The Larson Research Center

In the standard model (SM) of particle physics, quarks come in three colors, red, green and blue.  It’s not that the colors mean anything.  They don’t, but just as the three primary colors mix to white, the three quark colors mix to neutral color, which many times is conveniently glossed over in discussing quarks.  In Bilson -Thompson’s (BT) paper, his braids of three helons are uniquely ordered to represent the 3 possible colors for each of the four quarks.  We adopt a similar arrangement, as shown in figure 1 below:

 

Figure 1.  Three Colors of Quarks

To facilitate identification of the quark colors in the scheme, the S|T positions in the triplet have been numbered and the colors of the inner terms of the S|T units have been denoted as positive, negative, and neutral charges.  Notice that, in the red column of quarks, the “odd-man-out” (S|T # 3) is to the right of the two like-color charges, while, in the green column, the “odd-man-out” (S|T # 2) is in the center, and it (S|T # 1) is to the left of the like-color charges in the blue column.  This scheme is different than the BT scheme, but it correlates with our color convention of placing the color red “below,” to the left, of the color green, and the color blue “above,” to the right of the color red and green, just as 1 is lower than 2 and 3 is higher than both of them, so we put 1 to the left of 2, and 3 to the right of 2, to indicate the ascending order.

The first reaction to this concept of rearranging the locations of the colors in the triplet is that it is meaningless, since the orientation of the S|T units in the triplets is invariant under rotation, just as the geometric triangle is also. However, while this is true, it is also true that, relative to one another, the different positions have meaning; that is, while there is no absolute distinction, there is a relative one, which represents the relative orientation of the constituent S|T units of one triplet, relative to the orientation of the constituent S|T units in a second and a third triplet.

In the SM way of combining 3 quarks to form a first generation hadron, a proton or neutron, the resulting hadron must be color neutral. That is to say, one triplet in the hadron must have a “red” orientation, one a “green” orientation, and the third a “blue” orientation.  In the braids of BT, this is accomplished by “stacking” the braids.  He writes in his paper, A Topological Model of Composite Preons that the color interaction can be physically represented as a “‘pancake stack’ of braids” that can be added together as permutation matrices:

We may similarly represent colour interactions physically, this time as the formation of a “pancake stack” of braids. Each set of strands [in a braid] that lie one-above-the-other can be regarded as a “super-strand”, with a total charge equal to the sum of the charges on each of its component strands. If we represent braids as permutation matrices, with each non-zero component being a helon, we can easily represent the colour interaction between fermions as the sum of the corresponding matrices…Hadrons can therefore be regarded as a kind of superposition of quarks. 

In our theory, we combine the three triplets as a double tetrahedron (see previous posts below).  If we now introduce the quark color constraint that requires the hadron to be color neutral, we can combine them by summing the positions marked 1, 2, and 3 together; that is, we can sum the charges of the ones, twos, and threes of the three triplets, and, as shown in figure 2 below, the summed color charges in each position of the hadron are then sure to be identical.

              

Figure 2. First Generation Color-Neutral Hadrons

Notice that, while the color-charges add up to 1, positive, unit, in each position of the proton hadrons, they add up to 1, neutral, unit, in each position of the neutron hadrons, and that all three hadrons are color-neutral in each case, and, of course, that the electrical charges add up to positive 1 for the proton  (udu), and neutral for the neutron (dud).

Figure 3 below shows the double tetrahedron form of the grb neutron of the top row of figure 2.

                

Figure 3. GRB Neutron Hadron as Double Tetrahedron (Top and Bottom View)

Here, the inner terms of the inner triplet are numbered in the clockwise direction also, from 1 to 3, and the inner terms of the outer triplet are denoted with a and b designations to indicate that they are one and the same terms in the top and bottom view. The top half of the double tetrahedron (shown on the left) depicts the red up quark combined with the green down quark, while bottom half (on the right) depicts the same red up quark combined with the blue down quark, placing the up quark between the two down quarks. It’s interesting to note that, while this “stacking” of quarks is similar to the stack of braids in the BT model, the physical binding of the quarks into hadronic triplets in our model, like the physical binding of the S|T units into quark and lepton triplets, is not contrived, but has an actual physical basis.

