoretical physicists to cope with the incompatibility of discrete and continuous aspects of nature.  However, with the advantage of the RST, and the light it sheds on the fundamental nature of space and time, we can see that the effort to formulate a physical theory on the basis of vibration, or motion, is not misguided, just naive and uninformed.

In developing the new physics at the LRC, we first assume that the fundamental reality of the physical universe is nothing but motion, or the reciprocal relation between space and time. Then we identify the kinds of motion that exist in the universe of motion. Happily, we discover that these possible types of motion are determined mathematically, simply from the properties of numbers, where the three dimensions of observed magnitudes (four, counting zero) are shown to emerge from the dimensions of numbers 1 through 4.

Because we further assume that the constituent motions of the universe exist in three dimensions and in discrete units, a significant insight into the nature of higher dimensional numbers permits us to reconcile them with the three (four) dimensions of physical magnitudes. This new insight, based on the Bott periodicity theorem, shows how higher dimensions of numbers are actually related to greater scales of three (four) dimensions of magnitude, not extra geometric dimensions.  This is a great help in dealing with higher dimensions.

Before M₃ and M₄ motion were recognized, as legitimate forms of motion, separate and distinct from M₂ motion (see: the chart of motion discussions in previous posts below and in the New Math blog), it was not possible to correctly understand the simple mathematics of higher dimensional magnitudes. As a result, the erroneous concept of higher dimensions of space was employed in the LST development to develop higher dimensions of magnitude, in abstract “spaces” of logic, using complex numbers and the concepts of rotation, spin, and isospin.

In retrospect, however, we can now see that this approach amounts to an ingenious, if unnatural, adaptation of the principles of M₂ motion to the proper domains of M₃ and M₄ motion. Rotational motion is conspicuous by its absence in the Chart of Motion (CM), where all motion has only two modes, bounded or unbounded. Unbounded motion is unidirectional, while bounded motion involves a change in “direction.” The “direction” of motion is defined in the CM by the “bidirectional” property of operationally interpreted (OI), reciprocal, numbers: The reciprocal OI number, of a given OI number, is a value on the other side of unity, in the opposite “direction” of increasing value, with respect to one (1/1).

However, in the M₂, change of position, motion, a change of direction is possible that is analogous to the change in “direction” of the bounded motion of the CM, but this M₂ change of direction motion is not a change of “direction,” because it is not a change relative to unit, or 1/1, magnitude, but rather it is a change relative to a zero, or initial, position. This is a significant distinction, permitting us to clearly see that M₂ vibrational motion is “bounded” only in one dimension, and, as a result, the rotational version of it can be summed to two and three dimensions, as separate instances added together.

Thus, one complex number can represent one-dimensional magnitude, as in U(1) Lie groups, and two complex numbers can represent two-dimensional magnitude, as in SU(2) Lie groups. With some difficulty, this adaptation can be further extended to three-dimensional magnitudes as well, as in the SU(3) Lie groups.

Yet, the fact that these three Lie groups provide successful LST physical theories for the electroweak force (U(1) & SU(2)), and for the strong nuclear force (SU(3)), based on concepts of rotation and spin and isospin, doesn’t mean that RST physical theory can’t be even more successful, without resorting to the unnatural concept of rotation and complex numbers. Certainly, it remains to be seen, but the implication of the CM is that M₂ motion has been inappropriately applied, in LST physics, to develop concepts of n-dimensional magnitudes that are more appropriately understood in terms of M₃ and/or M₄ motion.

Without the benefit of the Reciprocal System of Mathematics (RSM), Larson too turned to the concept of rotational motion, in a similar attempt to adapt it to the development of n-dimensional scalar magnitudes, albeit without incorporating the use of complex numbers and Lie groups. He identified four types of motion in the universe of motion:

1. Translational
   
2. Vibrational
3. Rotational
4. Rotational vibration

Although these four types of motion clearly apply to M₂, or vectorial, motion, Larson’s idea was to apply them to concepts of scalar motion as well.  This enabled him to add a “2D rotation” to a 1D vibration, for instance, rotating the vibration in two dimensions simultaneously, which he was careful to insist is not two, one-dimensional, rotations, but rather one, two-dimensional, rotation; that is to say, unlike the initial 1D vibration, the rotation of the vibration is a two-dimensional magnitude. Subsequently, Larson obtained a three-dimensional magnitude by applying a second, optional, rotation of the rotating vibration.  This approach too is very successful, as far as it has been applied, to developing physical theory.

