To anticipate what we have to look forward to in what follows, here’s another quotation [3]: “The topologically iconic Moebius strip (MS) is a closed ribbon that incorporates a single half-twist but it can be generalized to any (integral) number of half-twists (NHT). Flattened Moebius strips (FMS) are generalized MS with a prescribed direction of traverse and an essentially two-dimensional configuration that can take the form of an elementary, triangular planform or the contiguous composite of such configurations. The composites result from an operation called fusion in which elementary configurations are combined to produce configurations with various values of twist. All values of twist can be realized in this manner but the process is degenerate; a multiplicity of configurations can exist with the same value of NHT”.

Composites will be addressed in the next section; here we introduce the kernel of our particle model: representations of four basic spin 1/2 fermions, that is four of the solitonic MS discussed in the preceding section, each of which has, however, been flattened into a triangular planform so that it is an essentially 2 + 1 dimensional entity with time being measured normal to the planform; these four will henceforth be referred to as FMS 3.

The very influential physicist Professor John Wheeler, master of the theoretical sound bite, is reputed to have proclaimed something like “Matter tells spacetime how to curve and spacetime tells matter how to move”, a quotation commonly used to encapsulate the nature of General Relativity. However, from the point of view of this model that’s not quite right. As solitons in and of the fabric of spacetime our particles are not actually discrete “objects” that exist distinct from spacetime and they are certainly not amenable to realistic depiction as such. Nevertheless, in the interests of reasonable exposition they will be so depicted—that is to say, represented—as shown in Figure 6 and in what follows. Also the labels A, B, C, and D will be retained until such time as identification with basic particles of the standard model becomes relevant.

As per the quotation in the previous section, the boundaries of B and C are folded-over unknots with n = −1 and +1, respectively, and, similarly, those of A and D are trefoil knots with, respectively, n = ∓ 3. An all-important result is the bilateral, mirror image symmetry between B and C and between A and D (Prior to the assignment of a direction of “traverse”). As we shall see, this bilateral symmetry as well as its breaking by the assignment of a direction of traverse are definitive in the development of the taxonomy which follows.

The selection of a direction of traverse allows the attribution of electric charge to each of the four basic fermions as follows: considering B and C to begin with and following the arrows we note that B features two folds down into the plane of the diagram and one fold up out of that plane. On the other hand, C is just the opposite with two up folds and one down fold. We see an immediate correspondence to the quark constituency of the neutron and proton of the Standard Model. On this basis we shall henceforth refer to the folds as quirks (not quarks!).

Now, if we go on to assign values of 2e/3 and −e/3 to up quirks and down quirks, respectively, the correspondence is complete with B and C corresponding to the neutron with a charge of 0 and the proton with a charge of +e, respectively, under the assumption that the charge of a basic fermion is the sum of its constituent quirks (just as the charges of the nucleons of the SM are the sums of their constituent quarks). Furthermore, fermion A with three down quirks is seen to have a charge of −e, identical to that of an electron.

Note that charge assignments are not arbitrary; since we have two kinds of quirks and four linear relationships connecting them to the charges of the four basic fermions, the assignment of charges to any two of the six entities, quirks or fermions, fixes all the charges. For instance we could have fixed the charges of B and C at 0 and +e, respectively and produced the above quoted quirk charges.

Clearly, there is a linear relationship between the twist and the charge of the four basic fermions as twist varies between NHT = −3 to 3 and charge from q = −e to +2e, namely,

(2-1)

which we plot in Figure 7.

Here q is the individual charge of any one of the four and Q is an average value for the involved particles, which we note is equal to e/2. Note that twist exhibits the bilateral antisymmetrical variation about NHT = 0 due to equal numbers of twist to the left and right, while charge is antisymmetrical about the offset e/2 because of the presence of the last fermion labeled D in figure1with charge 2e. We note the resemblance to the Gell-Mann/ Nishijima formula for the relationship between charge and isospin, namely

(2-2)

where I₃ is the (third component of) strong isospin and Y is the so-called hypercharge which in the case of the nucleons is Y = B +1 where B is baryon number. Thus, if we equate NHT with 2I₃ and Q with Y/2 there is a formal equivalence between the two formulas. For example, we have with Q = e/2, NHT = −1 and +1, and q = 0 and +e, for fermions B and C, respectively (also evident from the figure) which corresponds to equating B to the neutron and C to the proton. It turns out that there is also interest in considering only the first three basic fermions (without D) in which case the average charge is 0 but the average twist becomes −1.

Finally, we note that these four fermions (and their conjugates) are the only figures possible, given a triangular planform and two quirk labels. Although there are eight combinations of two labels taken three at a time, four of the combinations are redundant, assuming no corner is singled out to break the equality of all corners. For example, given freedom of rotation in their plane, these figures all represent the same fermion:

Later on, however, when we consider combinations, the situation becomes more involved with important consequences for the taxonomy.

The matter of flattening requires more discussion. It is well known that the number of twists, say T, and writhes, W, of a knot trade off to produce the invariant called linking number, NL = T + W, where both W and T can be either positive or negative. Given the knot/MS relationships discussed in the previous section we find this tradeoff to apply to MS as well [1]. Writhing in this case means that the twists are relaxed into loops, which in the process of flattening diminish in size, ending up as the folds (quirks) indicated in Figure 6. In the case of a completely unrelaxed MS there is no writhing so we have NL = T while in the case of an FMS we have NL = W. Thus, given the invariance of NL we must have W = T (Note that both T and W can be either positive or negative). In other words all the twist has been replaced by writhing or, in the in the limit of flattening, by quirks.

From another point of view, we recognize that the linear constraint imposed above between θ and ϕ in the torus knot representation need not apply to ph ysical MS in real space as long as θ is topologically quantized [2]. That is, as long as the condition is satisfied

(2-3)

where f(ϕ) = dθ/dϕ. In the case of an FMS the change in θ occurs in a discrete manner, namely at the quirks. In fact, suppose we consider the world of ordinary experience wherein an alternative to flattening a real, tangible MS is a synthetic approach in which an untwisted strip, ribbon, belt, etc. is folded at a discrete set of points before its ends are joined. Note, that θ can change only by ±π radians at each quirk because “fold”, here, means that the ribbon executes half a revolution about an axis in the plane of the resultant FMS. There is, of course, also a requirement for closure in the plane, namely

(2-4)

where Δϕ_i is the change in the ribbon’s bearing in the plane at the ith quirk and N_q Is the total number of quirks. The quantum condition becomes

(2-5)

where

and F_i is the unit step function at ϕ_i.

