The Larson Research Center

If you have been following these posts, you should understand how we are merging the two groups of operationally interpreted numbers, one infinite under addition, the other finite, under multiplication, to form a union of the two that constitutes a field under multiplication and addition.

The theoretical representations of these two groups are:

1. Representation of infinite group: The set of SUDRs, their inverses, the set of TUDRs, and the S|T identity element.
2. Representation of finite group: The set of positive S|Ts, their inverses, the set of negative S|Ts, and the S:T identity element

Recall that the S:T unit takes the form

S|T/S|T,

where the vertical pipe symbol “|” denotes the subtraction operation, and the diagonal slash symbol “/” denotes the division operation. We decided to use the colon symbol “:” to represent the division operation, and to abbreviate negative S|Ts as S and positive S|Ts as T.  A negative S|T unit is simply one that has more discrete SUDR units than discrete TUDR units, while in a positive S|T unit, the inequality is in the other direction; It has more discrete TUDR units than SUDR units.

The unusual aspect of this is that, while the identity element of the representation of the infinite group is 0, formed by the addition of any element in the group with its inverse, each such pair represents a unique form of the identity element that is a basis for a unique finite group of order |n|-1, where |n| is the quantity of discrete units denoted by the numerator and denominator of the identity element. For example, 1|1, 2|2, 3|3, …, n|n = 1|1 = 0.  Thus, we see that there is an infinite set of finite groups of order |n|-1, where n/n = 1/1 = 1 is the identity element of that group.

This is very different from the way integers are normally interpreted, where the addition of -n with its inverse +n results in a single value of the identity element, 0.  Yet, with these new reciprocal number (RN) groups, we can see this connection between the two groups, in the operation of their identity elements; that is, by selecting an identity element from the infinite group, we define a finite group of a given order, which is actually contained within the infinite group, as indexed by the unique value of the selected identity element.

To be more precise: Let a be an element of the infinite group paired with b, its inverse element; then, if c = a|b, is selected as the identity element of the finite group of order n, where n = (a -1), then there are (2n + 1) elements in the group.  For instance, if a = b = 2, then there are (2(|a|-1) + 1) = 2((2-1)+1) = 2+1 = 3 elements in the finite group, negative .5 and it’s inverse 2 (positive .5), and the identity element itself:

1/2, 2/2, 2/1 

If a = b = 9, then there are (2(9-1) + 1) = 17 elements in the group:

1/9, 2/9, 3/9, 4/9, 5/9, 6/9, 7/9, 8/9, 9/9, 9/8, 9/7, 9/6, 9/5, 9/4, 9/3, 9/2, 9/1

However, while the value of c = a|b, and the value of d = b|a, as identity elements of both the infinite group and the corresponding finite group of order n = a-1, are equal under both addition and multiplication, the values of their constituent magnitudes do not have the same sign, one is positive and the other is negative.  Hence, another, reciprocal, finite group is formed, when the identity element, selected from the infinite group, is d = b|a. The order n of this group is n = b-1, but to distinguish this group from its inverse group of order n, we denote its order as -n.

On this basis, pairing these inverse finite groups of order n and -n, we form the elements of a field under multiplication and addition, where

c/d * d/c = (c*d)/(d*c) = 0/0 * 0/0 = 0,

which is the identity element of the field (i.e. an element in the union of the infinite and finite groups) under multiplication. On the other hand, the addition operation gives the same result.

c/d + d/c = (c+d)/(d+c) = 0/0 + 0/0 = 0,

which is the identity element of the field under addition.

Hence, as long as a|b = b|a = a-b = 0, then the field identity element is zero, under both multiplication and addition, but if a|b is less than zero, then b|a is necessarily greater than zero, and vice versa. Therefore, in the non-zero case, c and d will always have opposite polarities. Then, in either case where a > b, or a < b,

c/d + d/c = (c+(-d))/((-d)+c) = 0/0 = 0..

In other words, under addition, the field element doesn’t change from 0, regardless of the size (n*1) of a and b, but, on the other hand, under multiplication, the field element does change:

c/d * d/c = (c*(-d))/(-d*(c)) = -(n*1)/-(n*1) = (n*1),

when a > b, and

c/d * d/c = (-c*d)/(d*(-c)) = -(n*1)/-(n*1) = (n*1),

when a < b. 

