The Larson Research Center

What we are doing can be described in a few steps:

1. Assuming the universal progression of space and time units
   
2. Assuming a local oscillation of “direction” in one aspect of the progression, creating SUDRs and the inverse TUDRs
   
3. Combining these into SUDR|TUDR (S|T) combinations of reciprocal space|time progression
4. Describing the progression of S|T units in terms of varying locality
5. Forming the bosons and fermions of the standard model from preon triplets, consisting of three S|T units each.
6. Equating the relationship of the three S:T triplet nodes to the reciprocal rotations of meshed gears
   
7. Quantifying the interactions of the nodes using the group properties of the reciprocal rotations of gears
   

Hence, analyzing the physical properties of the theoretical fermions consists in quantifying their three nodes of space:time progressions as reciprocal rotations, analogous to the rotations of two meshed gears. In the case of the neutrino, the rotation ratio is 1:1 in all three nodes, because each of its constituent SIT units is balanced, as indicated by the green color at the center of the three lines representing the constituent S|T units of the triplet, as shown in figure 1, below.

Figure 1. The Individual Space:Time “Rotation” Ratios at the three Nodes of the Neutrino Triplet

Notice that we use the notation S:T to represent the red:blue relation of two S|T units connected together at each of the three nodes, while the notation S|T is used to designate the SUDR|TUDR balance of a given S|T unit in the triplet.

The next question to be considered is how the three nodes relate to each other inside the triplet. The schematic of the triplet’s internal nodes of progression, as a triangle, indicates a spatial separation between the nodes, but the separation is a separation of both space and time, or motion. Therefore, the representation of the three space and the three time locations of the S|T units in the triplet model is different than the schematic representation. It consists of three, partially merged, space and time locations, which we represent graphically, as three, partially merged, spheres, as shown in figure 2, below.

Figure 2. Spherical Model of the S|T Triplets

In the case of the neutrino triplet, all three spheres, in figure 1 above, would be green; they would all be red for the electron triplet, and all blue for the positron triplet. These three would be the only stable triplets, as explained in the previous post below. However, since the S|T units are oscillating probability amplitudes, representing them as spheres is misleading time wise, because they only exist as spheres, at one point in time, at the moment of transition from outward progression to inward progression. At the opposite end of the cycle, where the transition is from inward progression to outward progression, the three spheres will have contracted to three points, and, if their oscillations are in phase, a spatial (temporal) separation will exist between them, at this point in the time (space) progression.

On the other hand, if the center locations of the spheres move together, as they contract, to prevent such a separation between them from forming, they will become one point and will subsequently re-expand as one sphere, changing them from a fermion to a boson, unless they move apart once more as they expand. If their oscillations are not in phase, then this phase difference would affect the situation accordingly, but, either way, it is misleading to model the triplet as three, static, locations, which sometimes are three, non-local, spheres, and sometimes three, local, points. Clearly, the model of these theoretical entities must be a dynamic, or time-dependent, model, not a static one.

This problem of modeling the theoretical S|T units, as expanding/contracting spheres and points, is avoided when we employ the numerical representation of the gear group, discussed in the previous post below. In this way, the expansion/contraction of each space|time (red|blue) node of the schematic triplet is a representation of the reciprocal rotation of interlocking gears, and it is the phase relationship, at a given node, that is a representation of the cyclical motion of the gears. This, in turn, enables us to represent the relation between the three nodes of the triplet in a similar manner, because we can represent the separation between the nodes, seen in the schematic triplet, as an abstract notion of compound, reciprocal, rotational, motion, rather than the somewhat misleading and confusing notion of spatial distance, which is constantly changing in three dimensions.

As indicated in the schematic triplet, the SUDR end of one S|T unit interacts (connects) to the TUDR end of the adjoining S|T unit, forming an S:T connection. So, we can represent the neutrino triplet, as a compound gear train, as shown in figure 3 below.

Figure 3. The Compound Gear Train of the Neutrino Triplet

The reciprocal rotations are indicated by the SUDR and TUDR colors. If we designate the red color of the SUDR as clockwise rotation, then the blue color of the TUDR indicates counter-clockwise rotation. Hence, we can see that, just as the colors alternate in the gear sequence, the direction of rotation does as well, maintaining our reciprocal locality|non-locality oscillations, as representations of the mathematics of reciprocal gear rotations.

In figure 3, all the gears of the compound gear train are the same size, hence the neutrino configuration is a representation of this 1:1 gear train, where the numbers of SUDRs and TUDRs, in each of the constituent S|T units, are equal. The total space:time ratio, then, is the product of the ratios in the compound gear train,

• S_n:T_n = [(1:1) * (1:1) * (1:1)] = 1:1.