However, again, it is clear to see that the use of rotation, as an M₂ adaptation for obtaining two and three-dimensional magnitudes of motion, while it may work to some degree, is, in all actuality, an inappropriate application of change of position motion, which is strictly confined to the domain of one-dimensional magnitudes of motion.  However, the fact that two and three-dimensional magnitudes of geometric space can be defined, using separate instances of M₂ motion, doesn’t mean that it should be adapted to define the true n-dimensional magnitudes of motion.  At least that’s the implication of the CM.

Probably the most convincing evidence of the unnatural application of the rotation concept is the enigma of the quantum mechanical “spin” concept, and its indispensable role in quantum theory.  Today, this concept plays a central and crucial role in physical theory, but after almost three quarters of a century, since its appearance, it is just as enigmatic as ever. Nobody can explain it, even though its effects have become more and more important in recent years, as significant and unexpected results have moved “spin studies” to the forefront of high-energy physics.  Indeed, the most recent data from the accelerators clearly indicate that high-energy interactions essentially depend on spin degrees of freedom.

With the advent of “spintronics,” in which electron spin states are exploited to carry information, many people think particle spin must be clearly understood, and it’s true that the so-called “spin space,” as a quantum mechanical degree of freedom, is very much a real space, but, yet, the rotation of the spin axis of a spin-1/2 particle doesn’t follow the ordinary rules of rotation in three geometric dimensions. Instead, these rotations follow the rules of the SU(2) group, which operates with a different sort of space, a space in which a rotation of 360 degrees does not constitute a complete revolution, yet a 180 degree rotation, in an orthogonal dimension, definitely does reverse its spin axis. 

Hence, while there is no doubt that the spins of particles are measurable quantities that play a fundamental and indispensable role in nature, the concept of a particle with no spatial extent, a point particle, can’t be regarded as possessing angular momentum, and the axis about which it spins can’t be regarded as rotating through 720 degrees to return the particle to its original state, but, nevertheless, that’s what we’re forced to believe.  Indeed, LST physicists don’t have a clue as to how to find a way out of this embarrassing predicament.

However, understanding that adapting M₂ motion, via rotation, to generate n-dimensional magnitudes that are described more appropriately by M₃/M₄ motion, is probably not the best way to proceed, in developing physical theory, liberates us to explore the development of a new approach, based on the newly identified types of motion.  Recall that, unlike M₂ motion,  M₃ motion defines magnitudes by a change of interval, not a change of position and that M₄ motion defines magnitudes by a change of scale.  What this means is that M₃ motion expands, or contracts, an interval between two or more points, so that the length of a line is transformed to a point, or the area of a surface is transformed to a line, through expansion/contraction, defining a magnitude of motion.  An example of this type of motion can be found in the transformation of a 2D rectangle into a 1D line, illustrated in figure 1 below:

Figure 1.  M₃ Motion

In the motion of the rectangle in figure 1, the angle at each of the four corners is 90 degrees, or 4 * 90 = 360 degrees total for the rectangle, as a whole.  If each of the four corners were hinged, pressing the top and bottom vertices together would merge all four sides into one line, collapsing the interior angles of the two horizontal vertices from 90 degrees to 0 degrees and expanding the interior angles of the two vertical vertices from 90 degrees to 180 degrees, for a total angle rotation of 360 degrees.  Subsequently, pulling the top and bottom vertices apart, expanding the collapsed line back to its original diamond shape again, rotates the interior angles a total of 360 degrees once more, but in the reverse direction.  Hence, the total angle rotation, which constitutes one cycle of contracting/expanding the diamond, is clearly 720 degrees.

Figure 2.  Equivalent 2D M₃ Motion in 3D M₄ Motion 

In a similar manner, expanding a 0D point, which by definition has no spatial extent, to a 2D circle, the spatial extent of which is measured as 360 degrees of rotation, is equivalent to a rotation from 0 to 360 degrees, while the contraction of a circle to a non-spatial, 0D point, is equivalent to a rotation from 360 to 0 degrees, or a total of 720 degrees rotation in one cycle of expansion/contraction.  In the case of the rectangle, with well-defined vertices, or poles, the inversion of the poles, represented by a 180 degree rotation of the rectangle, is equivalent to “flipping” the spin axis of a corresponding rotation.