This, then, is the reason that we can use the “meshed gears” analogy to quantify the numerical relationship of the S|T units, at the apexes of the triplets, although it’s now clear that increasing the diameter of the gears, as the ratios change, is not the correct representation of the ratio.  There is another way that corresponds to the field values we’ve just been describing.  To see this, consider the graphic below, which illustrates how the finite groups of order n and -n are embedded in the infinite group.

Figure 1.  Finite Group of Order 1 and Order -1

In the figure above, we can see that selecting -2 and 2, from the infinite group of RNs gives us two finite groups, one generated by the identity element c, where a|b = 2|2, and one generated by the inverse identity element d, where b|a = 2|2. However, the order of the two groups should be 3; that is, the order should be 2(|a|-1)+1 = 2(2-1)+1 = 3, but it isn’t.  Instead, we get a finite group of order 11. 

Of course, the order has increased because we are using the RNs, not their integer equivalents. In the case of RNs, two out of four units are required to reach a unit value of -2, from 0, but four of these same units are required to reach 2, from -2.  In other words, the distance between -2 and 2 is four units, or 4(.5) = 2.

Thus, the total distance from 0 to -2, and from -2 to 2, or vice versa, is 2+4 = 6. Plugging this number into our equation for order gives us

2(6-1) + 1 = 10 + 1 = 11,

the number of elements in the two finite groups shown above. However, this non-intuitive result stems from the mathematical properties of the group, because we have combined the inverse elements of the infinite group to get the identity element of both the infinite group and the corresponding finite group.  Moreover, we see that the identity element of the infinite group is 0, which is now understood as the reciprocal of the identity element of finite groups; that is, 0 is the reciprocal of infinity, and discrete is the reciprocal of continuous.   This stuff is mind-blowing, but it all proceeds from the symmetry of reciprocity, the most fundamental property of the universe.

In exploiting the gauge principle, we explained in the previous post that the LST physicists use the properties of complex numbers, which were invented by virtue of the imaginary number, ‘i’, to invent a set of infinite numbers, corresponding to the infinite radii, contained in the unit circle of the complex plane.  This infinite set of complex numbers constitutes a group under multiplication, wherein elements of the group correspond to the set of infinite points along the circumference of the unit circle, which is interpreted as an interval from 0 to 1, in terms of rotation. Figure 1, below, illustrates the idea:

Figure 1. The Unit Circle in the Complex Plane

This ingenious invention effectively transforms the natural set of numbers between 0 and 1, and 0 and -1, into a corresponding set of rotations and counter-rotations, but it was accomplished unawares, because the natural set of numbers in this set, the continuous set of operationally interpreted reciprocal numbers (RNs), was not, and still isn’t, recognized in the LST community. 

Nevertheless, now that we understand how the symmetry of reciprocity works to define the discrete units of integers, and the continuous units of non-zero rationals, in the form of the dual interpretations of RNs (see here for more), there’s no going back.  Obviously, physicists can’t continue to use an ad hoc invention to formulate our most fundamental theory of physics, when a purely inductive set of numbers has been found that is free of such free inventions of the human mind.

Clearly, however, not many of this generation will be convinced that the operational interpretation of RNs is so remarkable anytime soon, so we won’t hold our breath, waiting for the rush of graduate students and post docs to apply the new numbers in building a new physical theory, based on the new system of theory and the new system of mathematics.

Nonetheless, researchers at the LRC are committed to do so.  We have the daunting task to begin afresh in developing a new, RST-based, physical theory that promises to surpass what the geniuses like Bohr, Heisenberg, Pauli, Dirac, Schrodinger, and Einstein managed to accomplish in the 20th Century, which today is regarded as the greatest intellectual achievement of that century. Just articulating such a thing is enough to make anyone want to run home, jump into bed, and assume a fetal position.

Yet, we can’t shrink from the responsibility to press on. We must press on, regardless of the intimidating prospect of what lies ahead and simply trust that we are on solid ground here and that great things are brought to pass by means that seem small to men. So let’s cut to the chase: The dual interpretations of the operationally interpreted RNs, forming two sets of numbers, one of which constitutes an infinite group under addition, the other of which constitutes an infinite set of finite groups under multiplication, allow us to do two things:

1. Add (subtract) discrete units to form combinations of units.
2. Divide (multiply) the “size one,” or unit element of the group in 1 above into sub-units of arbitrary size.

The ability to do these two things is very relevant to the theoretical universe of motion, because the only existents in this universe are units of discrete motion, combinations of units of discrete motion, and relations between these.  Thus, to develop a unified theory of the structure of the physical universe, we must use these two groups, one discrete, but infinite, and the other continuous, but finite.