Moreover, since the double tetrahedron has five nodes, as opposed to the three nodes of a stack of three braids, two new degrees of freedom are introduced into these hadrons.  The implications of this are now being investigated and the results will be discussed in a future article.  Stay tuned!

 

I think it might be a good idea to backup at this point and recapitulate the physical theory we are developing, based on the two fundamental assumptions of Larson’s new physical system, the Reciprocal System of Physical Theory (RST) .  We start off with a unit progression of space and time that we can represent mathematically with the rational numbers of the Reciprocal System of Mathematics (RSM).  The RSM is based on an operational interpretation of rational numbers, and recognizes that the difference between the operational and quantitative interpretation of a rational number enables us to interpret a fraction, not as a part of a whole number, but as a whole number, and these whole numbers have two “directions” naturally; that is, they form a system of integers, where the negative integers are the inverse of the positive integers, represented as rational numbers, operationally interpreted.

On this basis, 1/2 is the inverse of 2/1, where 1/2 has a value of -1 and 2/1 has a value of +1, relative to 1/1. Of course, we can still reinterpret these numbers, quantitatively, when the need arises, by changing the reference datum of the value proposition from 1/1 to 0, where zero is the absolute value of nothing.  In other words, quantitatively interpreted, the value of 1/2 lies between 0 and 1, and this interpretation is appropriately expressed as a decimal number, .5.

The significance in taking the RSM approach to numbers is not to be understood in terms of advocating a change in the traditional way of using numbers in physics, but in opening up the possibility for investigating a new way of thinking about physical magnitudes.  In the RST, we assume that the entire structure of the physical universe is a structure of space and time; that is, the observed universe is not assumed to be a structure of space and time coordinates in which matter, radiation, and energy are contained and interact, via mysterious forces of interaction, causing objects to move, generating momentum and heat.  Instead, all constituents of the universe are regarded as forms of space and time, or motion, and there is no container.

The starting point for this universe of motion is a unit expansion of space and time that forms the unit datum of magnitudes in the universe.  This datum is dynamic, because it constitutes a ratio of change and, while it is represented by the number 1/1, it is a ratio of two rates of change, space/time = ds/dt.= 1/1.  However, to indicate the new operational interpretation of this ratio, as contrasted with the traditional quantitative interpretation, we drop the division symbol, “/”, and replace it with the pipe symbol “|” that indicates that this is not a numerical quotient, but a reciprocal relation between the expansion of space and the expansion of time.

Consequently, this approach is tantamount to redefining space as the reciprocal of time in the equation of motion and assuming that the entire structure of the physical universe proceeds from this definition.  Of course, this is not possible unless there is some possibility for a physical change in the unit ratio to occur, just as we would have no other numbers than 1|1, if both the numerator and denominator could only be changed together: The progression 1|1, 2|2. 3|3, …n|n is a uniform and unchanging unit progression.  The only way to derive anything interesting from this progression is to introduce a change of some sort.

For instance, If we change the numerator in such a way that it increases twice as much as the denominator, with each change, so that the progression is no longer a unit change, but is now 1|1, 4|2, 6|3, …2n|n,  then we have changed the rate of progression to ds|dt = 2|1, which is a positive unit of change in the progression rate. Similarly, to form a negative unit rate in the progression, we need only to change the denominator twice as much as the numerator, with each change, so that the progression is changed from 1|1 to 1|2, in the other “direction,” 1|1, 2|4, 3|6, …n|2n, which is a negative unit change in the progression rate, ds/dt = 1/2.

Physically, the only way to affect this change in the unit progression rate is to assume that the “direction” of the progression of one aspect of the motion changes by one unit each two units of progression, while the reciprocal aspect continues to increase uniformly.  Thus, to get a change from the unit rate of progression, we assume that one aspect of the progression reverses its “direction,” from increasing, to decreasing, and back to increasing, continuously alternating the “direction” of its progression, which in effect cuts in half its rate of increase, relative to the rate of its reciprocal aspect’s increase.

