The Larson Research Center

Last Spring, we started discussing an initial effort to develop the quantitative side of our RST-based theory (here), beginning with the fundamental magnitude conjecture (FMC).  The FMC simply states that there are three (four counting zero) fundamental numbers, or magnitudes (with quantity, dimension, and direction properties), from which all others are constructed. Raul Bott proved this true for the dimension property, and I’m simply conjecturing that, if it holds for one property of number, or magnitude, it probably holds for the other properties as well.

After discussing the FMC, we turned to the subject of force and acceleration (here and here), noting that Larson didn’t try to redefine these concepts in scalar terms, but only tried to clarify certain conceptual and dimensional aspects of them. For our purposes, however, we redefined them as scalar concepts, in terms of speed density (acceleration), and inverse speed density (force), varying by the inverse square law.

Then we really went out on a limb (here), by using these new definitions to treat the dimensional problems of force equations as misunderstandings of scalar quantities, which, if viewed in the context of the tetraktys and Bott periodicity, lead to a reconciliation of the infinities that plague the LST theories, reconciling the conflict between a point and a sphere.

But trying to stay on track with the quantitative development, we turned our attention to quantifying the SUDR and TUDR and the SUDR|TUDR combo, oscillations (here). We found that describing these oscillations in terms of the transition from local (point) to non-local (sphere), the changing “localness” property of space and time in the S|T combo acts like the conservation of energy between the reciprocal properties of kinetic and potential energy in a pendulum system of motion. But instead of using the sine and cosine functions, to mathematically describe the harmonic oscillations, we discovered (here) that the function that was needed could be represented by the reciprocal rotations of two, meshed gears, which led (here) to the discovery of the binary functions of compressed/stretched springs via the “gear group” of rotations, where the concept that “half as much” is the reciprocal of “twice as much” took us into the two, reciprocal groups, one under addition, and the other under multiplication, with all sorts of implications.

Of course, things were actually a little muddled for a while, as it wasn’t clear right away how the sine/cosine functions related to the binary functions of the gear group, so the discussion turned to the math aspects for a bit (here, here and here), which, in turn, led to discussing how the gear group relates to the LST’s gauge principles, using complex numbers of “size one” (here, here and here).

In the meantime, trying to explain some of these things, the crucial difference between the sine/cosine functions and the binary functions was finally recognized, which then led to the explanation of quantum spin (here). Needless to say, dealing with such fundamental concepts at this dizzying pace was difficult to handle, so I’ve had to take a break to regroup and try to see where all this leads and what should be done next. After all, the objective of the quantitative development is to get to the point where we can calculate the atomic spectra.

All of these concepts are no doubt going to play an important part in the quantitative analysis, but the question still is, where do we start? The answer is clear: We have something the LST community hasn’t had in a long time, a model of the photon, electron and other observed entities, as well as a foundation for understanding the principles of quantum behavior, such as uncertainty, spin, duality, etc.  Therefore, what we do now is test this model to see if the observed interactions of these entities can be predicted from the models and the quantum principles inherent in the 3D, space|time, oscillation.

The first step in this process is to relate the basic S|T unit to Planck’s constant h. Recall that in the early days, Millikan established the value of the charge of the electron and the validity of Einstein’s equation relating the kinetic energy of an ejected electron (photoelectron) and the scalar energy of the photon that ejects it, in the photoelectric phenomenon. In our RST-based model, this is an interaction between S|T boson triplets and S|T fermion triplets. Hence, it appears that a quantitative analysis of this interaction would be a good starting point for testing our conceptual model.

