The Larson Research Center

Now that we have seen that combining our SUDRs and TUDRs into SUDR|TUDR combos (S|T units) leads to theoretical entities with interesting QM-like properties such as probability densities, and measurement-dependent reality, and that these entities subsequently may be combined together to form preons, which appear to have the “directed” magnitudes of observed physical properties, such as photons, neutrinos, electrons, positrons, protons, and neutrons, as understood in the QM standard model of fermions and bosons, we want to delve into the quantitative aspects of these concepts.

The beginning point of our quantitative development is deciding what the natural units of the system are. Following Larson, we choose the Rydberg frequency of hydrogen and the speed of light, as the two physical constants we need to define the natural unit of the system. The reason that we don’t need three natural constants to do this is that we interpret the Rydberg frequency as a velocity, due to the central role of the definition of motion in the system, and this gives us both a space and a time constant, not just a time constant. Larson explains this in Chapter 13 of NBM:

From the manner in which the Rydberg frequency appears in the mathematics of radiation, particularly in such simple relations as the Balmer series of spectral lines, it is evident that this frequency is another physical manifestation of a natural unit, similar in this respect to the speed of light. It is customarily expressed in cycles per second on the assumption that it is a function of time only. From the explanation previously given, it is apparent that the frequency of radiation is actually a velocity. The cycle is an oscillating motion over a spatial or temporal path, and it is possible to use the cycle as a unit only because that path is constant. The true unit is one unit of space per unit of time (or the inverse of this quantity).

So, what we are assuming here is that the Rydberg frequency of hydrogen atom gives us the unit of time (space) it takes to complete an expansion/contraction cycle in a SUDR (TUDR); that is, it takes one unit of time (space) to expand the point in all directions to the unit sphere, and it takes another unit of time (space) to contract the unit sphere in all directions to the point, completing one cycle. Thus, “this is the equivalent of one half-cycle per unit of time (space) rather than one full cycle, as a full cycle involves one unit of space (time) in each [‘direction’],” as Larson explained it. In other words, since we are assuming that a cycle of the Rydberg frequency is the natural unit of space per natural unit of time once in each “direction,” the unit of time (space) of the frequency is twice the natural unit of time (space) in the velocity (inverse velocity). Thus, we need to double the value of the Rydberg constant, as Larson explains:

For present purposes the measured value of the Rydberg frequency should therefore be expressed as 6.576115 x 10¹⁵ half-cycles per second [2 x Ry = 3.28… x 10¹⁵].

The final step, then, is to take the reciprocal of this number, which gives us the natural unit of time, and to multiply the newly obtained natural unit of time times the speed of light, assuming the speed of light is the natural unit of motion, to give us the natural unit of space, as Larson explains:

The natural unit of time is the reciprocal of this figure, or 1.520655 x 10^-16 seconds. Multiplying the unit of time by the natural unit of speed, we obtain the value of the natural unit of space, 4.558816 x 10^-6 centimeters.

Of course, the accuracy of the measurements of these physical constants has improved since Larson’s day, and the new values will be used. They are

1. Rydberg constant for hydrogen, Ry = 1.09678 x 10⁵ cm^-1
2. Speed of light, c = 2.99792458 x 10¹⁰ cm/sec
3. Rydberg frequency of hydrogen, R_H = cRy = (2.99792458 x 10¹⁰ cm/sec) x (1.09678 x 105 1/cm) = 3.2880637208524 x 10¹⁵ Hz
4. Double R_H = 2 x 3.2880637208524 x 10¹⁵ Hz = 6.5761274417048 x 10¹⁵ Hz
   
5. Natural unit of time, t_n = 1/2R_H = 1.520652 x 10^-16 sec
   
6. Natural unit of space, s_n = c x t_n = 1.520652 x 10^-16 sec x 2.99792458 x 10¹⁰ cm/sec = 4.558799 x 10^-6 cm
7. Summary: t_n = 1.520652 x 10^-16 sec and s_n = 4.558799 x 10^-6 cm

What we discovered in the combination of SUDRs and TUDRs that leads us to the preons of the SM, consisting of combined S|T units, is that bosons consist of one compound space|time location, while fermions consist of more than one, adjacent, space|time location.  In fact, we are initially assuming that subatomic fermions consist of three adjacent space|time locations, but more or less than this may also be interesting.  In general, however, this concept has the following consequence:

If adjacent locations have the same space|time progression, they will remain adjacent, if not, they will separate. 