Thus, recognizing the existence of M₃ and M₄ motion permits us to make sense of the reality of “spin” magnitudes in a straightforward and unmysterious manner that is clearly physical, yet unrelated to the angular momentum concept of M₂ motion. Since an expanding sphere contains an infinitude of expanding lines and expanding planes, M₃ motion is contained in M₄ motion.  The consequences of this fact have important implications in the development of the new physics.

Last time I explained a little about the standard model of LST physics and how its underlying mathematical structure, in the form of Lie groups, suggests a relationship to our chart of motions. However, at the time I was thinking that since the groups are synonymous with magnitudes of force, the actual correlation ought to be with the three dimensions of magnitude in the chart, not the three bases of motion. In fact, I even tried modifying the chart in the previous post, but ran into trouble with formatting the concepts in an understandable way and decided to leave it be, due to time constraints.

Now that I have had some time to think about it, It seems I can get the idea across better, by separating the chart of motions into three charts, one for each base of motion. The three charts are:

Mechanical Motion (Base 2)

1. (2/2)⁰ N/A
2. (2/2)¹ U(1)
3. (2/2)² SU(2)
4. (2/2)³ SU(3)

Electrical Motion (Base 3)

1. (3/3)⁰ N/A
2. (3/3)¹ N/A
3. (3/3)² SU(2) + U(1)
4. (3/3)³ SU(3) + SU(2) + U(1)

Scalar Motion (Base 4)

1. (4/4)⁰ N/A
2. (4/4)¹ N/A
3. (4/4)² N/A
4. (4/4)³ SU(3) + SU(2) + U(1)

Figure 1. Lie Groups of the Standard Model and the Chart of Motions

To understand how this works, it’s best to start with the mechanical motion of base 2, the normal vectorial motion of objects. There are two modes of all motion, translation and vibration. In mechanical motion, the vibration mode is harmonic; that is, this motion is the oscillation between two positions, like the motion of pendulum in a gravitational field, or the motion of a crystal in an electrical field. In both of these cases, there is a 90 degree angular relationship between the potential and kinetic energy that corresponds to the sine and cosine of the angle between the horizontal and vertical positions of the pendulum. As the angle of a pendulum between the vertical position of the weight at the end of the arm, and the two possible horizontal positions of the weight, varies with time, so does the potential and kinetic energy of the mechanical motion.

When the 90 degree angle of the rotating arm of the pendulum is maximum, the sine of the angle is maximum, but the cosine is minimum and vice versa, which corresponds to the maximum potential energy at the top of the swing, on either side, and the minimum kinetic energy of the pendulum, since it is motionless at that point. However, when the pendulum is passing through the bottom of the swing, the speed is maximum, the kinetic energy is maximum, and the potential energy is zip. The kinetic energy due to the speed of the mass at the end of the arm also equates to a value of angular momentum that depends upon the weight of the mass, as well as its speed.

Of course, in the case of the harmonic motion of the pendulum, the angular momentum is not constant, because as the speed of the mass goes to zero at the top of the swing, so does the kinetic energy and the angular momentum. This simple physical relationship of potential and kinetic energy in the motion of the pendulum is important because it clearly demonstrates the relation between the principle of symmetry and the law of conservation of energy that is central to the mathematical analysis of physical phenomena. The motion of a pendulum (disregarding the effects of energy loss due to friction), in a gravitational field, is symmetrical; that is, it swings as far to one side of the bottom point, as it does to the other side, and in both cases, energy that is conserved as potential energy is transformed into kinetic energy and vice versa.

This would not be the case, if the symmetry of the pendulum’s swing was disturbed some how. Energy would be lost to whatever intervening force “broke” the symmetry of the swing. For instance, if the swinging motion of the pendulum was exploited to perform work like hammering a nail into a block of wood, the resistance of the wood to the nail’s penetration would constitute an intervening force that breaks the symmetry of the motion.

This same principle can be seen in the translational motion of a rotating pendulum; that is, if a pendulum were to be made to rotate in a plane perpendicular to the gravitational field so that its motion was perpetual in one direction of rotation around one position (again disregarding the effects of energy loss due to friction), rather than oscillating between two positions, the previous transformation of potential energy into kinetic energy, and vice versa, associated with the oscillation, would not occur. Thus, one would think that the kinetic energy and the angular momentum would be constant, and it would be, if it were not for the loss of energy due to the breaking of another symmetry.