Einstein’s photoelectric equation is

1/2mv² = hv - P,

where h is Planck’s constant, v is the frequency of the incident radiation, and P is the work necessary to accelerate the electron to escape velocity, so-to-speak. Since Planck’s constant is the unit of energy, dt/ds in our model, the total energy is some multiple of h determined by the number of cycles per second of oscillation inherent in the incident radiation v. The dimensions of the kinetic energy, mass, m, times the square of the velocity, v, are

(dt/ds)³ * (ds/dt)² = dt/ds

However, as noted above, we have redefined the concepts of force and acceleration in terms of speed, and inverse speed, density, from the scalar point of view, eliminating the need for the LST community’s ploy, wherein long ago it resorted to the concept of work (force time distance) to reconcile the dimensions of energy. That is to say, as long as the force vector is perpendicular to the distance (thus, there is no direction of displacement) vector, the inner product of the two (a scalar) is zero and no energy (work) is expended, but if a displacement takes place, then the scalar value of work (energy) is non-zero.

This ingenious workaround is necessary when dealing with mechanical concepts of energy, which depend on moving mass (momentum), but when dealing with the concept of rest mass, then Einstein’s other energy equation,

E = mc²

is applicable. Of course, in this case, we are not talking about a moving mass, and, therefore, there is no applicable force vector involved, as there is in electric, or gravitational, potential and kinetic energy. However, acceleration is involved, because inertial mass is equivalent to gravitational mass. Hence, in our new scalar view of force and acceleration, when we deal with rest mass, we must switch from the force concept of energy per square unit of space, (t/s)⁰/s², to the acceleration concept of velocity per square unit of time, (s/t)⁰/t².

But now we have the same problem as the LST physicists had with the definition of scalar energy, only in terms of scalar velocity; that is, we need to express a scalar quantity of velocity in terms of a vector quantity of acceleration and time, when it does work, like ejecting (accelerating) an electron from a metal surface, in the photoelectric interaction. 

Obviously, we have our work cut out for us (no pun intended! LOL)

Evidently, some are confused as to how one, 1D, extension/compression of a weighted spring, can be understood as a 2D rotation, as asserted in the previous posts below.  No matter how you look at the two, inverse, reference systems, they say, there’s still no way to get the two dimensions required for rotation, with only one, 1D, vibration.

Well, that’s true and that’s why what I’m pointing out has never been recognized before, because, when the 1D motion of the springs constitutes the SHM (given the proper phase relationship), it takes two, orthogonal, 1D spring vibrations, in the discrete reference system, to define a single rotation, in terms of the sine and cosine functions of the angle of rotation.  This is shown in Figure 1 of the previous post.

However, when the same two vibrations are viewed in the continuous reference system, the change from the fully compressed state of the springs to the fully extended state, and the inverse, from the fully extended state to the fully compressed state, becomes a double system of scalar SHM, defined by change in two, reciprocal states, instead of a single system of vector SHM, defined by two sets of reciprocal change of positions.

Two, orthogonal, 1D vibrations, as sines and cosines, define four lengths, as four radii of space in the unit circle, viewed in the discrete reference system, as the positive and negative positions of the orthogonal x, y coordinates. On the other hand, these familiar x, y positions of the 2D coordinate system cannot be defined in the continuous reference system. In the continuous system, the positions from 0 to 1, and the inverse positions from -0 to 1, represent something else.

In the continuous system, there are limits between -0 and 1, and between 0 and 1, which don’t exist in the discrete system.  In the discrete system, 0 is the point of “direction” reversal, and it’s the only such point, but in the continuous system, there are two points of “direction” reversal, 0 and 1 in the positive unit, and -0 and 1 in the negative unit.  There is no “direction” reversal at 1 in the continuous system.  In other words, there is no cross over point at the “origin” of the continuous reference system, from positive to negative, or from negative to positive, as there is in the discrete reference system.  This is a very significant difference between the two, reciprocal, reference systems.

In the springs, this change from the discrete reference system to the continuous reference system can be described as a change in the interpretation of the spring’s compression|tension property. If the compression|tension force in the spring changes “direction” at the equilibrium point, it provides the necessary restoring force that produces the oscillations of the weights attached to the springs, the motion of which has naturally been interpreted in terms of the discrete reference system, thereby coupling the SHM of two, 1D, vibrations to rotation, through the sine and cosine functions of the angle, as the position of the weight changes. 