Figure 8 shows the situation schematically for the case of n = +1. In the upper diagram we see a set of directed ribbon segments (physically, these can constitute the centerline of the ribbon) represented by a set of vectors connected by quirks (at the corners). The bottom diagram shows how θ varies with ϕ as a result of the impulsive rate of change that prevails at the quirks as shown in the middle diagram. In this case there are three quirks and, correspondingly, three connecting segments, which is the minimum number that provides closure. Consequently, the summation requires two up quirks and one down quirk in order to add up to n = NHT = +1. And, of course, two downs and an up for N = −1 and three identical quirks for n = ±3.

Figure 9 then shows what is meant by an “up” quirk and a “down” quirk; each vector (directed ribbon segment) can be viewed as impulsively reflecting off a wall in the plane (the plane of the FMS) with the direction normal to the plane.

To discuss composites we need one more definition: associated with each of the four fermions in Figure 6 is an antifermion, an example of which is the one associated with fermion C and labeled C* in Figure 10. As shown in the figure, antifermions are distinguished simply by a reversed direction of traverse. As per the discussion in the preceding section, we must also consider the corners of the antifermions to be antiquirks, each conjugate to the charge of its corresponding quirk. Thus the charges of the antiup quirk and the antidown quirk are −2e/3 and e/3, respectively so that the overall charge of an antifermion is conjugate to that of corresponding fermion.

As will be seen, this way to distinguish fermions from antifermions is essential for the development of a taxonomy because of the requirement for continuity of traverse in forming composites by combinations of fermions and antifermions. However, it is also implied by the well-known Dirac theory that indicates the existence of antimatter. (See Section 3.4). Furthermore it is found to be justified by the solitonic nature of (2, n) knots and the associated MS [4] (and see Appendix A) and, as shown below, by the well-known Wheeler-Feynman notion that an antiparticle looks like the associated particle moving backwardsin time.

By way of justification, to begin with, we reiterate a point of view mentioned in passing in the introduction of the model: our FMS are to be regarded as occupying a 2 + 1 spacetime wherein the out of plane dimension is time. As a consequence the discrete charges of the quirks are associated with steps in time. One of the ramifications of that point of view is that our FMS automatically manifest the invariance of the CPT product (charge, parity and time) to changes in orientation. Also, we can demonstrate the Wheeler-Feynman notion in terms of quirks and antiquirks.

To demonstrate CPT invariance, consider for example the d quirk portrayed in Figure 11 and the accompanying reference coordinate system. Parity here implies the orientation of the direction of traverse as portrayed by the arrows, counterclockwise, as shown is (+) and clockwise is (−). In terms of orientation, we will consider only rotations by 180 deg. about each of the portrayed axes. Axis x which represents time, here, is to be interpreted as pointing out of the plane of the diagram

1. It is reasonably clear by inspection that rotations around the x (time)-axis—that is in the plane of the diagram—have no effect on either C, P or T.

2. A rotation around the y axis produces Figure 12.

We see that the attributes C, P have changed but not T; time remains as defined. That is, if we list the changes as

and if we associate the value of −1 to a changed attribute and +1 to an unchanged attribute we see that the products C*P*T and CPT are equivalent—i.e., the rotation produces no change in the CPT product.

3. A rotation about the z axis produces Figure 13 and, upon inspection, we find exactly the same result: the new product is again C*P*T = CPT.

We conclude that the CPT product of a quirk, and therefore of an FMS, is invariant to orientation in spacetime.

There is a companion conclusion to these deliberations: consider, for example, the conjugate of Figure 11, that is a d* quirk, which we portray simply by reversing the traverse arrows as shown in Figure 14.

Now consider how that figure looks as seen from the back of the diagram: as per Figure 15 we see that it is also a d quirk (in fact the conjugate of Figure 11), which we interpret as a manifestation of the Wheeler-Feynman notion in terms of quirks and antiquirks (and consequently in terms of the fermions and antifermions of the model).

Another ramification is the relationship of quirks to instantons. The treatment here is strictly heuristic; we consider four kinds of instanton characteristics, labeled I, AI, RI and RAI in a two-dimensional (1 + 1) spacetime as shown in Figure 16. Although the mathematical considerations are more involved, in essence, the prototypical instanton characteristic labeled I here, is just a step change between two eigenstates (which we label here as residing in the spatial dimension) that occurs in a very short interval of time, and the label AI stands for the anti-instanton characteristic with a step in the opposite direction [12]. Customarily, only the two characteristics I and AI are discussed; here, characteristics RI and RAI are introduced strictly by reasons of symmetry, RI to portray an instanton progressing in negative time and, similarly, RAI to portray an anti-instanton progressing in negative time.

Now consider the same four diagrams relabeled as in Figure 17 such that t→ϕ and s→θ where ϕ and θ are the azimuthal and meridianal angles in the FMS discussion of Section 2; these four diagrams thus portray the two basic quirks and the corresponding antiquirks. That is we have the correspondences

In other words, our basic set of quirks and antiquirks are isomorphic to the above set of instantons and reverse instantons. One implication to be drawn from this correspondence is that, to the extent that our model of the basic fermions reflects reality, the postulated instanton characteristics, RI and RAI, would do so as well (and conversely).

Finally, as shown in [4] and summarized in Appendix A the result of investigating the behavior of a torus knot under the influence of General Relativity is that it obeys the Sine- Gordon equation

(2-6)

where to first order in r/R , R and r being the radii of the toroidal center line and the toroidal core, respectively andμ = n/m. The solution, well known to describe solitonic behavior (cf.[13]) is

(2-7)

Most of the variation in θ has the shape of a sharply rising S-curve, confined mainly to a length increment Δl = 1/η, which, as a fraction of the length of an actual MS, is Δl/2πR and is expected to be quite small.

Reference [4], goes on to “derive some measurable quantities” beginning with the conversion of the Sine Gordon equation into a dynamic form,

(2-8)

where m is a mass, a is an acceleration and F is a force (see the appendix for definitions of these quantities). However, what is of interest here is the relationship between traverse to the right and to the left in the figure of reference (Figure 1) in this case) in terms of what it is that does the traversing. As we see from the above, it is the solitonic torsional disturbance of limited extent that does so. Since, by implication, it is only traverse to the right (e.g., as per Figure 1) that has been the subject of the analysis leading to the Sine Gordon equation, we see that Equation 2-8 should really be expressed explicitly as

(2-9)

in order to emphasize that it applies only to traverse to the right. The corresponding equation for traverse to the left can then also be written in the same form, that is, to conform to the same dynamic formalism, with L substituted for R, i.e., as

(2-10)

However, as shown in Appendix A, this is true only if we stipulate that the two mass terms are related by

(2-11)

Thus we see here not only a physical manifestation of the concept of an antiparticle with negative mass but a physical justification for labeling an MS with traverse to the left as an “antiparticle”.