For instance, if the space aspect of the progression begins this alternating pattern at a given location, it would increase, or expand for one unit, and decrease, or contract, for one unit, while the time aspect increased uniformly for two units, or twice as many units.  Hence, the total units of progression is two units in each case, but only the time aspect has increased two units, while the space aspect has decreased and increased one unit, thereby changing the progression ratio of increase, from ds/dt = 1|1, to ds|dt = 1|2. 

By the same token, if the time aspect of the progression begins the alternating pattern at a given location, the space|time ratio at that location becomes ds|dt = 2|1.  These two, non-unit, rates of space|time progression are the fundamental units of the LRC’s RST-based, physical, theory. In this theory, the negative unit is designated the space unit-displacement ratio (SUDR), and the positive unit is designated the time unit-displacement ratio (TUDR). 

Our theoretical development differs from Larson’s RST-based physical theory, in that the space|time “direction” reversals are treated differently. In our new theory, these fundamental units of motion are not rotated in various ways, as they are in Larson’s development, but are simply combined together in various combinations, in a way that is very similar to the way that Sundance Bilson-Thompson, the Australian physicist, has combined twists of space|time “ribbons,” in his topological toy model of preons, the name given to LST theories of sub-entities, forming the fermions of the standard model (SM).

In the Bilson-Thompson (BT) model, the magnitude of the twists are measured in terms of ± π radians.  In this way, a “left” twist, rotating the ribbon - π radians, represents a magnitude of a half-cycle of rotation, while a “right” twist, rotating the ribbon + π radians, represents a magnitude of a half-cycle of rotation in the other “direction.”  Therefore, combined together, two left twists represent a magnitude of a full-cycle magnitude of rotation in one “direction,” while two right twists represent a full-cycle magnitude in the other “direction.”  Combining a left and right twist together creates an effective zero magnitude.

Lee Smolin and Fotini Markopoulou are now collaborating with BT in an attempt to take his ideas from a toy model of topography to a physical theory of spacetime quantum gravity with “local excitations that can be mapped to the first generation femions of the standard model of particle physics.” (see http://www.arxiv.org/ftp/hep-th/papers/0603/0603022.pdf)

However, the important thing for us is that the twists of spacetime “ribbons” in the BT model correspond to the space|time contraction/expansion, the SUDRs and TUDRs, in our theoretical model, where a half-twist corresponds to half of a contraction/expansion cycle of a SUDR or TUDR.  In the BT model, the twists, referred to as “tweedles,” can be combined into what BT terms “helons.”  He writes:

The three possible combinations of [tweedles] UU, EE, and UE = EU can be represented as ribbons bearing twists through the angles +2 π, −2 π, and 0 respectively. A twist through ±2 π is interpreted as an electric charge of ±e/3. We shall refer to such pairs of tweedles as helons (evoking their helical structure) and denote the three types of helons by H₊, H₋, and H₀.

Thus, clearly, the topological magnitudes of helons in the BT-model correspond to the space|time magnitudes of the SUDR (H_-), TUDR (H₊), and the SUDR|TUDR combo (H₀), in our theory.  However, in the BT model, only H_o helons are combined with H₊ and H_- helons, and the respective magnitudes are magnitudes of M₂ motion. The two positive and negative helons are never combined, because combining two instances of M₂ motion, produces no motion.  In our RST-based theory, on the other hand, the combination is a result of the mathematical relationship of the SUDRs and TUDRs, and, as we have shown below in the world line charts, the combination of a SUDR and TUDR produces the unit magnitude of (1|2)+(1|1)+(2|1) = 4|4, M₄, motion, not the zero magnitude of (-1)+(1) = 0, M₂, motion. 

Accordingly, it is the combination of SUDRs and TUDRs into the S|T combos that produces the S|T entity, which is analogous to the H₀ helon in the BT model, but differs in that it represents unit motion.  Adding SUDRs and/or TUDRs to the fundamental S|T combo then creates the three types of S|T combos, as balanced in two “directions” (green), as unbalanced in the negative “direction” (red), and as unbalanced in the positive “direction” (blue).

Combining the three colors of S|Ts into triangular triplets then produces the first generation of fermions and bosons of the SM, as discussed in the previous posts below, following the patterns of helons in the BT model, developed with the help of group theory.

