The Larson Research Center

It is really Heisenberg’s concept of isospin, and Gell-Mann’s concept of hypercharge, that form the basis for the quantum mechanical observables that are the foundations for the quarks and gluons in the standard model (SM), which Gell-Mann was first able to formulate, via the properties of the SU(3) Lie group. These two concepts led to Gell-Mann’s “eightfold way,” which provided the initial breakthrough for classifying the myriad particles observed in high-energy particle interactions. However, the concept of isospin is simply an abstract mathematical space, an “internal symmetry,” with no connection to real physical space, and the fact that it is conserved, and that its use has been so crucial to the success of the SM, greatly astounds many physicists, let alone philosophers, to this day.

Actually, there are two concepts of isospin used in the SM. One is based on SU(2) symmetry, the concept of weak isospin, and one is based on SU(3) symmetry, the concept of strong isospin, if you will, which has evolved into the internal symmetry space of “color charge,” where the “color” of quarks is preserved through charge interactions. In the previous post below, I introduced the S|T triplet versions of the first generation red, green, and blue quarks, which combine to form the first generation baryons of the SM, the proton and neutron.

In Heisenberg’s concept of isospin, the neutron and proton were seen as nucleons, with the value of nuclear isospin, determining the degree of “protoness,” or “neutroness,” that the nucleon, as a whole, possessed. However, this original concept of isospin eventually had to be extended, until the current concept of three color charges, working via QCD, and its eight differently colored bosons, called gluons, managed to explain how quarks are confined to nucleons, through the principle of asymptotic freedom, and, in the process, demonstrated the power and beauty of gauge theory.

At some point in time, in The New Math blog, I hope to be able to explore more deeply the connection between the mathematical principles of the Reciprocal System of Mathematics (RSM), and those of Lie groups, used in gauge theory, that are involved in the SM.  The connection is extremely interesting and informing, because, in the RSM, we don’t have recourse to the ad hoc inventions of the real and complex numbers, which are the foundation of quantum mechanics and gauge theory, but, of course, we still have to be consistent with observation and mathematics. 

Nevertheless, right now I would like to continue with the discussion of the S|T triplets, which we have been discussing in this blog, in order to show how the conceptual logic of our RST theory might be extended to the combinations of protons and neutrons, as nucleons, in terms of the S|T triplets that represent the quarks and leptons of the SM. 

Again, however, I would like to stress the tentative nature of these developments.  Naturally, everything is subject to change and revision, as we learn more about the RST, but the fact that we can still proceed in a logical manner, to develop the consequences of our fundamental assumptions from the beginning, and that we are able to “discover” theoretical entities, with the observed properties of the SM entities in doing so, is compelling to say the least.

When we look at the double tetrahedron form of the proton and neutron combinations of S|T triplets, discussed in the previous posts of this blog, we see that we can identify three, external, “faces.”  Accordingly, we continue to refer to the topological features of these combos, but all the while insisting that such features do not exist in extension space, as it were, but that they only represent the relative space|time magnitudes of the constituent S|T units, just as the “distance” between different numbers is a delta magnitude, not an actual n-dimensional line, area, or volume of space.

In figure 1 below, we denote each “face” of the a and b tetrahedra, the two halves that comprise the double tetrahedron, as faces 1, 2, and 3, as shown, which we can then treat separately, for purposes of clarification.

Figure 1. Three “Faces” of S|T Tetrahedra

Given the identification of the three faces in figure 1 above, and the separation of these three faces of both the a and b parts of the S|T double tetrahedron, we can then separate and denote the six S|T triplets that emerge from the combination, as shown in figure 2 below.

Figure 2. The Six Faces of the S|T Double Tetrahedron as Six S|T Triplets 

The three circles, representing the middle, or inner, term of each triplet’s three constituent S|T units, representing the three sides of the triplets shown in figure 2 above, are omitted for the sake of clarity. However, it’s clear that an additional S|T triplet can be added to the center of each of these six S|T triplets, forming three more “faces” in each case, and that this addition of triplets can continue, ad infinitum.  Indeed, we see from this that the “geometry” of these combos is clearly fractal, where, again, the “distance” is purely numerical, not spatial. Therefore, the magnitude of the triple “surface,” in each iteration, is not reduced, as must be the case with the area of the geometric representation of these combos.

The important point to notice at the moment, however, is that each S|T triplet, constituting one of the three quarks in the proton or neutron, creates the necessary “space” to add three more quarks to the S|T combination, each of which, in turn, makes “space” for three more, and so on.  Of course, what this implies is that these additional “spaces” make it possible to combine protons and neutrons.  However, the problem with this is that the geometric truth that makes this possible (the fractal geometry) requires that we treat the S|T magnitude “spaces” as actual spaces, when, in fact, these faces are only spatial representations of combinations of space|time ratios, or magnitudes of speed.

This has to give us pause at this point, because, if we proceed, we will find ourselves in the same kind of predicament as the LST physicists who are not able to provide a physical interpretation for the mathematical concept of isospin, or color charge “space.”  In our case, we start with the unit space|time progression, and then we logically proceed to the SUDR and TUDR space|time displacements of this progression, which we find corresponds to the positive and negative, operationally interpreted, rational numbers of the RSM.  Moreover, once we have these discrete, reciprocal, units of M₄ motion, we see that the nature of the space|time progression produces the possibility that they will combine, forming the S|T combo that then progresses in both space and time at unit speed, but as a union of two, reciprocal, physical entities, which looks an awful lot like a photon, when we plot the combo on the space|time world line chart (The world line chart was discussed in previous posts below, but far enough below now that it’s obvious that I’m going to have to start archiving these posts so that I can provide direct links to them, which is something that I promise to do soon.)