First, we selectively recall that part of the quotation that introduced the previous section:

The all-important topic of degeneracy will be treated in detail later on but here we begin with a discussion of the fusion operation, which as we shall see, is the main ingredient in developing a taxonomy of the composites mentioned above. Fusion always joins a fermion and an antifermion in a manner that maintains continuity oftraverse. More specifically, it can take place only between a quirk of the fermion and a corresponding conjugate antiquirk of the antifermion.

Figure 18 presents detailed views for both the free and the fused conditions for both allowable situations, that is either d-d*mating (call that an “x” junction) or a u-u*mating (call that a “y” junction) with a “plan view” at the top and an “edge view at the bottom. In the free condition the edge view shows the connection being made “vertically” (out of the plane of the planform) and in the fused condition it is made horizontally. For fusion to occur the vertical connections must be disenabled and the horizontal connections enabled. Similarly, the freeing-up of an originally-fused pair, an operation we refer to as fission, implies the converse operations of enabling and disenabling of the vertical and horizontal connections, respectively.

We note the fundamental dichotomy—connections are either horizontal or vertical—a situation reminiscent of the Kauffman Bracket [8] which calls for a short digression: first we define state functions for the horizontal and vertical connections of a quirk-antiquirk pair, <Ψ_α> and <Ψ_α>, respectively, where the value of α is either d or u. Also, we define A and A^† to be the disenabling and enabling operators4, respectively. Then, the fusing of a junction between an originally free FMS and antiFMS pair can be expressed in terms of a state function as

(3-1)

Also, we define the value of <Φ_α> to be (z_α) such that z_d = x and z_u = y, respectively. Similarly, the operation of fission can be expressed in terms of a similar state function as

(3-2)

These functions turn out to find expression as coefficients in a tensor formalism [2] that describes the operations of fusion and fission, which implicate this model’s identity as a Hopf algebra [14]. Appendix (C) discusses that connection in more detail but, briefly, we need to associate the process of fusion with the Hopf algebraic operation of multiplication and the process of fission with the operation of comultiplication. This gives us two essential components of a bialgebra, and with the inclusion of what’s known as an antipode, the essential elements of a Hopf algebra [14].

For comparison, here is the Hopf algebra’s diagrammatical way to picture the two operations of multiplication and comultiplication: Diagrammatically, the two sets of operations—those in Figure 18 and Figure 19—really say the same thing: on the one hand two entities combine to make a third entity and on the other, one entity splits in two.

In the Hopf algebra these operations are representative of corresponding tensor operations on members of vector spaces [14]. From the point of view of our model (See section 3b, below), the inputs f and represent fermions and antifermions, also members of vector spaces V and V*, respectively, and the Bs are bosons, members of a certain matrix, M. One requirement for relating our model to a Hopf algebra is therefore to model the fusion and fission operations in a tensor formalism. As carried out in [2] that task is summarized in Appendix C with the result that fusion and fission can, indeed, be recognized as manifestations of the Hopf algebra operations of multiplication and comultiplication, respectively.

The antipode is also discussed in the appendix; while its definition and translation from Hopf language into model language is somewhat involved, it turns out that the model version of the antipode involves the composites resulting from fusion in a straightforward manner. The conclusion is that our model qualifies as a Hopf algebra (or, equivalently as a quantum group).

As to those compositional products, two examples of what we shall designate as first order fusion are shown in Figure 20 (in this case it turns out that these are mesons, or more explicitly, pions).

On the left we see fermion C fused with the conjugate of fermion B and we note that the continuity of traverse is automatically maintained; we see traverse circulating around the “figure eight-like” configuration that results in each case, counterclockwise around the fermion and clockwise around the antifermion. The composite particle on the right is the conjugate of the composite on the left. That is we have the usual algebraic relationship

(3-3)

and, geometrically, the pair also illustrate the Wheeler-Feynman notion with regard to motion either forward or backward in time.

Since, as readily demonstrated, both algebraic twist andcharge arepreserved in fusion, the result is a boson with even values of twist and spin. Looking ahead a bit, if we equate B and C with the neutron and the proton, respectively, the particle on the left, representing the composite CB*, is the way the pion π⁺ is modeled herein, namely as a bound particle composed of a proton and an antineutron. Similarly, the one on the right represents the π^- as composed of a neutron and an antiproton. Consequently, our model yields charge values of +e and −e, respectively, as it should, and twist values of 0 for both, also as it should if we associate twist with isospin as discussed in the previous discussion. The algebraic reason that each of the two pions has a vanishing twist is that the composite is the sum of two components with opposite twist in each case. The physical reason is that, although each composite can be formed by fusion (as per the above quotation) each can also be regarded as an untwisted closed band that has been subjected to an additional half twist. 5

The matter of creating bound particles and, in particular pions, is also interesting in a historical context: in 1949, Enrico Fermi and C. N. Yang wrote a paper entitled “Are mesons elementary particles” [15]. The introduction to the paper contains the following statement: “We propose to discuss the hypothesis that the π-meson may not be elementary, but may be a composite particle formed by the association of a nucleonand an antinucleon. The first assumption will be, therefore, that both an anti-proton and an anti-neutron exist, having the same relationship to the proton and the neutron, as the electron to the positron. Although this is an assumption that goes beyond what is known experimentally, we do not view it as a very revolutionary one. We must assume, further, that between a nucleon and an anti-nucleon strong attractive forces exist, capable of binding the two particles together”.

Once again, our model has been anticipated, although this time by only half a century; thus, our figures are graphic manifestations of what Fermi and Yang were talking about, in this case, the π-plus and the π-minus, mesons, respectively.

The fact that traverse in our model does, indeed proceed one way through the fermion component and the opposite way through the antifermion highlights how yet another historically iconic item, in this case the Dirac theory, has direct bearing on our model. According to all accounts, the theoretical concept that antiparticles exist emerges in the combination of Quantum Mechanics and Special Relativity via the Dirac theory. To set the stage for certain conclusions, pertinent to the matter of fusion as implied by that well-known theory, we abstract it as follows: First, the equation for a particle with rest mass m and spin 1/2 is succinctly stated by the Dirac equation

(3-4)

where ψ is the quantum mechanical state vector, D is the Dirac operator given by

(3-5)

and summation over μ = 0 to 3 is implied with 0, and 1, 2 and 3 corresponding to the time t, and the spatial variables x, y and z, respectively. Also, a factor of ħ = h/2π in the first term is set equal to 1.

Dirac imposed the SR compatibility constraint by demanding that

(3-6)

be equal to the Klein-Gordon equation,

(3-7)

which is the quantum mechanical equivalent of the Lorentz invariant

(3-8)

with E and p being the energy and momentum associated with relativistic bosons and

(3-9)

the complex conjugate to the expression in Equation (3-5).