However, as far as I know, neither BT nor his collaborators have extended their work to the second and third generations of fermions as yet, but, as S|T combos are easily extended into higher dimensional objects, doing this in our “preon” theory, looks very promising.

Obviously, combining the entities of the S|T version of the SM, such as quarks and leptons into hadrons and atoms of the periodic table, so that they come out with the appropriate qualitative and quantitative properties of these subatoms and atoms, is the task of the LRC’s Microcosmic Research Division.  Working with the consequences of these developments in the field of condensed matter physics is the task of the Macrocosmic Research Division, and working with the consequences with all of these in the field of cosmology is the task of the Macrocosmic Research Division, with all of these divisions depending upon the work of the LRC’s Mathematics Research Division.

Hence, regardless of whether or not a verifiably correct theory of both continuous and discrete physical magnitudes (so-called “theory of everything”) will eventually emerge from the work of the LRC, we can now clearly see that a new science has been born, an inductive science of physics that is testable and predictive, that is at once exceedingly simple and complex, and, while it is consonant with the LST science of the past, it is truly revolutionary in its concepts, systems, and results. The excitement is palpable.

 

Now that we’ve thought about how one quark might be converted into another, via W bosons, in terms of S|T triplets (actually, I didn’t mention the W⁺ boson conversion, because it’s the same, only in a different direction), we ought to go ahead and ruminate a little bit about combinations of quarks in terms of S|T triplets. Such combinations are called hadrons and the hadrons, belonging to the first generation fermions of the standard model (SM), are the familiar proton and neutron, the only stable hadrons known.

Curiously, however, the free neutron is not stable, but undergoes beta decay in about 15 minutes, outside the atomic nucleus (see previous post below). Not only this, but the quarks are never observed outside the hadrons either, but are “confined” to their bound state that constitutes the hadron. Nobody knows why this is so yet, but the observation is that the energy required to separate the quarks of a hadron exceeds the energy to produce a new quark|antiquark pair (meson). Yet, evidently, the quarks are “free to move inside” the hadron (i.e they have properties of motion, such as force, momentum, etc.), even though the “force” that binds them together becomes very strong if one attempts to separate them. This is called “asymptotic freedom” and it is part of the LST community’s quantum chromodynamics (QCD) theory for the SM.

In QCD, two up quarks and a down quark constitute a proton hadron, while two down quarks and an up quark constitute a neutron hadron. In terms of S|T triplets, these two combinations of the three quarks can take the form of a double tetrahedron, where the three nodes (ABC) of one S|T triplet each become combinations of the four SUDR - TUDR components of four SIT units, while the two opposed nodes in the third dimension are combinations of three SUDR - TUDR components, forming the two apexes of the double tetrahedron. The illustration in figure 1 shows the general idea of a double tetrahedron.

+ =

Figure 1. Two Tetrahedra Combined as One Double Tetrahedron

In our S|T combinations, the first tetrahedron is formed when a base S|T triplet is combined with its inverse, where the inverse is formed by splitting one end of a W^- boson and joining the three separated magnitudes to the ABC vertices of the base triplet. Then the same process, repeated with a W⁺ boson, on the opposite side of the base triplet, forms the double tetrahedron. However, it has to be stressed, again, that these are geometric analogs of motion magnitudes, at this point. There is no distance, or spatial length, involved in these combinations of magnitudes of motion, other than the numerical “distance” between magnitudes of opposite “direction.” Therefore, physically, these S|T combinations constitute one expanding/contracting sphere of unit radius, with superimposed combinations of space|time magnitudes, depicted by the geometric magnitudes.

Moreover, even though a geometric tetrahedron is so called, because it is a figure of four surfaces, and, therefore, the double tetrahedron is a figure of eight surfaces, for our purposes, the geometric tetrahedron represents a combination of two S|T triplets, corresponding to an even numbered hadron, the quark/antiquark pair, or meson, while the double tetrahedron is a combination of three S|T triplets, corresponding to an odd numbered hadron, or baryon. In order to more clearly illustrate the nodes of these combinations, we will alter the previous graphic colors of the triplets, by using the color magenta to represent the red|blue combinations of the ABCD nodes of the double triplet, and the RGB colors to represent the net color balance of a triplet’s constituent S|T units.  Figure 2 below illustrates the meson doublet in the new graphic representation.