Obviously, the S|T units, since they progress in both space and time, come into contact with other SUDRs and TUDRs, but what about coming into contact with other S|T units?  How can this happen?  See the problem is that, if a discrete location of space and a discrete location of time are plotted on the world line chart, as a combined space|time location, the location’s position on the unit progression line is unique; that is, different locations on the line cannot “move” relative to one another: They are eternally separated, as shown on the world line chart in figure 3 below.  

Figure 3. Two S|T Unit Locations on World Line

The S|T units plotted on the chart in figure 3 above, where p is the xyz space location, and t is the xyz time location, are, by definition, propagating as separate locations in both space and time. Combining them is tantamount to eliminating them as separate units.  Therefore, the question is, how can we combine three S|T units, as three constituents of a triangular triplet, since the three vertices of the triplet must be separated somehow in space|time?  Surprisingly, the answer not only assures us that it is possible to do this, without violating the world line chart, but, at the same time, it explains why two, or three S|T units can combine, but not more than three, and, consequently, why there are two types of hadrons, the baryons and mesons of the SM, and also why bosons can easily combine, but fermions cannot.

To understand how this works, it is necessary to recognize that only one of the three space coordinates, or only one of the three time coordinates, in a given S|T unit, need differ, in order to separate them.  Therefore, to plot these separate locations, it’s only necessary to convert the two-dimensional world line chart into a three-dimensional chart.  This permits us to plot the space|time progressions of three S|T units in parallel, as shown in figure 4 below.

Figure 4. World Line Plot of Parallel S|T Units 

In the 3D plot (here, we’ve omitted the green unit progression line for clarity), the z component of the p coordinate of a given S|T unit may differ from the z component of an adjacent unit.  Similarly, the z components of the t coordinates of two units may differ. The difference constitutes a relative offset of two locations, in one direction only, just as the locations of the center point of two adjacent spheres can only be in one direction.  Of course, for every possible offset position, there is a corresponding offset position diametrically opposite it.  Consequently, there only exists two potential z “slots” on either side of any given location, where the x and y coordinates are identical.  In other words, if we fix any two of the three components of a coordinate location, there can be only two, reciprocal, directions that the non-fixed component can take, in an adjacent location. This is the geometric version of the “odd man out” principle.

However, just as many spheres may surround a given sphere, there are many possible locations where one of the three components of the p and t coordinates of the associated locations may vary, but, in each case, there is only one set of these locations, and there is always a corresponding set that contains locations that are diametrically opposite to any location in the reciprocal set.  Moreover, in the case of S|T magnitudes, instead of lengths of distance measure, these two, reciprocal, sets correspond precisely to the reciprocal red and blue S|T magnitudes.

Therefore, in the universe of motion, it turns out that there is a physical reason why there can only be a maximum of three colors, and it is a mathematical one: any S|T magnitude, except the minimum, which happens to be a 1|2 or 2|1 (-1 or +1), is a sum of smaller values, in two, reciprocal, “directions.”  Again, it’s the ancient idea of balance.  There are only three discrete possibilities:

1. Balanced postitive and negative magnitudes
2. Negatively unbalanced magnitudes
3. Positively unbalanced magnitudes

Here, then, is the explanation for why there are boson and fermion S|T units: bosons consist of combinations of S|T units with all three constituent S|T unit magnitudes on one, or the other side, of unity, or they consist of equal magnitudes (balanced magnitudes) of both sides, but, in addition to this, the sum of the SUDRs and TUDRs that constitute the S|T units all occupy the same space, or time, location; that is, just as the pans on either side of a balance may be balanced or unbalanced in two “directions” with any conceivable number of units on either side, that may be arbitrarily subdivided into pairs of balanced and unbalanced units, many bosons can be combined into one boson for the same reason. They are the analog of a combination of n one-dimensional lines that are easily bundled together,

However, fermions consist of combinations of S|T units where one of the three components of the constituent S|T units is allowed to vary in one of two “directions.”  Thus, the locations of the constituent S|T units are not coincident, but offset in a given direction, as explained above, hence the fermion S|T triplets are not easily combined, unless they are oriented in a special way, just as planes can only be easily stacked, if they are oriented so that they are parallel.  In the case of spatial lengths, we don’t make this distinction, because one-dimensional lines also have to be parallel in order to be bundled, but, in the case of S|T magnitudes, one-dimensional combinations of magnitudes have no freedom of orientation; that is, S|T units are spherical, and thus invariant under rotation. 

The two-dimensional combinations of fermions (two-dimensional as in the two dimensions of a triangle), however, are orientable relative to one another! that is, the S|T unit positions 1, 2, and 3, in the triplet, may coincide, or not coincide. If they coincide, the orientation is equivalent to parallel planes, but if they don’t coincide, the orientation is equivalent to non-parallel planes. In Bilson-Thompson’s braid model, there is no provision for the non-coincident case, but, we recognize that it exists in the S|T model, though we won’t pursue the consequences at this time.  Only suffice it to say now that since the constituent quarks, in a proton, or neutron, or combination of them, are set, certain constraints should follow from this fact.

Certainly, we must conclude, in view of all of these developments, that there is yet much more to say. Yet, clearly, it is easy to see, from what has emerged so far, how the physical principles of symmetry, and thus the mathematical principles of Lie groups, can be used to describe the laws of conservation of “charge,” or what we would term “displacement,” that are inherent in all of this, which then obviously makes for a consistent system of physics, albeit one much more complex than what we are dealing with here.

The direct and inescapable implication is that there exists a much simpler system of constructing physical theory, a new system of physical theory, if you will.