The resulting requirement is that the gammas must be constant 4 × 4 matrices that conform to the definition of a Clifford algebra. This leads (it turns out) to the reexpression of the Dirac equation as two, coupled, vectoreigenvalue equations, namely

(3-10)

where ψ_α = (ψ₁, ψ₂)^T, ψ_β = (ψ₃, ψ₄)^T and

These equations have traveling wave solutions with exponents proportional to linear combinations of the time and space variables—i.e., to , being the radial frequency and momentum eigenvector, respectively, which implies the equivalent form

(3-11)

where

(3-12)

is essentially a spin matrix. Especially noteworthy is the sign reversal, a break in symmetry of the mass term, that is the eigenvalue, in the second of the two coupled equations above. Thus, what we have on the RHS of Equations (3-10) or (3-11) is a four-component vector whose upper half, ψ_α, constitutes a positive energy representation of a two-component spinor and whose lower half, ψ_β, constitutes a negative energy (an antiparticle) representation of a similar spinor.

What is intriguing for present purposes is the coupling in either Equations (3-10) or (3-11) between ψ_α and ψ_β, the two-vector halves of the four component state vector, in particular, its circular nature. That is, we see that ψ_α depends on ψ_β (as modified by the spin matrix) and conversely, ψ_β depends on ψ_α in the same manner. It turns out that this dependence implies a relationship between our particle model and the Dirac theory and to exhibit that, we begin by rewriting Equations 3-11 as

(3-13)

We can then demonstrate the circular nature of this pair of equations, explicitly, by the operational diagram of Figure 21.

In view of the preceding discussion of fusion, the topology of this diagram is seen to be identical to that of the MS model of a boson: a bound state comprised of a spin ½ “fermion” on the left with counterclockwise traverse and positive mass, and its conjugate “antifermion” on the right with clockwise traverse and negative mass. Or, to put it another way: the MS model is also a manifestation of this emergent Dirac formalism. Additional implications are discussed in Section 7.

We are now in a position to discuss the fusion of the complete set of basic spin 1/2 fermions with the associated conjugate set. At this level of taxonomical organization, it is expedient to consider an abstract, group theoretic approach, which bypasses the detail but summarizes the general architecture of the taxonomy. In this approach (cited in [1] as per the treatment in [5]), a particle hierarchy is developed as the direct product of vector spin spaces.

Of course, the actual, physical process, corresponding to the direct product, is fusion . Correspondingly, the abstract result is expressed as the direct sum of subsidiary spin spaces, the Clebsch-Gordan decomposition. As discussed previously, the return to an original condition requires two traversals of the MS for all odd NHT (which we recall is equal to n) but only one traversal for all even NHT. For FMS, higher values of twist assume the form of a composite of triangular planforms. Since each member of the composite with odd NHT must also be traversed twice, we assign spin1/2 to it and assume that spin is additive. Thus, FMS with odd values of NHT (fermions) are characterized by odd multiples of spin1/2 and those with even values of NHT (bosons) corresponding even multiples of spin1/2. In effect, the number of spin1/2 multiples in a given composite is the same as the number of its basic triangular planforms except where the composite is actually an excited state (see below).

With the additional assumption, per our previous discussion, that the applicable group structure is that of the gauge group SU(2), the result of the direct product of vector spin spaces with spin s1 and s2 is given in [1,5] as

(3-14)

which equals V₀ + V₁ for the case of first order fusion, that is , for s1 = s2 =1/2.

Specifically, the direct product,

(3-15)

of, respectively, the vector of four basic spin1/2 fermions and its conjugate vector of antifermions, V = (A, B, C, D)^Tand V* = (A*, B*, C*, D*)^T is the matrix of sixteen two-element composites shown in Equation 3-16. In analogy with quantum mechanics, we note for future reference that M can be viewed as an operator. Also, the equivalent matrix in standard model nomenclature is shown in Figure 40 of Section 7.4.

(3-16)

Notice the two elements in square brackets in the corners; in reality, these elements cannot exist because no common quirk-antiquirk combination exists in either case. Also, V₀ consists of only two elements, namely, CB* and its complement, BC* the ones discussed above and shown in Figure 20. To reiterate, although these two composites can be formed by fusion, topologically, they are really just doubled-over versions of the trivial, zero-twist MS. In other words they are excited states of the basic untwisted state and, in fact (we recall), in each case the algebraic sum of the twists of the fused constituents is zero. The other twelve bosons are all V₁ vector bosons in their ground state and can also be formed either by fusion or directly by a twist whose NHT is also the sum of those of its constituents.

Note that twist increases from left to right in matrix M for fermion components and upward for the antifermion components. Thus the loci for composite twist are lines parallel to the principle diagonal as we indicate in Figure 22 and the gradient is directed along the orthogonal diagonal. Each dot in the figure represents a boson whose associated twist is the sum of the indicated abscissa and ordinate values. It is instructive to reexamine the twist-charge (Gell-Mann/Nishijima) relationship and we begin by restating it in simplified form as

(3-17)

where upon twist and charge become

(3-18)

indicating that twist and charge gradients are mutually orthogonal.

As indicated, twist loci are antisymetric about the principle diagonal, which, we note for future reference, also functions as the gradient of charge which is directed downward and to the right, normal to the twist gradient (as per Equation (3-18) above). Thus the associated charge loci are antisymmetric about the orthogonal diagonal. There are seven members in each set of loci. The two sets intersect in sixteen points that can be grouped as discussed below.

First, the reason we are focusing on twist is twofold: its relationship to isospin as noted above, and its relationship to the topic of degeneracy, which we select to be the determinant for the next lower taxonomical level after spin (in good part because of its relationship to isospin). We recall that the notion of degeneracy is customarily associated with the invariance of a fundamental attribute shared by a multiplicity of otherwise-distinguishable versions of a physical system. In what follows, that attribute will be twist. The structure of the associated degeneracy can be described in terms of combination, permutation, composition and contingency, defined as follows:

Briefly, more than one combination of integers can add up to a specified value of twist at a given order of fusion.

A given combination of integers can be ordered in more than one way to generate a permutation.

A given permutation can be realized in more than one way as a consequence of the detailed composition of the constituents; and finally.

The possibilities for a given higher order of fusion hinge on those for the previous fusion—thus, contingency.