But in reality, this is accomplished through the interaction of the spring’s compression|tension property, with the momentum of the attached weights. The interaction is just a physical mechanism for producing the oscillations of SHM. Nevertheless, in terms of numbers, the change in one spring is from -1 (fully compressed) to 0 compression|tension, and from 0 tension|compression to 1 (fully tensioned).  This is the same relation of space and time displacements that exists in the RST.

Equivalently, however, this same range of motion, from maximum compression, to no compression, at the relaxed, or “equilibrium” position, and from no tension to the maximum tension (fully stretched) position, can be seen as one range, bound by two limits (0 and 1). In this view of the magnitudes involved, the “direction” reversals actually take place at the limit of the fully extended point, and at the limit of the fully compressed point, not at the mid point of equilibrium, if the factor of a restoring force’s interaction with momentum is not relevant.

We can see this clearly, if we place the spring between our hands and squeeze it, until it’s fully compressed, and then slowly release it so it can fully extend again. If we then pull its ends apart, we can stretch it to its maximum and then slowly release it so it can once again return to its relaxed position. As far as the numbers alone are concerned, the rate at which we compress, or stretch the spring is immaterial.  Starting from the fully compressed state and stretching the spring to its limit, constitutes a single change of state from -0 to 1. The reciprocal of this state is formed by starting from the fully stretched state and compressing the spring to its limit, from +0 to 1.

When we realize this, then it’s obvious that there is a correlation between the reciprocal states of the compressed, and tensioned, spring and the reciprocal phases of rotation; that is, compressing the relaxed spring is equivalent to 90 degrees of rotation, in a single dimension, from 0 to 90 degrees, and stretching it is equivalent to another 90 degrees of rotation, in that same dimension, but in the opposite “direction” from 90 to 180 degrees, so stretching it from the fully compressed state, to the fully tensioned state, is equivalent to 180 degrees of rotation, in a single dimension, regardless of the changing “direction” of the compression|tension force at the mid point. Conversely, compressing it from the fully tensioned state, to the fully compressed state, is equivalent to 180 degrees of rotation, in a single dimension, in the opposite “direction,” regardless of the changing “direction” of the tension|compression force at the midpoint.

In this case, the fully compressed and the fully tensioned states of the spring, become the limits of the system, which represent one full unit, between them, which is divisible into a continuous spectrum of magnitudes of a given size, where these are the magnitudes of the continuous group, of a given order, that lie within the bounds of a single unit.

What we have done is combine the two, discrete, units of the two opposed radii, described by the sine, or cosine, of the angle of rotation, into a single diameter of the circle, to form one continuous unit, from two discrete units. The question now is, how can we couple this single diameter to a 2D rotation?  In the discrete system, since such a coupling requires four discrete units, the positive and negative units of the sine and cosine, which are equivalent to two, orthogonal diameters, the question becomes, “Don’t we need two diameters in the continuous system as well?”  

The surprising answer to this question is no, we don’t.  The reason why we don’t, is because, in changing from the discrete reference system, to the continuous reference system, the vectorial motion of the spring, from one point to another is no longer the defining property.  It’s not the changing position of a weight, attached to the spring that is represented by the change of compression|tension, but the ratio of the spring’s compression|tension itself, a scalar value.

At the point of equilibrium, the compression|tension ratio is 2|2 = 1|1 = 0, because, from this point, the two units, out to the fully compressed limit and back, and the two units out to the fully tensioned limit and back, are two potential, reciprocal, units. When the spring is fully compressed, the potential ratio is altered, from 2|2 = 0 to 1|2 = -1, and when it is fully tensioned, the potential ratio is altered from 2|2 to 2|1 = 1. These magnitude changes represent the discrete elements of the group under addition, the discrete group.