What’s surprising is that the elements of the infinite group (integers) are discrete (“size one”), while the elements of the finite group (non-zero rationals) are continuous (variable size).  Of course, in the LST view, there are gaps in the set of non-zero rationals, which must be filled in with irrationals, but, as we have discussed before (see here), this too is an ad hoc invention, which in the new inductive science of the LRC, must be excluded. 

Obviously, Larson’s unit speed-displacement, which is a discrete unit of scalar motion that takes two, reciprocal, forms, one as the space|time progression ratio of ds|dt = 1|2, and the other as the space|time progression ratio of ds|dt = 2|1, corresponds to the positive and negative elements of the discrete group. We label these two ratios as the space unit-displacement and the time unit-displacement, respectively, or SUDR and TUDR.

Since the SUDRs and TUDRs are elements of the discrete group, as an infinite number of positive and negative elements in the group, they can be combined under addition, forming new units of positive and negative elements of the group, and, since for every combination of positive units, there is a corresponding combination of negative units (satisfying the inverse requirement of a group), the combinations of these inverse elements forms the identity element of the group.  Moreover, adding the identity element of the group to any element of the group, including the combination units, doesn’t change the element, and adding positive or negative elements of the group, to the identity element of the group, creates new elements of the group (i.e. subtraction works too).

In short, all the operations of combining SUDRs and TUDRs into SUDR|TUDR combo units (S|T units), are justified by virtue of the fact that the set of these discrete units of motion constitutes a group under addition. However, since, physically, SUDRs are fixed spatial locations, relative to one another that are progressing in time, and TUDRs are fixed temporal locations, relative to one another that are progressing in space, combining them creates a physical unit that is progressing in both space and time, and therefore cannot remain fixed in space or time.

Thus, the motion of S|T units is unit motion, relative to the fixed spatial and temporal locations of their constituent SUDRs and TUDRs, which can be understood in the world line charts that we’ve discussed previously (see here).

We color the SUDRs red, and the TUDRs blue, and when they are combined together in an S|T unit, we change the color to green, indicating the S|T combo, which is the identity element, but where identity is understood as meaning that an equal number of SUDRs and inverse TUDRs constitutes the combo. Hence, all equal combinations form the identity element, but this means that 1|1, 2|2, 3|3, …n|n are all the identity element.  In other words, the identity element of the group itself, 1, is infinite, but under the operational interpretation of the discrete group, it is also 0 at the same time.

This union of infinity with zero is very fortuitous and has many implications that are beyond our scope right now, but suffice it to say that the value of the same identity element, under the operational interpretation of the continuous group, is 1, not 0. The binary operation of the continuous group is multiplication (actually it’s division), but remember that the size of the unit elements of this group is not “size one,” as in the discrete group, but are the continuously variable units that can be defined inside one discrete unit, from -0 to 1, and from +0 to 1.

Thus, we see both the similarity and the disconnect between the RST groups viz-a-viz the LST groups.  However, the major, and most significant, difference between the two groups of operationally interpreted RNs, which we are using to develop an RST-based physical theory, and the two (actually three) “unitary” groups of rotationally interpreted complex numbers, used in the LST development of the standard model, U(1) and SU(2) (and also SU(3)), is that the RST-based groups are inductively derived, while the LST-based groups are free, ad hoc, inventions.

The discrete group of RNs is an infinite group under addition (actually subtraction) and the continuous group of RNs is a finite group under multiplication (actually division), but the rotational groups of the complex numbers are infinite, continuous, groups that are equated to two infinite, continuous, groups of rotation, R(2) and R(3).  What the heck is going on here?

It behooves us to understand why the LST community needs to invent an infinite, continuous, “complex space” of rotations that is equated to the infinite, continuous, “real space” of rotations, in order to get the infinite, continuous, group they need, in developing their quantum theories.

However, this is not just a question for us outsiders.  They themselves do not have the answers, which is great clue that we are on to something, in taking a fresh look at their motivations and thinking on this.  Indeed, Peter Woit has specifically pointed this out for us when he explained in a recent presentation that (see our discussion on this here):

1. The mathematics of the [standard model] is poorly understood in many ways.
2. The representation theory of gauge groups is not understood.
3. The unification of physics may require the unification of mathematics.