Composition and contingency require a more detailed consideration and will be so discussed in Section 6. Clearly, aggregation in terms of twist for first order fusion can be accomplished simply by inspection—that is by assembling the entries along each diagonal locus of matrix M (or, equivalently Figure 22). However, noting that the loci illustrate the relationship n₂ = n – n₁, we are led to the formal notion of symbolic convolution; where the direct product matrix M presents the requisite information visually, convolution does the same thing algebraically. For the case of first order fusion, we therefore define the operation of (symbolic) convolution to be

(3-19)

where α _μ, β_v and γ_λ are to be identified with the set of four basic letters, the set of four basic conjugate letters and the twist-valued headings, n, in the output of the convolution operation, respectively. That is,

(3-20)

It helps to picture what’s happening operationally as shown in Figure 23.

Note that the summation indicated in Equation (3-18) really denotes assembly of products rather than numerical summation as we see in the results of convolution shown in Figure 24.

Especially noteworthy is that, with the replacement of letters A and B by D and C, respectively, the figure exhibits bilateral mirror symmetry, essentially a perpetuation of the situation noted above in the case of the basicfermions. Correspondingly, note that the sum, over the figure, of the individual values of twist is zero. Also, charge sums to zero in each twist assembly (each column). Finally, for future reference, we note that there is a total of 16 two-letter “words”—that is, bosons (including the two unrealizable ones) which may be trivially viewed as assembled into permutation groups, in this case 4 singlets and 6 doublets, the latter occurring in charge conjugate pairs.

Full disclosure: our alternative model is inherently ambiguous in that a given label (one of the capital letters) may represent more than one fundamental particle of the Standard Model depending on the elementary particle interaction that is being modeled. The ambiguity is summarized in Figure 35. As indicated, the nucleons are represented in the first row and the leptons in the next three rows. The labels employed heretofore in this paper are in the bottom row and the idea of the figure is that, a priori, such a label may represent any one of the particles in its column. However, the label C represents the proton unambiguously and the label D likewise represents only that enigmatic particle whose existence is required by symmetry (but whose role is problematical). On the other hand, there is a triple (leptonic) ambiguity associated with label A and a quadruple ambiguity associated with label B which can represent either the neutron (as in Section 3.3 above) or one of the neutrinos.

In practice, this kind of ambiguity is not really a problem: to quote [1], “—the ambiguity can be resolved by invoking the fundamental rules that constrain which interactions are realizable. Some of these are derivable from basic principles (conservation of charge is one and a limitation on the sum of decay product masses to less than that of the original particles is another) and some are not. Among the latter is conservation of baryon and lepton numbers.”

Such ambiguity resolution will be illustrated presently in terms of well-known interactions, but in the meantime, we note that, in the reverse situation—that is, where the Standard Model elements are specified to begin with—the corresponding alternate model elements are indicated unambiguously.

For example, consider the creation of delta particles by the excitation of nucleons operated upon by pions (or conversely, delta decay into nucleons) as summarized in Figure 36 (cf.[1]). This figure is essentially that presented in [16]) with the exception that there are two versions, π^0R and π^0L, shown here of the neutral pion rather than the one, π⁰ usually shown in the SM.

For convenience we reproduce the boson operator matrix (Equation 3-16).

Now, recalling the previous discussion of Section 3.2 in which the two entries of Figure 20 were equated with π⁺ and π^-, respectively, note that if n is equated with B and p with C, then, as per this matrix (recall that fermion twist increases to the right and antifermion twist upward) we see that

(7-1)

Thus, viewing these pions as mesons operating on the nucleons as per previous discussion, the delta particles are expressible as

(7-2)

In terms of the nomenclature that we found in Figure 29 in the discussion of taxonomy, Δ⁺ and Δ^- are two members of the last triplet in the NHT = −1 column of that figure and Δ⁺⁺ and Δ⁰ are two members of the last triplet in the NHT = +1 column. However we can also express two of the delta particles as

(7-3)

which we recognize as the remaining member of the NHT = +1 triplet and the remaining member of the NHT = −1 triplet, respectively of Figure 29.

Figure 37 then shows what, as a result of mediation by the pions, our alternate model deltas look like using SM notation.

We note for future reference that Figure 36 and Equations 7-1 to 7-3 have reference to the discussion in Section 9 regarding a quantum mechanical connection.

As per [1], we can also establish a correspondence between the SM neutral pion and the two alternate model versions shown here by defining the superposition

(7-4)

in terms of the quirks available after formation of the composites. Upon eliminating the common factor, u*u from each term we are left with

(7-5)

which “is the accepted SM composition (also viewed as a superposition) for the π⁰.” In the same vein, we note that the two pion versions can also form the superposition for the η particle, viz:

(7-6)

Experimental evidence for two varieties of the neutral pion is discussed in [17].

Modeling weak interactions is quite different: in the case of the beta decay of the neutron, we begin, again, with its excited states as shown in the NHT = −1 column of Figure 29, noting specifically the terms AC*C and CC*A (which are operationally identical for our purpose) in the uppermost triplet of the −1 column. Using AC*C, Figure 38 shows the first stage of an alternative model version of neutron Beta decay—that is, of the process

(7-7)

where, as defined previously, x and y represent the dd* and uu* junctions, respectively.

On the basis of the quirk structure, C and C* translate unambiguously to p and p*, which conserves twist but not baryon number and charge. Thus, we need to increase baryon number in the diagram by one and cancel the lepton number and charge of the A. Furthermore, on the basis of mass, A should translate to e rather than μ or τ (which are too heavy) in which case we need to change the C* to something that translates to v_e (for its lepton number).

All this can be accomplished if we just change the indicated u* quirk into a d* quirk! In [1] this is referred to as the “beta switch” and can be rationalized in knot theoretic terms (see Appendix D)6. Thus, we go from AC*C, an excited state of the neutron, to AB*C, an excited state of the electron as shown in Figure 39. In terms of twist, we go from NHT = −1 to NHT = −3 by changing the u* quirk with NHT = +1 to the d* quirk with NHT = −1. The meson operator is AB*, our model’s version of the W^- particle (see the boson matrix M, Equation (3-16)) which then splits into an electron (i.e., A) and its antineutrino (i.e., B*)

Muon decay is a similar process; we begin again with an excited state, this time in the NHT= −3 column of Figure 29, specifically noting the terms AC*B and, equivalently, BC*A of the sextet. Using AC*B we see a diagram very similar to Figure 38 and it goes through an identical beta switch process.

On the basis of mass, the A can only translate to an e and, similarly, B must be a neutrino rather than a neutron. Again, there is a baryon number and charge mismatch across the equal sign due to the C* (which translates to a p*) and, again, it is corrected by the “beta switch” which changes the C* into a B*. In turn, that B* translates to an antineutrino which must be a v_e in order to cancel the electron’s lepton number. And finally, the B of the second fermion must translate to a muon neutrino v_μ in order to match lepton numbers, whereupon we end up with AB*B (see Figure 41) which translates into W^-v_μ which splits up into an electron and its antineutrino and the muon’s neutrino.