But now we know that the interpretation of the reciprocal operator, from the difference interpretation, to the quotient interpretation, transforms the elements of the discrete group under addition, to the elements of the continuous group under multiplication.  When we make this change in interpretation, then 1/2 = .5, 2/2 = 1/1 = 1, and 2/1 = 2, and we are now in a new reference system, the reference system contained within the limits of two, reciprocal, units.  In this system, .5 is the negative limit, defining the unit between -0 and 1, while its inverse, 2, is the positive limit, defining the reciprocal unit between 0 and 1, a fact that the chirality of the system disguises, but that the inverse product operation of the continuous group exposes, as .5 x 2 = 1, where 1 is the identity element.

Consequently, under this interpretation, the spring’s compression|tension property is the reciprocal of its tension|compression property, and, while it can take either form, it cannot take both forms simultaneously.  Hence, in order to have both a negative unit and a positive unit, we must have two springs, where, at any given moment in time, the state of one is the inverse of the state of the other, with regard to these reciprocal properties.  This is just another way of saying that, in a coupled system of scalar SHM, one spring must be compressing, while the other spring is tensioning, or, in other words, there must be a 180 degree phase difference, not a 90 degree phase difference, between the two springs. Obviously, a single spring cannot be compressing and tensioning at the same time!

Hence, since, in the continuous reference system, the relative phases of the two reciprocal springs in scalar SHM is not the 90 degrees of the discrete reference system, but the 180 degrees of the continuous system, as shown in the SUDR|TUDR graphic, in the previous post, and shown in a more compact form below, the scalar SHM of the system is derived from a double set of parallel springs, not two independent, or orthogonal, springs.

Figure 1. The Reciprocal States of the Continuous Reference System

The key here is not to think in terms of the lateral motion required to compress and tension the two springs, but rather to think in terms of the change of state in the springs themselves, from a compressed (local) state, to a tensioned (non-local) state. Such a change is a continuous change analogous to the scalar change from point to sphere (expansion), and from sphere to point (contraction), which we use to map the reciprocal oscillations of the S|T units to the reciprocal rotations of two, meshed, gears.

The relevant point here is that one cycle of scalar magnitude, from local to non-local and back, is the equivalent of 360 degrees of rotation, and, when two of these are combined, as shown in the graphic of figure 1 above, the two constitute the equivalent of  720 degrees of rotation in one cycle of oscillation, just as the counter-rotations, of two, meshed, gears constitutes 720 degrees of rotation per cycle.

After several telephone conversations have convinced me, I’ve decided to redraw the graphic of the previous post.  In the previous graphic I didn’t have room for two springs of the discrete group, those that are mapped to the sine and cosine, but I thought people would not need two of those.  However, it’s much clearer, if both are included, so I’ve made two graphics, one for the sine/cosine interpretation, which uses two discrete groups, and a separate one for the two continuous groups.

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Figure 1. The Discrete Interpretation of Vibration

Figure 2. The Continuous Interpretation of Vibration  

It should be clear that the relative phases are the same in both cases; that is, in the reference systems of both groups, 1, and -1, is the inverse of 0, or -0.  While the concept of -0 may seem strange to those not familiar with the RST, it’s an elementary concept in the new system.

The difference that the new found groups make, one under addition, the other under multiplication, is the two, inverse, reference systems they constitute.  When the discrete reference system is used, the units are displaced from unity, so 0 is interpreted as zero displacement, to the left (negative) or to the right (positive) of unity.  When the continuous reference system is used, however, units are displaced from 0, and there is no left of negative 0, or right of positive 0 (even though these “directions” were invented by mathematicians, but that’s another part of the story), so 0 should be interpreted as 0 parts of a whole (i.e. “positive” and “negative” parts don’t make sense in the continuous reference system.)

In the SHM of the springs, the point of equilibrium is naturally interpreted as a 0 point of the discrete reference system (its identity element), where either positive or negative displacement is possible, because this kind of reference system is necessary to identify the SHM of the two springs, with the two functions of the angle of a single rotation.  However, we can now see that a point of equilibrium only applies to a system with a restoring force, supplied by the spring tension in this case, and by gravity, in the case of a pendulum.