One indication of the problem with string theory [is that it is] not formulated in terms of a fundamental symmetry principle. What is the group?

implying that there is an accepted consensus in the LST community, which recognizes that the closure, identity, inverse, associative, etc. properties of the mathematical group are essential to the descriptions of correct physical theory.  What he only dimly understands is that, while string theorists don’t have the necessary group, the group that the particle theorists have, and that they have used as the basis for developing the standard model, is an ad hoc invention that mixes up the infinite with the finite, the discrete with the continuous, to form one unclean and confused concoction of numeric concepts.  Is it any real wonder that they don’t understand it? Duh!!!

Of course, understanding how they use their invented group to get the success of the standard model will be an important part of this investigation and in our efforts to obtain the same level of success, and beyond, in our new theory.  These guys are really, really smart. What they did can only go so far, as they are now finding out, but it’s just amazing to see how clever they are in inventing ad hoc concepts to keep their theories together.

Ironically, it turns out that we need one of their invented concepts, the concept called the gauge principle, but in a different context that transforms it from an invention to a geniune principle of scalar mathematics.  Hopefully, we will be able to see why in the next post or two.  

In the development of quantum field theory (QFT), one of the most important advances is the development of so-called gauge theories. Gauge theories are described in Wikipedia as “… a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally.” Normally, applying a scale factor to a system applies globally, to every component of that system. Otherwise, if the change of scale applies locally, where it affects only some of a system’s components, a distortion results. For example, a square remains a square when the scale of the 2D spatial coordinate system, in which it is defined, is doubled, or halved, for some reason, but if only the scale of the x axis in the system is affected, the square will be distorted and its symmetry lost.

However, a local change of scale that is a relative change can leave the symmetry unchanged, For instance, when the scale of the reciprocal B and E fields of an electromagnetic wave equation change, the change in one, is compensated by a change in the other. Such a symmetrical change in a system is viewed in physics as an immunity to change, in the words of Joe Rosen; that is, it’s a change that is not really a change, or we can say that it is the possibility of making a change that leaves some aspect of the physical state of a system unchanged, in the words of Alexandre Guay (see here).

According to Guay, in QFT, the application of gauge, or local, symmetry principles, permits certain transformations in the fields that have no empirical consequences, but the trouble is that, while these changes are the foundation of successful theory, there is no way to tell if they have any physical meaning, because they are based on unobservable quantities. In other words, the question, “What is changed under a gauge transformation?”, in these QFT  theories, cannot be definitively answered.

This state of affairs is unacceptable, because it implies that the QFT formalism, used to model the micro-structure of the physical universe, may have superfluous mathematical structure. Guay quotes Ismael and van Fraassen on this who write:

Formalisms with little superfluous structure are nice, of course, because they reflect cleanly the structure of what they represent; they have fewer extra mathematical hooks on which to hang the mental structure that we project onto phenomena.

Fortunately, in constructing an initial “formalism” of our RST-based theory, it appears quite clear that we have no superfluous structure, even though it seems that we too must incorporate a gauge principle in our theory. Whether or not this implies that the qauge theories of QFT have an actual physical meaning, we can’t say, because the motivation is not the same in both cases, but we can say that they most definitely have a physical meaning in our theory.

The motivation for introducing gauge theory into QFT has to do, naturally, with rotation.  Rotation, in LST theory, has to do with complex numbers, where the complex number represents a vector on the unit circle, a radius, if you will.  Hence, rotation in the complex plane consists of an infinite set of radii, which are specifiable in an infinite set of complex numbers, of “size one” (i.e. (a² + b²i)^1/2 = 1) and it is a set of one-dimensional numbers. Therefore, using complex numbers of “size one,” the circumference of the unit circle can be arbitrarily sub-divided, as an infinite continuum of complex numbers between 0 and 1. 

This is important, because the group properties of this set of numbers permits the binary operation of the group to be exploited to form combinations of rotations that are elements of the group, called U(1).  Since the binary operation of this group is multiplication, this means that the product of these complex numbers is actually a one-dimensional equivalent of successive rotations (a sum), in a two-dimensional coordinate system of real numbers, R(2).  In his book, Deep Down Things, Bruce A. Schumm describes the two equivalent concepts as follows:

The designation U(1) may be a bit obscure. For the set R(2) of rotations in two (real) dimensions, we read the “R” as “rotation in real space” and the “2” as “in two dimensions.”  In the case of the complex group U(1), however,  we saw that we admit the possibility of something mathematically equivalent to rotation with just a single complex number - a single complex dimension.”  Furthermore, these “rotations” involve themselves solely with the complex numbers of size one - of unit length. Thus, for U(1), we read the “U” as “the set of unit-length numbers” and the “1” as “in one complex dimension.”