For reference, here is the bosonic operator matrix one more time:

and here (Figure 42) it is translated into Standard Model nomenclature mainly according to the interactions we have been reviewing:

The empty corners signify the previously mentioned impossible combinations, and in fact, the upper row and right hand column—six items in all—are also combinations that are not involved in actual SM interactions. There is an important functional symmetry between the lower left hand quadrant and the overlapping, next diagonal quadrant: Those operators involved in “electroweak” interactions, namely the gauge vector bosons γ, W^-, W⁺ and Z⁰, are in the lower left quadrant and in the (diagonally) next, overlapping quadrant are the pions involved in “strong” interactions: π^-, π⁺, π^0L and π^0R. In the SM these two groups of operators enact parallel roles in terms of weak and strong isospin, respectively. That is, the manner in which the pions in the second quadrant act upon the nucleons is identical to the manner in which the vector bosons in the first quadrant act upon the electron/(anti)neutrino pair of leptons. In [1] a model, reproduced below in Figure 43, shows pions mediating Yukawa type exchanges between nucleons to maintain Deuteron stability in what may be characterized as strong isospin space.

In this process, the π^- splits into a free neutron and an antiproton which fuses with the original proton to make the π^0R and, similarly, the π⁺ splits into a free proton and an antineutron which fuses with the neutron to form the π^0L. That is we have the interactions

(7-8)

such that we always have a free proton and neutron pair.

Now if we substitute “electron” for “proton” as well as “neutrino” for “neutron”, we can construct precisely the same diagram for the stability of the electron/neutrino pair, with the W⁺ splitting in analogy with the π^- and so on to perpetuate the electron/antineutrino pair. That is, we begin by making the correspondences:

(7-9)

which imply the additional correspondences

(7-10)

Thus the above interactions translate into the interactions

(7-11)

In other words, the vector bosons of the model mediate lepton pair exchanges in “weak” isospin space in the same way that the model’s pions mediate nucleon exchanges in “strong” isospin space, thus manifesting the relationships of the Standard Model.

Finally, we recall the discussion of the Dirac theory in Section 3 where Figure 21 shows the output to be basically an uncharged, bound particle composed of a fermion and its corresponding antifermion. In terms of the boson matrix of Figure 42, this configuration could represent either a photon, a neutral pion or a neutral vector boson, the Z⁰.

So far, our model has been discussed in terms of a single generation of particles whereas the SM, as is well known, actually consists of a family of three generations. The object of this section is to display a corresponding triplication of our model, in fact a family structure consisting of three and only three generations, most of which is demonstrated in detail in [1]. Here, while omitting much of the detail, we shall, nevertheless, go a bit further in terms of the symmetries inherent to the model. Following [1] we therefore arrange the four basic fermion representations, in order of twist, as shown in Figure 44, recalling the progressive replacement of d quirks by u quirks, and the corresponding increase in twist from −3 to + 3 and charge from −1 to +2. Even though it plays no part in either the taxonomy or interactions we include fermion D for reasons of symmetry. The possibility that it plays an active role in some way is discussed in Section 11.

These four objects can also be viewed as four points on a line in what we define as “label space”, shown in Figure 45 as a function of twist. The line is anchored by labels D and U, which, here, stand for quirk structures of ddd and uuu, respectively (please accept our apologies for the confusing switch in notation).

The idea here is to initiate an organizing rationale for incorporating further generations. Thus when we consider an additional quirk, denoted by the letter s (in analogy with the SM strange quark), we also introduce its representation, capital letter S, in label space such that it interconnects as in Figure 45 with the other two labels. The result is the triangular array shown in Figure 46 which illustrates the replacement progression from a composition of three d quirks to one of three s quirks along a line anchored by capital letters D and S and similarly for the progression of replacing letter u by letter s. So far this is quite similar to most elaborations of the SM way to organize multiple generations

Note that the triangular display accommodates seven appropriately placed intermediate points for a total of ten points, to each of which we can associate a particular member of a particular set of alternative model particles. In other words, we can view such a geometrical structure as a scaffold, the indicated points of which are loci upon which to emplace orderly arrangements of basic alternative model fermions or bosons, up to and including a decuplet arrangement.

Note also the isomorphic relationship of the DUS triangle to the defining representation of the gauge group SU(3), the triangular array featuring quarks d, u and s in the development of the historic “Eight fold way” of Gell-Mann and Zweig. Thus, Figure 46 can be considered as the prototypical manifestation of a “flavor” SU(3) format which we will see replicated as we proceed to erect a family tree. The inclined dotted lines are loci of constant charge and the horizontal lines are hypercharge loci in each case so that twist, charge and hypercharge increase as shown in the accompanying coordinate system, all in direct analogy to the SM.

The key feature here is that of symmetrical interconnectivity of capital letter labels; if we impose that as a requirement for the incorporation of additional flavors it implies, to quote [1] “that the associated geometrical edifice must be regular. In particular, if there are N flavors, each flavored vertex must extend a line to N − 1 distinct other flavored vertices. Thus adding another flavor, the (c) quirk—produces 4!/2!2! = 6 lines, which combine in the tetrahedral structure” of Figure 47. Note that this structure also automatically subsumes 4!/3! = 4 contiguous triangular arrays.

However, accommodating the addition of another flavor (for a total of five) by trying to fit to another regular three-dimensional structure is precluded, as discussed in [1], by the well-known Euler constraint,

(8-1)

(where v, f and e stand for vertices, faces and edges). On the other hand, adding two flavors, for a total of six, in other words, three complete generations, can be modeled by invoking the symmetries of the icosahedron whose vital statistics, known since antiquity, are (in consonance with Euler’s equation) 12 vertices connected by 30 edges which form the boundaries of 20 equilateral triangular faces.

In Figure 48, we show, for reference, a representative unlabeled icoshedron, together with an example of a capital letter labeling scheme. Note that each vertex is surrounded by five other vertices and labels are applied with the constraint that each labeled vertex is surrounded by each of the five remaining labels. In the indicated scheme, what is shown as a pentagonal outer perimeter is actually the representation of the single vertex labeled D*.

Note that there are indeed twenty faces represented, five whose corners bear fermion labels, five with antifermion labels, five with two fermion and one antifermion label and another five with one fermion and two antifermion label. Of course each label is associated with five other vertices and an associated five other labels so that there are indeed 2 × 6 = 12 unambiguously labeled vertices and 2[6!/(2!4!)] = 30 unambiguously labeled edges. In other words, all icosahedral vertices, edges and faces are accounted for as are, correspondingly, all six fermionic labels and associated antifermionic labels; a perfect fit!