 

      +                 =               

Figure 2.  Doublet of S|T Triplets Combined as One Tetrahedron, Representing the quark|antiquark Pair of the Meson

Notice that while the arrangement of the lines of the second triplet are the geometric inverse of the arrangement of the lines of the base triplet (the up quark), the colors of the second triplet are also the inverse colors of the antiquark, which is the relevant information.  Since it will be too difficult to accommodate the five nodes of the double tetrahedron in a single graphic that is clear enough, we will opt to illustrate it with two separate tetrahedron graphics instead.  Figure 3 below illustrates the baryon triplet that constitutes the proton, as two tetrahedra.

+ +   =     +                    

Figure 3.  Three S|T Triplets Combined as Double Tetrahedron, Representing Three Quarks of Proton

In figure 3, two up quarks and a down quark are combined into a proton hadron, represented by two halves of a double tetrahedron, not two separate tetrahedra.  In other words, the three, outside, magenta colored nodes of the two halves are one in the same nodes. They are the ABC nodes of the down quark, drawn twice, for clarity. The inside magenta colored nodes, in the two halves, are the diametrically opposed separate nodes of the two up quarks, which would be drawn in the 3rd dimension, normal to the screen, if the graphics were drawn with perspective.

Consequently, the total number of nodes, in this split illustration of the double tetrahedron, is five, the three shared magenta colored nodes, on the outside, and the two separate magenta colored nodes, on the inside, which we will denote A, B, C, D, and E respectively, where D and E denote the two separate nodes. The color balance of the nine constituent S|T units, comprising the proton, is indicated by the RGB colors of the circles on the edges of the tetrahedra, representing the inner terms of each constituent S|T unit.

Notice that the net color balance of the ABC nodes is dependent on four red|blue inputs, while the net color balance of he D and E nodes is dependent on three red|blue inputs, and that the possible combinations of magnitudes comprising these colors are constrained by the fact that each of the inputs has an inverse at the diametrically opposed node.  Investigation of the consequences of perturbing these additional degrees of freedom will be the subject of future articles.

Combining two down quarks, with an up quark, produces a neutron hadron.  Figure 4 below illustrates this combination in terms of combining S|T triplets.

+ + =  + 

Figure 4.  Three S|T Triplets Combined as Double Tetrahedron, Representing Three Quarks of Neutron

What I hope to be able to do next, or at least eventually, is show how the proton is charged positively, relative to the electron, but the neutron isn’t, and how the free neutron undergoes beta decay, while the free proton never decays.  I don’t mind admitting that this seems to be a scary and daunting task to me, but, then, I would never have believed that we would get this far with an RST version of the SM.  No, not in a million years!

One of the most heralded accomplishments of the standard model of physics (SM) is the unification of the electromagnetic and weak nuclear interactions into one electroweak interaction. According to the Wikipedia article on it, the weak nuclear interaction is mediated by the so-called W and Z bosons:

The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four fundamental interactions of nature. In the Standard Model of particle physics, it is due to the exchange of the heavy W and Z bosons. Its most familiar effect is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. The word “weak” derives from the fact that the field strength is some 10¹³ times less than that of the strong force.

Yet, the W, Z bosons are only theoretical particles. They are so short-lived that their existence can only be surmised, though this assumption works very well in calculations. According to the same Wikipedia article quoted above, the unification of these theoretical entities, with the observed photon, can be explained through the existence of another theoretical entity, the Higgs boson:

The Standard Model of particle physics describes the electromagnetic interaction and the weak interaction as two different aspects of a single electroweak interaction, the theory of which was developed around 1968 by Sheldon Glashow, Abdus Salam and Steven Weinberg (see W and Z bosons). They were awarded the 1979 Nobel Prize in Physics for their work.