The reason that the “size one” is important really comes to bear in the case of two-dimensional rotation, which requires two complex numbers, because there are three, “size one,” generators of two-dimensional rotations in the complex plane (SU(2)), which makes its algebra equivalent to the x, y, z rotations, in three, real, dimensions (R(3)).

All of this, like a particular path through any complex maze (no pun intended), has a rather convoluted explanation, but once understood, it is not that difficult to comprehend. The key to understanding it, in the context of the RST and its mathematical counterpart, the RSM, is the clarification of the confusion introduced by the historical failure to understand the fundamental nature of reciprocal numbers (RNs).

However, with that course correction under our belt (see the latest post in The New Math Blog here), the great reverence paid to the discovery of gauge theory, otherwise known as Yang-Mills theory, after the two guys who first suggested it, is understandable.

What it amounts to is that rotations with real numbers is only a group under addition, but, with complex numbers, that group can be transformed into an equivalent group under multiplication.  Both groups are infinite groups, which is what the LST physicists needed, but without the magic of complex numbers, the only set of real numbers that is a group under multiplication is the set of non-zero rationals, which is a finite group of a given order (actually an infinite set of finite groups), but this set is not understood in the LST community as a group under multiplication, but rather as a field under addition and multiplication, due to the failure to recognize the dual RN interpretations. 

Nevertheless, once it is understood that the duality of nature, reflected in the union of discrete and continuous magnitudes, that is so difficult to resolve in terms of LST concepts, is easily understood as two, reciprocal, interpretations of the ratio of RNs, things begin to fall into place much more easily.

In the RSM, there are no such things as complex numbers, in the usual sense that incorporates the ad hoc invention of imaginary numbers. Yet, the simple concepts of our RST-based theory are able to unravel the mysterious features of quantum mechanics, and explain in simple terms, what all the shouting is about, because there is a one-to-one correspondence that can be made between the physical and mathematical logic of the RST and the RSM.

In this sense, we are not trying to construct a mathematical “formalism” that cleanly reflects the physical structure of the universe, without introducing superfluous structure, but we are rather seeking to unravel the common basis of both!   In the next post, I’ll try to develop the details more.

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! 

In the previous post below, I introduced what I call the gear group of rotations, which has a representation in the set of S|T units, which comprise the preons of the standard model, defined previously. The gear group is a mathematical group, expressed as two, reciprocal, rotations, which act just like mechanical gears. The requirement for a set of numbers to qualify as a mathematical group is that it meets a short list of criteria. Wikipedia offers this definition:

In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition.

The group axioms are:

1. Closure: The result of the binary operation with elements in the set, must also be a member of the set.
   
2. Associativity: Changing the order of a sequence of binary operations must not make a difference in the result.
   
3. Identity element: There must be an identity element in the set.
4. Inverse element: Each element in the set must have an inverse element in the set.

The set of integers and the set of rational numbers are examples of mathematical groups. The Wikipedia article shows a proof that the set of integers satisfies the group axioms under the binary operation of addition:

A familiar group is the group of integers under addition. Let Z be the set of integers, {…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …}, and let the symbol “+” indicate the operation of addition. Then (Z,+) is a group.

Proof:

• Closure: If a and b are integers then a + b is an integer.
• Associativity: If a, b, and c are integers, then (a + b) + c = a + (b + c).
• Identity element: 0 is an integer and for any integer a, 0 + a = a + 0 = a.
• Inverse elements: If a is an integer, then the integer −a satisfies the inverse rules: a + (−a) = (−a) + a = 0.

This group is also abelian because a + b = b + a.