In combining adjacent space|time locations into one, partially merged, location, as we do in the preon triplets, we introduce the notion of distance between locations. Hence, if the space|time magnitude of the constituent S|T units of a given S|T triplet are equal, the space|time progression of adjacent space|time locations, defined by the constituent SUDRs and TUDRs of the S|T units making up the triplet, is equal.  Therefore, like three aircraft flying in formation, through a climbing turn, the distance between the relative locations of the three S|T units remains the same.

Of course, the relative distance between the aircraft of the flight formation is the space aspect of M₂ motion, whereas the relative distance between the S|T units of the S|T triplet is the space aspect of M₄ motion.  Nevertheless, the concept of preserving the relative space difference of adjacent locations is the same in both instances, disregarding the dimensions of motion involved.  In the case of the aircraft flight formation in a climbing turn, the 1D altitude is changing at the same time the 2D heading is changing, which means that all three dimensions, x, y, and z, are affected.  As every pilot knows, if the changes in altitude and heading of the three aircraft are not carefully coordinated by the pilots, the distance between the adjacent aircraft, in the dimension where a difference exists, will change accordingly, and if it is great enough, the integrity of the flight formation, as three adjacent flight paths, will be lost.

Similarly, if a difference in the 3D scalar motion of the S|T triplets exists, the integrity of the S|T triplet, as three adjacent locations in the space|time progression, will be lost. Thus, we immediately see that the only triplets that would normally remain intact are those where all three S|T units in the triplet possess displacement in the same “direction,” with the same magnitude.  Assuming the same magnitude in each S|T unit, the only stable fermion triplets, then, are the neutrino, electron and positron (see here), those where all three center colors are the same.  The quark triplets would not be stable.

If all the S|T triplets, corresponding to theoretical quarks and anti-quarks, are unstable, this would theoretically explain why the physical quarks are never observed in isolation, something the LST physicists explain in terms of “asymptotic freedom.”  This is a concept wherein the strong force gets stronger, as the distance between quarks increases, and, therefore, the energy required to separate them is equal to the energy required to create a new set.  Thus, quarks are never observed in isolation, only as constituent triplets of baryons.

However, in the development of our theory, the challenge is just the opposite of that of the LST physicists who need to explain why quarks can only be observed together.  Our challenge is to explain why quarks can exist together.  We know that they can’t exist apart, because the “directions” of the speeds of their three progressing locations don’t match.  For example, the “direction” of the speed magnitude of two S|T units in the down quark is neutral (green), while the “direction” of the third S|T unit in the triplet, is an “odd man out;” that is, the “direction” of the third S|T unit of the down quark is negative (red), which means, over time, the space|time location of that unit will move away from the adjacent locations of the other two S|T units, destroying the integrity of the triplet.

If we look at the constituent speed displacements of the triplet in terms of speed, we can see this more clearly. For example, assuming unit displacement in each unit of the neutrino, we get (ignoring the inward component of the motion):

1. A = (1|2) + (2|1) = 3|3 = 1|1 = 0
   
2. B = (1|2) + (2|1) = 3|3 = 1|1 = 0
   
3. C = (1|2) + (2|1) = 3|3 = 1|1 = 0
   

where A, B, and C, are the three combined space|time progression ratios of the three merged space|time locations of the neutrino triplet, represented by the apexes of the triangle.  In this case, the change in time and the change in space is unit change in each instance, thus maintaining the triplet “formation.” From the perspective of the unit progression, the magnitude of the outward space progression is equal to the magnitude of the outward time progression; that is, it is a one for one progression, and therefore the three space and time locations are synchronized, as the progression proceeds.

However, if one of the S|T units of the triplet picks up an additional SUDR, which would change the color of that unit from green to red, changing the color combo of the neutrino triplet to that of the down quark triplet (2 green, 1 red), the speeds of the nodes change to

1. A = (2|4) + (2|1) = 4|5 = -1
   
2. B = (1|2) + (2|1) = 3|3 = 0
   
3. C = (1|2) + (2|1) = 3|3 = 0
   

assuming that the A node is the SUDR connection of the affected S|T unit. Now, we see that the progression of the A node, from the unit progression perspective, changes.  It changes, from unit space|time progression, with no speed-displacement (3|3 = 0), to non-unit space|time progression, with a negative unit of speed-displacement (4|5 = -1).