According to the electroweak theory, at very high energies, the universe has four identical massless gauge bosons similar to the photon and a scalar Higgs field. However, at low energies the symmetry of the Higgs field is spontaneously broken by the Higgs mechanism. This symmetry breaking produces three massless Goldstone bosons which are “eaten” by three of the photon-like fields, giving them mass. These three fields become the W and Z bosons of the weak interaction, while the fourth field remains massless and is the photon of electromagnetism.

Although this theory has made a number of impressive predictions, including a prediction of the mass of the Z boson before its discovery, the Higgs boson itself has never been observed. Producing Higgs bosons will be a major goal of the Large Hadron Collider being built at CERN.

In the LST community’s SM, the theoretical Higgs field is a universal quantum field, the quantum of which is the Higgs boson. The concept of the Higgs field is, in some ways, similar to the unit space|time progression of the RST. It is the fundamental field of the SM, permeating the universe of matter, just as the unit space|time progression is the fundamental motion of the universe of motion. However, whereas the symmetry of the unit progression is “broken” or “hidden” by the “direction” reversals of the space, or time, aspect of the unit progression, the symmetry of the Higgs field is broken by the Higgs mechanism, which is a complex treatment of the principles of symmetry, as energy conservation laws, leading from one, high-energy, field quantum, to four, low-energy, field quanta.

The thinking is that the four so-called “fundamental” forces are unified at high energy, but, at low energy, the symmetry is broken and the Higgs boson is manifest as four separate bosons, the three W, Z bosons, plus the photon boson. Thus, on this basis, everything is related to a fundamental quantum of energy. In contrast, the RST approach treats matter and energy as discrete units of motion, or the reciprocal relation of the space and time aspects of a universal progression, where, at a given location, the forward progression of one aspect, or the other, is displaced relative to its inverse aspect, creating a discrete unit of matter or energy, at that location.

Consequently, in the RST, everything in the universe is a discrete instance of motion, a combination of these, or a relation between them. Force, by definition, is a property of these motions, combination of these motions, or relation between these motions. It cannot exist independently, or autonomously, as it is currently treated by the LST community, where it is viewed as an autonomous agent of change. However, by the same token, interactions between theoretical entities, resulting in the merging of separate entities into new combinations, or the separation of combinations into constituent entities, must take time, and/or space, to accomplish. Hence, such changes in the configuration of units of motion, or interactions between units, or combinations of units, or relations between units, are themselves units of motion.

A good example of this is the so-called negative beta decay, where the spontaneous decay of a neutron, into a proton, electron and antineutrino, is characterized in the SM as a result of the weak nuclear interaction, which is described as the conversion of a down quark into an up quark, when a W^- boson is emitted from the neutron that subsequently decays into an electron and antineutrino. Using the S|T triplets (see previous posts below), we can clearly see that the changes to the involved combinations of quarks and leptons are consistent, but only if the neutrino is present to explain the time and space required for the change to take place.

To see this in the triplets, we need to understand that, from the material sector perspective, the energy equivalent of the SUDR motion (red color) is “lower” than the energy equivalent of the TUDR motion (blue color), just as, from the perspective of negative numbers, -1 is a “lower” value than +1, even though, from the perspective of 0, one isn’t any lower, or higher, than the other is. Thus, as shown in figure 1 below, adding SUDR (red) motion to TUDR (blue) motion, “lowers” it towards unit (green) motion, while adding SUDR motion to unit motion “lowers” it to SUDR motion, just as subtracting weight from one side of a balance will imbalance it, if it is balanced (green), or balance it, if it is unbalanced (all things being equal).

Figure 1. Triplet Transformations in Beta Decay

In figure 1 above, the three nodes of the down quark are “lower” than the three nodes of the up quark as indicated by the color of the inner terms of the respective S|T units, because red is “lower” than green, and green is “lower” than blue. Therefore, adding red (SUDR) motion to the TUDR motion of the up quark, in the form of the W^- boson, lowers the blue nodes of the up, to the green nodes of the down quark, and the green node of the up, to the red node of the down quark, which is to say, the down quark must emit a W^- boson to convert into an up quark (again, all things being equal).