But consider the case of two, counter, rotations. In this instance, no restoring force is involved. One complete rotation, from 0 to 1 and back to 0, takes place in the continuous direction of rotation, with no reversals, and if there are two such rotations, rotating in opposite directions, the inverse phase relationship constitutes SHM just as surely as it does in the discrete reference system, but it does so as a double rotation, not a single rotation.

Therefore, we conclude that SHM comes in two, reciprocal, forms.  In the discrete form, 0 constitutes the origin of a single reference system, and in the inverse of this form, 0 constitutes the two limits of a double reference system. The reciprocal nature of the two reference systems is illustrated in the graphic below.

Figure 3. Inverse Reference Systems of Discrete (top) and Continuous (bottom) Groups.

Larson was the first to identify these two reference systems and to apply them to the development of RST-based physical theory, but he did so without the benefit of the knowledge of their respective mathematical groups, and the tremendous power they afford the developer. With this much understood, we are prepared to move forward like never before. 

Update, Jul 23:  I’ve updated the graphics to show the correct phases and edited out the text explaining the incorrect phases.

Note: I originally posted this as an addendum to the previous post below, but decided to make it a separate entry instead.

I finally got to sleep. Now that I’m rested, I have the energy to make a graphic to illustrate the point better. In the figure below, you can see that the key is understanding the difference between the two representations of the reciprocal groups.  Historically, the oscillation of two springs, 180⁰ out of phase, has been interpreted in terms of the sine and cosine, the discrete group. The oscillation of one spring is mapped to the sine function of the angle of rotation, while the other is mapped to the cosine function of the angle of rotation, but, now, we can see that this hides a very important fact: The duality of operationally interpreted RNs provides for a reciprocal interpretation of the magnitudes involved.

Since the continuous group is the inverse of the discrete group, like the “young lady” | “old lady” picture, while the two magnitudes of the spring oscillations can be mapped to the discrete group, they are actually not magnitudes of the discrete group, but magnitudes of the continuous group. Therefore, the two oscillations are equivalent to a total 720⁰ of rotation, mapped to 360⁰ by virtue of the fact that the inverse relationship of the discrete and continuous groups are not recognized!  Well, until now that is. You heard it here first, folks!

Figure 1. Explanation of Quantum Spin’s “Inexplicable” 720⁰ “Rotation”

Of course, I’m not exactly putting on my tuxedo, because there is this huge gulf between the LST community, and the RST community, that will prevent any invitation to Stockholm, during my lifetime. However, I want to take this opportunity to publically thank a most generous and kind man who has believed in me and supported this work from the beginning.  His name is Jesse Grier, and if it weren’t for him, and his unflagging support, this moment would have never arrived.

Thank you Jesse.

I know that I promised to finish with the posts on the gauge principle, but in the meantime, I’ve been distracted by a couple of things. They are related.  One is the discussion going on at the ISUS discussion forums, and the other is, as a result of the ISUS discussion, I’ve started to write the book on all this stuff.

Now, however, it’s almost midnight and I can’t sleep for the thoughts about spin that are running through my head.  With the new understanding of the discrete and continuous number systems that are the result of the two groups of RNs, one group under addition, isomorphic to the integers, and the other group under multiplication, isomorphic to the non-zero rationals, the explanation of quantum spin, which is so enigmatic to the LST community, turns out to be just as simple as it can be.

During the course of the ISUS discussion, I tried to illustrate the connection between the scalar vibrations of the SUDRs and TUDRs and rotation in the complex plane, using the equations of GA, implemented as a simulation that I borrowed from a GA website.  I showed 135 degrees of bivector rotation, in the graphic below.

Figure 1. Expanding/Contracting Sphere in Geometric Algebra 

At first, I thought this was really a convincing demonstration of the equivalency of SHM and the spherical expansion/contraction cycle, but it turns out that it isn’t, not at all.