What this speed-displacement difference between the nodes translates to is a difference in the length of the cycle of the A node relative to the B and C nodes.  The speed of the change in the SUDR side of the affected S|T unit is still the same, because two units of space quantity in each “direction” in four units of time quantity in one “direction”, is the same speed, or space|time ratio (2|4 = .5c), as one unit of space in two units of time (1|2 = .5c), but it takes twice as long to complete one cycle of oscillation.  It’s the same as saying that the speed of two cars traveling two different distances is the same, but if one has to travel twice as far as the other, at that speed, the duration of the longer trip doubles, relative to the shorter trip, but not the speed of travel.

Equivalently, we can say that the frequency of node A has decreased relative to the frequency of nodes B and C, because one of the two S|T units, connected to node A, picked up an additional SUDR.  Therefore, the relative frequency “distance” between the nodes, which was zero, is now non-zero. However, we are not accustomed to thinking of frequency in terms of space|time progression. After all, we don’t think of two separate piano keys as moving away from one another over time, because their frequencies differ.  Hence, we need to quantify the difference in a more useful manner.

Fortunately, there is a great way to do this, I think, but it requires a subtle notion of representation.  In fact, I think it is the same mathematical idea of a “representation” of a group, which LST physics uses in connection with Lie group theory, but I may be mistaken. Someone more proficient in abstract algebra and group theory might be able to enlighten us on this.

Regardless, taking the notion of the 3D, or scalar, oscillation of SUDRs and TUDRs, from a space (time) point to a space (time) sphere and back to space (time) point again, as a reciprocal relation of space|time locality|non-locality in the S|T unit (see previous post below), we can construct a mathematical representation of this scalar motion in a vector space; that is, the set of mathematical values that the reciprocal oscillations of the S|T unit take, in our RN equations, form a symmetrical group that acts on the complex vector space of rotations in the same way that the solutions to Maxwell’s equations form a group that acts on the rotations of the SO(3) group (or the other way around!).

To tell you the truth, I don’t know a thing about these obtuse mathematical concepts, and I’m sure I have the concepts all mangled up here, but intuitively, I think the following concept is “isomorphic” (LOL) to group representations: The S|T unit is a representation of rotation similar to the SO(3) group, because the “oscillation” from spatial local to non-local is the reciprocal of the “oscillation” from temporal local to non-local. I don’t know if the rotation group has a name, or if it’s simply isomorphic to a well known existing group, but I can identify the rotations: they are gear ratios!

I know this must sound like the height of crackpottery to professional mathematicians and physicists, but if it is, we will soon find out, and I’ll stand corrected.  However, as things now stand, this conclusion is inescapable.  As shown in figure 1 below, the identity gear ratio is the identity element of the group. 

Figure 1. The Identity Element of the Group of Reciprocal Rotations, or the “Gear Group”

Clearly, just as the “direction” of the space “oscillation” of the SUDR, is always the reciprocal of the “direction” of the time “oscillation” of the TUDR (i.e. outward in time is equivalent to inward in space and vice-versa), the direction of the rotation of the SUDR gear is always the reciprocal of the direction of the rotation of the TUDR gear; that is, if the SUDR gear rotation is clockwise, the TUDR gear rotation must be counterclockwise, and vice-versa.

In figure 1, the unit ratio of the identity element is expressed by the equal radius of the two gears.  The reciprocal, gear, ratio is 1:1, just as it is in the reciprocal, unit, SUDR|TUDR ratio; that is, the 1|2:2|1 S|T unit ratio is the equivalent of a -1:+1 ratio of reciprocal speed-displacement, where the signs of the magnitudes represent the inverse magnitudes of operationally interpreted rational numbers.

The phase of the “oscillation” is represented by the position of the orange dots on the gears.  In the illustration of figure 1, the phase indicates that both components are local (points), or both 0, at the same time, which is convenient for comparing the relative number of revolutions of the reciprocal gears, but it is not what we want, since this would not represent the reciprocal phase of symmetry that constitutes the law of conservation, according to the Noether theorem. 

Thus, we must modify the relative phase of the representation, as shown in figure 2, below.

Figure 2. S|T Representation of the Identity Element of the “Gear Group”

Of course, to show that the “gear group” meets the criteria of a mathematical group, is going to take some explaining, which I will undertake in the next post, but it’s clear that the operationally interpreted rational numbers of the new reciprocal system of mathematics (RSM), should enable us to do just that, since we already know that they form a group under binary addition.

However, combining the S|T units, as conserved space|time speed-displacement ratios, will necessarily lead us to consider a group under binary multiplication.  I hope we can work the required RSM math operation of multiplication out, but I really don’t know if we can, at this point. I haven’t given it any thought, because, before the discovery of the “gear group,” I didn’t know why we needed it, but now that has changed.