Moreover, the all-red motion of the W^- boson is identical to the all-red motion of the electron, as far as the three constituent S|T units go, except that the connections of the three nodes are different; that is, the three red magitudes and the three blue magnitudes of the boson triplet are summed into one red and one blue node, or two nodes all together, while the red and blue magnitudes of the fermion are paired together into three nodes, forming the three vertices of the fermion triplet. Thus, the space|time magnitude required to make this change in the configuration, or to convert the motion combination from a two-node boson triplet, to a three-node fermion triplet, is accounted for in the magnitude of the antineutrino triplet.

At least, that is what appears to be the case, but to be sure, we have to make the calculations, which we can do in terms of natural units as follows: Let A, B, and C denote the triplet nodes as before (see previous post below), then, assuming the minimum unit displacements,

T_d = T_u + T_w- =
(A + B + C)_d = (A + B + C)_u + (A + B + C)_w- =
{[(1|2)+(2|1)] + [(1|2)+(2|1)] + [(2|4)+(2|1)]}_d = {[(1|2)+(4|2)] + [(1|2)+(4|2)] + [(1|2)+(2|1)]}_u + {[(2|4)+(2|4)+(2|4)] + [(2|1)+(2|1)+(2|1)]}_w- =
[(3|3)+(3|3)+(4|5)]_d = [(5|4)+(5|4)+(3|3)]_u + [(6|12)+(6|3)]_w- =
(10|11)_d = (13|11)_u + (12|15)_w- =
10|11 = 25|26,

which are equal magnitudes displacement-wise, because the one red displacement unit of 10|11 is the same as the one red displacement unit of 25|26. The difference between them is only a difference in scale, or a gauge difference if you will, where

25|26 = 10|11 + 15|15,

the balanced (green) motion of the neutrino and antineutrino. However, just as in the LST gauge theory, 15|15 is indistinguishable from 1|1. Hence, the equation balances, but not without the antineutrino. The only difference between the antineutrino and the neutrino is the assigned dimensions. Where the space|time dimensions of the neutrino are space/time, the dimensions of the antineutrino are time/space, but given the balanced magnitudes of both neutrinos, the difference is academic - in actuality they are one in the same, as far as can be determined at this point, but it’s early in the game.

Some will undoubtedly find this hard to swallow, even though it seems straight forward enough.  However, gauge theory is the basis of the LST SM, and it appears that if one wishes to take issue with these developments, where the same principle is applied to our space|time triplets that is applied to the complex numbers of gauge theory, albeit in a different context, then, to be consistent, the same objection would have to be raised against the SM.

Regardless, it should be clearly recognized that we are not simply replicating features of the LST theories of the SM here.  The similarities are instructive, and illuminating, but the differences are profound indeed.  It is the difference between a universe of matter, existing in the framework of space and time, and a universe of motion, existing as a complex relationship between discrete entities of space and time.  

 

 

 

 

In the previous post, I introduced the S|T combos as preons, the LST community’s historical name for theoretical entities, composing fermions of the standard model (SM).  Sundance Bilson-Thompson has recently extended the concept to include the bosons of the SM, as well, publishing a paper on it last Summer. Lee Smolin happened to read the paper and realized that it was just what he and his colleague, Fotini Markopoulou, were seeking in order to explain the braid-like spacetime structures in loop quantum gravity (see story in New Scientist).

Smolin was so excited with these developments that he’s quoted as saying, “I’ve been jumping up and down about these ideas,” which is also an apt description of my reaction when I read about it, because, just as Smolin had been looking for something to explain his spacetime braids, I had been looking for some way to combine S|T combos.  For months, the only way I could see that they could be combined did not produce anything interesting.  Combining four S|T units into a square, or a cube, did not get us very far, because a square has as many vertices as it does sides, and the equation for the net inward or outward motion didn’t change as a result.  I just couldn’t see how to get started.

However, when I saw the braided triplets of Bilson-Thompson’s preon-inspired concept of helons, I realized I could use groups of three S|T units, instead of groups of four, and follow his lead, as I explained in the previous post below, in modeling the SM with S|T triplets.  Nevertheless, I knew that this would only give us a start, because the braided helons can’t explain the origin, or nature, of spin, mass, or Cabibbo mixing, whatever that is, and the S|T combos have to do this, or else we have nothing worth pursuing. 