The symmetry that is broken in the case of rotation is the symmetry of direction; that is, the angle tangent to the rotational path of the weight represents a continuous change in one direction, a continuous change to the “inside’ direction, if you will, and just as before, the breaking of the symmetry is accompanied by the loss of energy. If the lost energy is not replaced, as it is, for instance, in the case of orbiting planets or moons, by the gravity of the mass being orbited, the rotational motion of the system will eventually stop.

This was, of course, the problem facing the Bohr model of the atom, at the turn of the 19th Century, when quantum mechanics was devised to solve the problem of an electron orbiting the nucleus of an atom. There was obviously no way that the energy associated with the gravitational attraction of the mass of the atomic nucleus can supply the lost energy of the orbiting electron, to keep it orbiting perpetually, in the way stars, planets, and moons maintain their orbits.  At the same time, the energy associated with the Coulomb attraction between the nucleus and the electron acts differently than the gravitational attraction, because, unlike accelerating masses, accelerating charges radiate energy.  Thus, the electron could not orbit the nucleus without continually radiating, or losing, energy due to the changing direction of its path in the circular orbit.

At the same time, however, Bohr’s idea was the only game in town that explained the quantization of energy: that only an integral number of wavelengths fit into a circumference of a given radius around an atomic nucleus. This concept of quantization was needed for explaining the recently discovered fact that the energy of electrons, and the radiation from atoms, are quantized and inter-related. Thus, a new mechanics, or concept of motion, was born. A concept that has been mysterious and enigmatic ever since, because, while its foundations are unknown, it works really well for calculating the frequency of atomic spectra and formulating a theoretical foundation of other atomic phenomena.

The key to the new mechanics was ultimately found to lie in the use of complex numbers, where the ad hoc invention of the imaginary number, the square root of -1, fortuitously came to the rescue. However, when most people think of quantum mechanics they think of Schrodinger’s wave equation, which describes the behavior of a charged particle in a field of force. There is the time-dependant equation, used for describing progressive waves, applicable to the motion of free particles, and then there is the time-independent form, used for describing standing waves. The imaginary number i doesn’t appear directly in the time-independent equation, but solutions to this equation come in two forms: the oscillating and the rotational form of the pendulum that we have been discussing.

In other words, the quantum mechanical wave equation relates a periodic mathematical function (wave function) with the associated physical principle of energy conservation and symmetry, in the form of the potential and kinetic energy transformations of the swinging pendulum, to the rotational form of the same conservation principles. Of course, In the case of the oscillating form, the energy to resupply the lost energy is supplied to the mechanical instance of the pendulum system by a gravitational field, when the plane of oscillation is parallel to the force of the field, while in the case of the rotational form, the energy would hopefully come from the electrical field, associated with the Coulomb charges, in the absence of a suitable gravitational effect, at the center of rotation.  Of course, since the orbiting electron would radiate away this energy in time (10^-9 seconds!), this model is unstable.

Nevertheless, it is still the rotational form of the solution to the time-independent equation that is used in quantum mechanics, but only because the concept of the complex number, in describing the motion of the electron rotation, can be exploited to provide the solution to the energy conservation problem.  Mathematically, the two solutions to the equation have opposite signs, one is positive, the oscillation solution, and one is negative, the rotation solution. Since the electron is negative, and is envisioned as rotating around the atomic nucleus, it is the negative, rotation, solution that is used in QM. Specifically, the Schrodinger equation for ψ(x) can be reduced to the form:

d²ψ(x)/dx² +/- k²ψ(x) = 0 ;

a two-dimensional equation, where k is a real number. The form of the solution is determined by the sign of the k² term. If k² is taken to be positive, then the equation has the same form as the two-dimensional harmonic oscillator:

ψ(x) = A cos(kx) + B sin(kx) ;

but, if the k² term is taken to be negative, then the solution has the two-dimensional general form:

ψ(x) = Ae^kx + Be^-kx

which, if written as a complex number, using the polar form of coordinates, instead of rectangular form, where

z = ρ(cosθ + i sinθ),

it can be rewritten, using Euler’s Formula, where

cosθ + i sinθ = e^iθ,

which is a rotation expressed as a complex number.  In other words, the complex number has the form