If we extend this example further by considering the integers with both addition and multiplication, which (sic) forms a more complicated algebraic structure called a ring. (But, note that the integers with multiplications are not a group)

Note that the integers do not form a group under the binary operation of multiplication. The reason for this is that the integers include zero, and thus there is no identity element under multiplication. However, in the RSM, the operationally interpreted (OI) rational numbers, which are equivalent to the set of integers, replace zero with 1/1, the identity element. Nevertheless, we have not defined the multiplication operation for OI numbers, because we couldn’t see a need for such an operation. Given SUDRs and TUDRs, we combine them into SUDR|TUDR combinations by addition. For example, one SUDR combined with one TUDR (its inverse), results in the identity element, under addition,

ds|dt = 1|2 + 2|1 = 3|3 = 1|1,

but multiplying one SUDR by one TUDR,

ds|dt = 1|2 * 2|1 = 2|2 = 1|1,

while it also results in the identity element, didn’t make any sense, from the perspective of the definition of the binary operation; that is, in scalar mathematics, multiplication simply is a short hand for stating how many instances of a number are to be added together, like stating 1 of x, or 10 of y, or 3 of z, but the idea of -1 of x, or -10 of y, or -3 of z, is just nonsensical.

Fortunately, however, the so-called non-zero, quantitatively interpreted (QI), rational numbers do form a group under multiplication, according to Wikipedia.  So, it appears that, in order to introduce the multiplication operation into the relations of S|T units, as a representation of a group under multiplication, we need to revert to the ordinary QI rational number.  At first glance, it appeared to me that this would be counterproductive, given the central role of the OI rational number in our theoretical development to this point, and so I balked at the idea, initially.

However, I soon realized that when we consider the S|T units as a representation of the gear group of rotations, we are not combining SUDRs and TUDRs individually, but only combinations of SUDRs and TUDRs, as S|T units. Since these combination units have both positive and negative elements (SUDRs & TUDRs), simultaneously, we can add additional SUDRs, and/or additional TUDRs, to existing S|T units, and we can add multiple S|T units together, because they are all elements, or sums of elements, which are included in the representation of the group of OI rational numbers (integers), under addition. Consequently, the set of summed SUDRs and TUDRs, as S|T units, represent a subgroup within the group of SUDRs and TUDRs, under addition.

But if we want to change the binary operation of this subgroup to multiplication, then the set of S|T units must qualify as a representation of the group of QI rational numbers (non-zero rationals), under multiplication, in its own right. In other words, if the gear group of rotations actually satisfies the four axioms of a group, under the multiplication operation, which we’ve yet to show, then the S|T unit representation of this group opens up an entirely new range of possibilities for describing relations between S|T units, and combinations of S|T units, such as preons.

Recall that the identity element of the gear group is the unit gear ratio, 1:1, as shown in figure 1 of the previous post. Clearly, increasing the unit gear ratio from 1:1, to 2:1, or decreasing it, to 1:2, is equivalent to adding a unit positive, or unit negative, number, respectively. To do this, we double the size of one, or the other, of the two gears. For example, we can double the SUDR gear, as shown in figure 1 below. Doubling the SUDR displacement in the S|T unit, from ds|dt = 1|2 = -1, to ds|dt = 2|4 = -2, is equivalent to doubling the diameter of the SUDR gear.

Figure 1. Unit Gear Ratio Reduced from 1:1 to 1:2

In the unit gear ratio, illustrated in the previous post below, one revolution of the SUDR gear is equal to one revolution of the TUDR gear, but decreasing the gear ratio from 1:1, to 1:2, by doubling the size of the SUDR gear, as shown in figure 1 above, means that for every revolution of the SUDR gear, now the TUDR gear must revolve two times. If we use the space and time units of the SUDR and TUDR, as a representation of the elements of the integer group, to calculate the net result of this relationship, we get (ignoring the inward units for simplicity)

ds|dt = 2|4 + 2|1 = 4|5 = -1,

which indicates that the sum at the apex is unbalanced by one displacement unit of progression in the SUDR (or negative) “direction.” In other words, it indicates that the SUDR “gear” is twice as big as the “TUDR” “gear,” at the apex connection.  But this is in terms of the ratio of the number of individual space units and time units of the OI rational numbers (integer) group, which represent net space|time displacements. If we express this same relationship in terms of the ratio of the number of SUDR units (2) and and the number of TUDR units (1), instead of in terms of the space and time units that make them up, we get a different number altogether, even though they are both expressing the same value.  The new number is a ratio of ratios.

For example, if we consider the value at the A apex of a triplet, the S|T units are reciprocal ratios of SUDR units and TUDR units, which are units of unit negative and unit positive space|time displacements, not reciprocal ratios of space units and time units. Yet, the two expressions are equivalent. Hence, we can write the value of the A apex with two SUDRs connected to one TUDR as,

S|T = 2|1 = [((1|2) + (1|2)) + (2|1)] = (2|4 + 2|1) = 4|5 = -1.