Fortunately, it appears that the S|T combos (obviously, all this is very tentative, of course) should go far beyond the braids of Smolin and company, in explaining the origin and nature of spin, mass, and, who knows, maybe even Cabibbo mixing, whatever that is!  Here’s how it might work out for mass. If we denote each of the three vertices of the S|T units forming the triplet, as A, B, C, and the red (SUDR ) end of a given S|T, as “r,” the  blue (TUDR) end as “b,” and the middle term as “i” for “inward,” then we easily see that

1. A = a_r + c_b
2. B = a_b + b_r
3. C = b_b + c_r

where a, b, and c denote the three S|Ts in a given triplet.  Figure 1 below illustrates the scheme:

Figure 1.  Labelling the S|T Triplet 

Recall that the minimum value for these magnitudes is one unit.  For the blue units this is one TUDR, or ds|dt = 2|1, and, for the red units, it is one SUDR, or ds|dt = 1|2.  Each S|T unit constitutes an arbitrary combination of SUDR and TUDR units, possibly corresponding to different frequencies of bosons.  If the number of SUDRs and TUDRs in the S|T combo are equal, then the middle term is balanced and colored green, but, if they are unequal, then the middle term is colored red or blue, depending upon the direction of the imbalance.  Hence, we see that the possible values for the properties of a given triplet are determined by its quantum degrees of freedom, which depend on the inherent displacement values of its nodes, rather than on any spatial orientation.  In other words, we are attempting to use the fundamental degrees of freedom, used to describe space and time magnitudes in our RST theory, to give rise to matter, with properties, as described in the LST theory of the standard model (SM).

We can list a few of the initial displacement values in a table, as shown below.

Table 1. First Five Possible SUDR|TUDR Combinations

Using table 1 above, we can select values corresponding to the red and blue nodes of the S|T units in a given triplet, and then calculate its color.  For example, for the triplet in figure 1 above, we can pick the following values, which yield an all red combination:

1. A = a_r + c_b = 4|8 + 6|3 = 10|11
2. B = a_b + b_r = 4|2 + 5|10 = 9|12
3. C = b_b + c_r = 2|1 + 2|4 = 4|5

Since the summed values of the nodes, constituting the A, B, and C vertices of the triplet, are all less than unity, we must color them red, as shown in figure 2 below:

 

 

Figure 2.  A Red S|T Triplet

Then, the space|time value of the triplet is the sum of the values of its vertices:

T = A + B + C = [(10|11) + (9|12) + (4|5)] = 23|28,

which is the same displacement value we get if we calculate the middle value of each S|T and then sum them

1. a_i = a_r + a_b =  4|8 + 4|2 = 8|10
2. b_i = b_r + b_b = 5|10 + 2|1 = 7|11
   
3. c_i = c_r + c_b = 2|4 + 6|3 = 8|7
4. T = a_i + b_i + c_i = [(8|10) + (7|11) + (8|7)] = 23|28,
   

where, again, the subscript “i” indicates the inward magnitude of the middle term of each S|T in the triplet. In other words, this all-red triplet has a net inward space|time displacement, ds|dt = 23|28, which we are want to tentatively identify with five, natural, units of mass, but who knows.  The problem then becomes, what is charge? One possibility is that unit charge is due to the phase difference between the nodes of opposite color; that is, an electron’s charge differs from a positron’s charge, because, when its expanding, the positron is contracting, but this is going to take some work, given that there are three possible red and three possible blue nodes in each triplet.  It seems to require that all the electron’s red nodes have to be in sync, while all the positron’s blue nodes also have to be in sync, but, at the same time, opposite in phase relative to the electron. 

The question is, what happens when one of the nodes of a triplet is “the odd man out” color wise, but either in or out of phase with either its companion nodes, or the anti nodes of an antiparticle? Things really get complicated in a hurry, with just a few degrees of freedom.

Of course, we realize that we are very fortunate indeed to even have candidates for all these properties.  I think the LST community would be envious, if they knew what we have. I’m sure Lee Smolin would be jumping for joy.