Furthermore, as noted in [1] “this is as far as we can go in label three space with the high degree of symmetry implied by the combinatorial requirements of a family structure”. The implication is that there can be three generations but no more on the particle family tree. Of course this is really a rationale not, in the ontological sense, an explanation for the existence of three generation.

However, we assert, this scheme goes further than the Standard Model in that regard and, in a somewhat speculative vein, we can do better. To begin with, we can delve further into the munificent geometry of the icosahedron. Although not all of that is germane to this paper, it is pertinent to point out that vertices, edges and faces each come in antipodal pairs. Thus, looking directly down upon the vertex labeled D in Figure 48, one sees a pentagonal outline for the assembly of 5 faces formed by the connections between vertex D and each of the five surrounding vertices, labeled clockwise from the top as B, T, C, U and S. Also visible is the antipodal pentagonal outline formed by the connections between the antipodal vertex D* and each of the associated antipodal vertices, labeled clockwise from the bottom as B*, T*, C*, U*, and S*.

Note that the two pentagonal outlines (and their included faces) are in relative rotation about the DD* axis by 36 degrees which is half the central angle of an edge as measured in the pentagonal plane. Antipodal edges can then be related in pairs to form rectangles whose shape is such that the length, h, of the long side and the length, ℓ, of the short (edge) side are (it can be readily shown) in the “Golden ratio” and diagonals UU* and DD* have lengths of times the edge length. An example is the rectangle with edge UD connected to antipodal edge D*U* as in Figure 49.

Following Baez [17] we shall henceforth refer to the rectangles as “Duads” and since there are 30 edges, there are 15 such duads.

The duads can be assembled into groups of three called Synthemes” [17] and the Synthemes can be assembled into three larger groups of five Synthemes each called “Pentads” [17]. One such pentad is the assembly of “true crosses”, listed below, so called because each of its five synthemes consists of three, mutually orthogonal duadic planes [17]. In this list, each duad is represented by the labels of one of the edges with the opposite edge to be understood (for example, UD represents the rectangle shown in Figure 49, above). Note that there are fifteen such duads in the list, each of which occurs once and each label occurs five times, that is, paired in a duad with each of the other five labels.

Of particular interest because of its correspondence with the three generations of the SM is the first syntheme which is illustrated in Figure 50. The three planes are indeed mutually orthogonal but also encompass all twelve icosahedral vertices. Thus, they completely define the icoshedron and its labeling. Alternatively, the icoshedron may be defined by the associated orthogonal coordinate system shown in Figure 51 where coordinate axes are the normals to duads UD, TB and CS, respectively.

These three mutually orthogonal duadic planes and the associated set of orthogonal axes define the geometry of the icosahedron as a framework for the six generations of the family. They also implicate a corresponding group structure, the gauge group SU(3), actually manifested earlier (see discussion re Figure 46). Beyond that, however, suppose we view each of the three duadic planes as a kind of internal space, each peculiar to interactions that take place within and only within a corresponding generation, one plane to a generation. For example, in duad UD the first generation might be exemplified by the interactions discussed in Section 7. (Note that this view augments that of the external organization we outlined above in which the three generations occupied the twenty faces of the of the icosahedron. We shall return to the details of that occupation below).

We are not yet done with the “speculative vein” adopted above: given the duadic planes, the role if any of the associated orthogonal coordinate system comes into question and, in that regard, we recall, once more, our identification of the out-of-plane coordinate for individual FMS with time as in the immediately preceding sections. Here we have three duadic planes, with, in the sense posited above, a given duadic plane uniquely constituting the internal space for the FMS of a corresponding generation. The next step is not inevitable but neither is it unreasonable: we simply associate a unique “time”label to the particular coordinate vector associated with (normal to) each duadic plane. In other words, the spacetime of reference—the one inhabited by the particles of our model—that spacetime has become a 3 + 3 dimensional continuum.

Although spacetimes of more than four dimensions are nowadays commonplace, these are generally in the Kaluza-Klein tradition where the extra dimensions, being space-like, must be “hidden from view” in a “Calabi-Yau” manifold in some way, shape or form. Our “triplication of time” is not like that; it is more like an upgrading of the familiar time coordinate of our macroscopic existence into a time manifold with the triplicated time coordinates forming its orthogonal basis. In effect, time is promoted to be on the same dimensional footing as space7.

However, since real time would not necessarily always correspond precisely to any one of the three orthogonal coordinates, we might construct something like a linear combination of coordinate vectors. Or, expanding upon that notion, we would construct the equivalent Kobayashi-Maskawa (KM) matrix of the Standard Model one that produces transition probabilities between generations, in which case we would need to establish a correspondence to the latter. One way to approach such matters is discussed in Appendix E in terms of the algebra of quaternions, the rationale there being that from the point of view of MS/FMS traversal there is only one dimension of space that matters. In this view, spacetime is effectively 1 + 3 dimensional, an entity that lends itself to that kind of treatment in the same way that our ordinary 3 + 1 Minkowskian spacetime does. On that basis, a direct correspondence to the KM matrix is indeed found to emerge.

There are of course problems that come to mind almost immediately. For example, on the family specific side, one such is that empirically, the three generations are very unequally experienced; it takes a lot more energy to find (what is interpreted in the SM to be) a top quark than it does a down quark. Thus, the three-dimensional symmetry that our time triplication manifold inherits from the icosahedral structure that gave rise to it in the first place must be broken. And looming over such issues is a fundamental problem; validating the notion of time triplication would introduce a radical new symmetry into fundamental physics, a novel generalization of the symmetry of spacetime over and above the point of view that underpins relativity (either special or general). At this point, additional discussion of the ramifications of time triplication is postponed to future correspondence.

We now return to the detailed particle occupancy of the icoshedral surface. In particular, to round out this section, we need to illustrate what it is that is emplaced upon loci in the triangular faces of the scaffolds discussed above. As mentioned, a prototypical triangular face can accommodate up to a decuplet of detailed particle models but before we show an example of such a decuplet, suppose we ignore the three corner locations of the face as shown in Figure 52.

Nominally we would be left with the intermediate seven. However, in the process of descending from the top row to the second row, by selecting a d quirk for replacement by an s quirk, the detailed symmetry of the alternate model is broken, in particular that of the fermion labeled “n” for neutron. This allows the central location of the second row to harbor two distinct particles, our model’s version of the Σ⁰ and Λ⁰ particles, with the overall result that we realize, as per [1], the SM’s spin ½ baryon octet.