So, now that we see that we have equivalent, but different numbers, representing the same magnitude, the question is, are the properties of the two different numbers the same?  In particular, we want to know, if we have two numbers, representing S|T units, can we multiply them together, and will the result be another S|T unit that is also an element in the set, and will there be an identity element and inverse element, and so forth, forming a representation of a group under multiplication?

Of course, the reason this question is so important is because we must multiply two gear ratios, to get a new gear ratio. We only add them in determining the relative sizes of the gears. So, we need to know if the results of the multiplication operations with S|T numbers qualify them as a representation of the group of reciprocal rotations we call the gear group.

I think that the answer to this question is yes, but I can’t demonstrate it completely, yet. However, I can say some things about the differences between the two kinds of numbers, though.  Notice that the sign of the OI polarity, or OI “direction,” in the S|T unit is reversed, relative to the “direction” of the space|time ratio; that is, while more SUDRs than TUDRs is an imbalance in the negative “direction,” the greater number is on the left of the symbol, “|”, used to denote the reciprocal relation in OI numbers. This never happens in the space|time ratios, where the smaller number is always on the left, or on what we designate as the negative side of unity. Therefore, we will not use the same symbol in forming both kinds of numbers from now on. Instead, we will use the customary colon symbol, “:”, to denote the reciprocal relation of S:T numbers, instead. On this basis, we can see the equivalency of the two kinds of numbers, as follows:

1. S:T = 1:1 = ds|dt = (1|2) + (2|1) = 3|3 = 0, and
2. S:T = 2:1 = ds|dt = 2(1|2) + (2|1) = [(1|2)+(1|2)] + (2|1) = 4|5 = -1, and
3. S:T = 1:2 = ds|dt = (1|2) + 2(2|1) = (1|2)+[(2|1)+(2|1)] = 5|4 = +1.

However, multiplying two S:T numbers results in a higher, or lower, ratio, which is not the same as a higher displacement, just as a higher, or lower, dimension, obtained through multiplication, is not the same as a higher quantity, obtained through multiplication.  So the multiplication operation, while consistent in terms of the appropriate group, is not consistent across groups.  What this amounts to is the necessity of switching from the OI reciprocal number (RN) (integers in the RSM), to the QI RN (rationals in the RSM), in the S:T representation of the gear group. This is an unexpected, but welcome, surprise, because I didn’t even know that there were rationals in the RSM!

However, just as the ordinary, non-RSM based, integers are different in some important ways from the ordinary, non-zero, rationals, the QI RNs are different in some important ways from the OI RNs.  For example,

S:T = 2:1 * 2:1 = 4:1 = -3 ≠ (-1) * (-1)  = 1,

even though,

S:T = 4:1 = [((1|2)+(1|2)+(1|2)+(1|2)) + (2|1)] = 4|8 + 2|1 = 6|9 = -3.

That is to say, in the multiplication operation of the elements of the S:T group, 2:1 ≠ 2|1, even if the reverse “direction” of the polarity of the S:T number is noted. The reason for this is that the necessity of using the QI RN, in lieu of the OI RN, to accommodate multiplication, changes the value of 2:1 to the QI value of “.5”, from the OI value of “-1”.  Hence, with QI RNs,

S:T = 2:1 * 2:1 = 4:1 = (.5)*(.5) = .25,

not -3, as it would with OI RNs. Yet, at the same time, if we have a 1:2 SUDR unit, multiplied by a 2:1 TUDR unit, we also get the identity element:

S:T = 1:2 * 2:1 = 2:2 = (.5) * 2 = 1 = 1:1,

and the rest of the properties of the group axioms, under multiplication, are likewise satisfied, I believe.

It turns out then, that I think that we have discovered something new and startling: The OI RN is the reciprocal of the QI RN; that is, as Hestenes points out, following Grassmann and Clifford, it is true that there are two interpretations of number possible, but what is more, we have found that the quantitative and operational interpretations of number that they identified, are actually only two, reciprocal, aspects of the same thing.  The truth is that “half of much of something” is the reciprocal of “twice as much of something,” and any RN may be interpreted either way, depending upon the desired outcome. 

This is a profound and fundamental concept, and I will treat it in more detail on The New Math blog, as soon as I can get to it.  In the meantime, I think we have found the way to multiply our RNs. We simply need to use the “half as much” interpretation, rather than the “twice as much” interpretation, when we multiply them.