However, when I realized why it wasn’t, everything fell into place, and the long mystery of what quantum spin actually is was solved at last.  You see, the problem with using GA equations, where the inner and outer products represent the sine and cosine of the angle of rotation, is that the trivector goes through the contract/expand cycle four times as the bivector rotates through 360 degrees.

When I first realized this, I was really embarrassed, and hoped that nobody would catch it on the ISUS site.  I was really puzzled though, what is wrong here? I thought I knew that the SUDR cycle had to be one cycle per revolution, synchronized with the bivector rotation; that is, one complete expansion over the first 180 degrees of rotation, and one contraction over the second 180 degree rotation, or one cycle per 360 degrees of bivector rotation, not four cycles! 

Then it dawned on me. The GA equations are equivalent to the complex rotations, but the SUDR cycle isn’t.  It’s equivalent to one half of SHM! You can easily see this, when you compare the SUDR expansion, from a point to the unit sphere, to the extension of a spring mounted weight.  As the SUDR expands from the point to the sphere, the spring goes from fully compressed, to fully extended, and as the SUDR contracts from the sphere to the point, the spring goes from fully extended, to fully compressed.  When the SUDR and TUDR are combined as an S|T unit, the two, inverse, oscillations constitute SHM, just as two springs do, when their oscillations are 180 degrees out of phase.

One complete, expansion/contraction, cycle, then, is equivalent to one 360 degree rotation, and it takes two, inverse, rotations, coupled together, to constitute one cycle of SHM, just as it does for the extension/compression cycle.  The reason this is so is now very clear as well: The 720 degree spin is the equivalent to 360 degree rotation, but it is continuous motion, not discrete motion.  Recall that discrete motion has two components, each passing through zero, between positive and negative one, while continuous motion has two components, but each passes through one between positive and negative zero. Duh!  One is the inverse of the other, as we’ve been discussing.

The set of meshed, counter-rotating, gears is the perfect representation of this, as I’ve already noted, but I didn’t make the mathematical connection with rotation in the complex plane, until this came up, but look how easy and plain it is to see:

1. 1 complex rotation of unit circle: -1 <——> 0 <——> 1 (sine)  -1 <——> 0 <——> 1 (cosine)
2. 1 SUDR|TUDR exp/cont cycle::  - .5 <——> 1 <——> 2 (SUDR)  -2 <——> 1 <——> .5 (TUDR)
3. 1 spring1|spring2 ext/comp cycle: - .5 <——> 1 <——> 2 (sprg1)  -2 <——> 1 <——> .5 (sprg2)

The expansions of the spheres, like the extensions of the springs, never pass through 0, at 90 degrees, as they do in the graphic of figure 1 above.  Instead, at the half-way point, they have doubled their initial value (their “0”), from .5 to 1, and the contractions of the spheres, like the compressions of the springs, never pass through 0, at 90 degrees. Instead, at the half-way point, they have halved their initial value (their “0”), from 2 to 1. Meanwhile, the sine and cosine have both passed through 0 twice, once each for every 180 degrees of rotation.

So, unlike the trivector/bivector relationship. or the sine/cosine relationship, the expansion of the SUDR to a spatial sphere is equivalent to 180 degrees of rotation, not 90 degrees, and the contraction of the TUDR to a temporal point, at the same time, is also equivalent to a 180 degrees of rotation. Thus, quantum spin is not two 360 degree vector rotations around the unit circle, but two simultaneous, inverse, scalar expansions/contractions, totaling 720 degrees of equivalent rotation, in every cycle.

Moreover, now that we can understand quantum spin, in such simple terms, we should be able to cut through quantum mechanics like a hot knife through butter. Not only this, but we can also see that comparing amplitude modulation to frequency modulation of a carrier signal, in terms of octave changes, now provides us the same type of association between them. I suspect that this will enable us to tie M₂ and M₄ motion together, through M₃ motion, just as the Chart of Motion suggests we should be able to do; that is, we can get one integrated picture of classical and quantum mechanics.

Man this stuff is exciting. No wonder I can’t sleep.