However, in the case of the electron’s constituent S|T units, the number of SUDRs is greater than the number of TUDRs. Assuming the minimum possible number of each, the SUDR “gears” of the electron gear train must be twice as large as the TUDR “gears,” as shown in figure 4, below.

Figure 4. The Compound Gear Train of the Electron Triplet

Again, the total ratio of the compound gear train is the product of all the gear ratios,

• S_e:T_e = A:B:C = [(2:1) * (2:1) * (1:2)] = 4:2 = 2:1.

Notice that the gear ratio of the last S:T gear (last term) is “the odd man out” and its “direction” must be reversed in the equation, relative to the other two ratios, just as it must be in the gear train illustrated in figure 3 above and in the triplet schematic. The “directional” reversal is indicated in the equation by denoting the ratio in bold print.

In the case of the positron, there are twice as many TUDRs as SUDRs (again, assuming the minimum number), so the blue “gears” are twice as large as the red “gears,” and the corresponding equation is the inverse of the electron’s equation,

• S_p:T_p = A:B:C = [(1:2) * (1:2) * (2:1)] = 2:4 = 1:2,

which indicates the relative inverses of the positron and electron gear trains.

In the case of the quarks, the symmetry of the gear trains found in the stable particles is quite broken, especially in the case of the down quark, as shown in figure 5 below.

Figure 5. The Up Quark (left) and Down Quark (right) Gear Trains

Since there are six “gears” in each quark and three quarks in each hadron, for a total of eighteen “gears,” in a proton or neutron, it would be too difficult to try to illustrate the hadronic configurations in a single graphic. Instead, we will illustrate the configuration in a separate graphic for each of the five nodes of these hadrons.

Each hadron has one up, or one down quark, with two of the opposite quark types. The neutron has two down quarks. The proton has two up quarks. We’ll take the neutron first. It has an up quark between two down quarks. As regards the neutron’s up quark, we will designate the left apex of the triplet as the A node, and to its right, at the top of the triplet, is the B node, and to the right of that is the C node, as shown in figure 6 below. The S|T unit between the A and B nodes is S|T unit 1, designated U1. Unit 2 is between the B and C nodes, designated U2, while U3 is between the C and A nodes.

Figure 6. The A B C Nodes of the Neutron’s Up Quark Triplet

The three constituent S|T units of the two down quarks connect to each of the ABC nodes of the up quark, one in front, and the other behind the up quark, forming the double tetrahedron of the neutron. We will designate the front node, as node D, and the rear node, as node E. The S|T units of node D are designated left to right, DF1, DF2, and DF3. The S|T units of node E are designated DR1, DR2, and DR3, but, viewed from the back, the positions of the A and C nodes are swapped.

Figure 7. The D and E Nodes of the Down Quark Triplet in the Neutron Hadron

Nodes D and E each have three connected S|T units, while nodes ABC each have four S|T units connected to them (two up quark units and two down quark units.)  Hence, nodes A, B, and C have three blue “gears” and one red “gear,” while nodes D and E each have two red”gears” and one blue “gear,” in the configuration shown in figure 7 above.  Other configurations may be possible, but this is the most straightforward, it seems to me.

Since space to space, or time to time, is not motion, the “gears” must be arranged in space to time ratios, or red to blue (blue to red) “gears;” that is, red “gears” mesh only with blue “gears” and vice versa, indicated by S:T, or T:S symbols.  On this basis, the A, B, C, D, E node equations of the neutron are (red space units on the left, blue time units on the right)

1. S_A:T_A = [(U1:U3) * (U1*DF1) * (U1:DR3)] = [(1:1) * (1:1) * (1:1)] = 1:1
2. S_B:T_B = [(U2:U1) * (DF2:U1) * (DR2:U1)] = [(1:2) * (2:2) * (2:2)] = 4:8
3. S_C:T_C = [(U3:U2) * (U3:DF1) * (U3:DR3)] = [(1:2) * (1:1) * (1:1)] = 1:2
4. S_D:T_D = [(DF1:DF2) * (DF1:DF3)  = [(1:1) * (1:1)] = 1:1
5. S_E:T_E = [(DR1:DR2) * (DR1:DR3) = [(1:1) * (1:1)] = 1:1
   

 and the S:T compound gear ratio of the neutron is therefore

• S_N:T_N = (A:B:C) = [(1:1)*(4:8)*(1:2)*(1:1)*(1:1)] = 4:16 = 1:4.

The proton has one down quark and two up quarks, and the relative orientation of the constituent S|T units in a down quark, of the same chirality, is reversed.  Therefore, the ABC nodes of the proton’s down quark have a different S:T configuration, as shown in figure 8 below.

Figure 8.  The A B C Nodes of the Down Quark Triplet

Likewise, the configuration of the two up quarks in the proton’s D and E nodes has the opposite orientation, as shown in figure 9 below. 

Figure 9. The D and E Nodes of the Up Quark Triplet in the Proton Hadron

Thus, the A, B, C, D, E node equations of the proton are

1. S_A:T_A = [(U3:U1) * (DF1:U1) * (DR3:U1)] = [(1:1) * (1:1) * (1:1)] = 1:1
2. S_B:T_B = [(U1:U2) * (U1:DF2) * (U1:DR2)] = [(1:1) * (1:1) * (1:1)] = 1:1
3. S_C:T_C = [(U2:U3) * (DF1:U3) * (DR3:U3)] = [(1:1) * (1:1) * (1:1)] = 1:1,
4. S_D:T_D = [(DF1:DF2) * (DF1:DF3)  = [(1:2) * (1:2)] = 1:4
5. S_E:T_E = [(DR1:DR2) * (DR1:DR3) = [(1:2) * (1:2)] = 1:4

 and the S:T compound gear ratio of the proton is therefore

• S_P:T_P = (A:B:C:D:E) = [(1:1)*(1:1)*(1:1)*(1:4)*(1:4)] = 1:16.

However, the problem with this analysis is that the five nodes are not connected as a compound gear train, with an output shaft and an input shaft.  Arranging the gears at each node as planetary gears enables us to align the nodes sequentially, but the interconnections get really wild. Nevertheless, it’s not the gear ratio per se that we are interested in, but rather the properties of the constituent motions themselves; that is, we have no input and no output, as you would with an actual gear train of mechanical gears.  What we are interested in, instead, is the phase relationship of the probability amplitudes, which, in this case, are tantamount to the local|non-local state of the constituent SUDRs and TUDRs.

To describe these relationships we can use calculus because the positions of the marks on the “gears” are represented by the corresponding changes in the “locality” property of the expanding/contracting SUDRs and TUDRs, which have the same reciprocal relationship as that of the sines and cosines of the changing angles in the “gears,” as their reciprocally placed marks revolve in time.  In other words, its a phase relationship that can be described in terms of the 2pi radians of rotation, just as is done in quantum mechanics. 

The details of this development are still being worked out, but the reader can get a good idea of what the equations will describe from this remarkable animation by Dale Meier found on Bob Palais’ website.  Be sure to click the “Multi” box:

http://www.math.utah.edu/~palais/daledots.swf 

It’s especially enlightening to understand these motions in terms of what Frank H Makinson terms “double pi,” which seems to me to be a very apt description, because, while the rotation of the relative positions of the gears is a change of position, or vectorial motion, the relative change of the associated sines and cosines is actually a description of scalar motion, even though it is always depicted as two, orthogonal, vectorial vibrations.  Frank writes in a comment posted in the ISUS Discussion forum:

Dirac’s constant is one of the examples that I think of when 2pi is used to extract characteristics of the physical universe. The definition of that constant allows it to use either frequency [changing sine and cosine] or angular frequency [changing position], rather than arriving at the conclusion that the numeric value might actually represent the composite of the two, which I call double-pi.

Two times Archimedes’ Constant, pi, is not the same as 2Pi defined as frequency or the 2pi defined as angular frequency. …[The fact] that double-pi represents the composite of 2pi as a frequency and 2pi as angular frequency will be a strange concept to most people. 

However, I think Dale’s animation belies this last statement.  Given the animation, the double, or composite, nature of 2pi as a frequency of two, reciprocal, aspects of angle, and, simultaneously, as a rotation of a location on the circumference of a circle, is not going to seem like a strange concept to most people, when the two are seen for what they really are.  On the contrary, this picture of composite M₂ and M₃ motion (and M₄ motion, if it were in 3D), in the form of compound rotations, ought to seem beautiful beyond words to most people.  It certainly does to me. 

Clearly, Dale’s animation graphically shows the true nature of quantum mechanics through the use of classical equations, provided that one understands the RST meaning of the changing sines and cosines (seen as linear vibrations), in the animation:  They are not to be understood as changing positions along intersecting diagonals lines, or M2 motion, but rather as changing values of probability amplitudes, the projections, or analogs of M₂ motion.

Geez, this stuff’s exciting!