Returning to the decuplet, Figure 53 shows the spin 3/2 baryons beginning with the delta particles, as portrayed in the top row. Only three quirks actually take part in the replacement process so the nonparticipating quirks are shown enclosed by circles. We no longer have the triangular symmetry featured by the neutron in the octet of Figure 52 so all transitions in a given line are distinguishable. Thus, except for the three corners, all labeled loci are degenerate so that the figure really represents a multiplicity of diagrams, which, as pointed out in [1], mirrors SM practice for this decuplet.

As a final example, we show in Figure 54 another octet format, this time the spin 0 meson octet of the SM, beginning the replacement process with the neutral pion pair as portrayed in the alternative model. Again, nonparticipating quirks are indicated by circles but this time only a single quirk- antiquirk pair remains in each meson model to take part in the replacement procedure. This time the procedure also moves upward replacing d* antiquirks by s* antiquirks as well as downward where d quirks are replaced by s quirks as before.

At this point it is necessary to point up an inconsistency in the quirk/ quark correspondence for particles beyond the basic ones. In this regard we note first that the quirks described above as “nonparticipating” in the above interactions correspond, in fact, to quarks that do not make up part of the corresponding particles in the standard model taxonomy. Thus where mesons in the SM consist of a quark and an antiquark (e.g., pions), mesons in our model consist of two quirks and two antiquirks. Similarly, spin 3/2 fermions (e.g., delta particles) consist of three quarks in the SM whereas in our model they consist of four quirks and an anti quirk. In Section 11.3 some experimentation is discussed that appears to relate directly to this fundamental difference in portrayal. That discussion indicates that our model may have predictive capability.

Nowadays, there is a large and growing literature [8,21] on the connection between knots and physics, not the least of which involves the connection between knots and Topological Quantum field Theory (TQFT) but the string theory connection comes to mind almost immediately if for no other reason than the connection between the words “knots” and “strings”! However, there is more to the string theory connection than that. For one thing, the coefficient of the kinematic term in the Lagrangian yielding the Sine-Gordon equation is identifiable as string tension. And, in terms of concatenating torus knots, we note that, by definition, each is explicitly wound around a torus, which amounts to “toroidal compactification” as in string theory. Also, as noted in [1], a couple of decades ago, an historic shift in attention from ten to eleven dimensions was associated [22] with the extension of primal strings to two –dimensional membranes—“two-branes”—as a function of the increase in the string coupling parameter. In fact, as is well documented, string theory has developed into something much more encompassing, an entity often referred to as M theory with a number of theoretical “manifestations”. However, there appears to be something like a “core” M theory associated specifically with five-branes, gravitons, and, more specifically, two-branes, which with a twist, are our MS.

Two promising candidates for connection to physics and in fact, to mathematics as well, are the relationship of the alternative model to the subjects of Topological Quantum Field Theory and Quantum groups/Hopf algebra. In both cases, specific requirements put the operations of fusion and fission under detailed scrutiny. In the Hopf case a further requirement (the existence of an “antipode” [8]) turns out to be satisfied in a unique way specific to the model. The Hopf connection is summarized in Appendix C.

With regard to TQFT, apparently there is more than one approach to the subject and more than one way to characterize what it means. Broadly speaking however, and without regard to dimensionality, a realization of a topological quantum field theory features parameters that are quantized simply on the basis of its topology and are invariant to topologically invariant distortions. From that perspective alone, our particle model certainly qualifies as a TQFT featuring, as discussed in the introduction, quantized electric charge, spin, isospin and CPT invariance.

Furthermore, Witten’s work in developing the notion of a topological quantum field theory [23] and his interpretation of the Jones polynomial for knots in that light [24] are well known as is Atiyah’s axiomatic approach to the formalization of TQFT [25]. In particular, as stated in the Atiyah frontispiece; “The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to topological quantum field theory in 2 + 1 dimensions.” This, fortuitously, is also the dimensionality of the FMS in our alternative. Also pertinent is the Atiyah statement [25]: “The Jones polynomial has been generalized in a variety of ways.” And “(a) fundamental way involves choosing a compact Lie group G and an irreducible representation.”—“The original Jones polynomial corresponds to taking G = SU (2).” And further in that regard, we note that, in making a necessary choice of pertinent gauge group, Witten was also led to select the group SU(2). Thus, our model appears to coincide with the Witten/Jones/Atiyah parametric basis.

Although a comprehensive discussion of the model’s role as a TQFT must await further publication, we can summarize an approach as follows: With reference to Figure 56, the main idea behind the upper branch is to show that MS/FMS ontology, especially as employed in the development of the model taxonomy, does indeed embody the fundamental Atiyah requirements for a TQFT as detailed in [25] which is essentially a mathematical “legitimization” and generalization of Witten’s interpretation of the Jones polynomial as a TQFT. On that basis, (undocumented) preliminary work indicates that the model does indeed qualify as a legitimate TQFT.

With regard to the lower branch of the figure, the equivalence between the Jones polynomial and the Kauffman bracket polynomial is well known [8] as, in particular, is the KBP of (2, n) torus knots [8] and the relationship between such knots and the MS as discussed in this paper. Thus, given the putative validity of Witten’s interpretation of the Jones polynomial as a TQFT, the path is closed and the implication is that no further discussion is necessary; the FMS model is, ipso facto, a TQFT.

Although that logic leaves out the demonstration of a common ontology, it turns out that the existence of a correspondence between composite FMS and composite cobordisms as 2 + 1 manifolds emerges in the detailed analysis of the axioms mentioned above, a correspondence also readily demonstrated on the basis of diagrammatic comparison.

That analysis will not be repeated here but there is a suggestion that a KBP in a form that reflects such commonality would be useful. Such a formulation is described in Appendix F of this paper. It involves a weighted sum of convolutions, each associated with a particular knot and therefore of a particular FMS component of fusion, or, in terms of TQFT formalism, with a particular component in a union of component cobordisms. The implication is that we can invoke the algebra associated with (2, n) torus knots or links to study the algebra associated with FMS fusion and fission and, by implication, composite cobordisms.

In summary to this point, on the basis of the preliminary analysis carried out to date, the judgment that our model qualifies as a topological quantum field theory lacks only formalization and documentation. On the other hand, also lacking in such judgment is a certain desirable amount of perspective. We note that the subject of cobordism alone has been investigated in considerable depth as has been that of TQFT in dimensionalities other than 2 + 1. From a broad perspective, TQFT is really one aspect of a triumvirate of approaches to understanding the fundamental nature of nature, along with Quantum Field theory and Quantum Gravity. It would be instructive to embark upon a more comprehensive investigation of TQFT that brings out our model’s place in such a wider context.

The difference between how bosons and higher spin fermions are portrayed in our model and in the SM was discussed in Section 9. It evokes some experimentation that as mentioned in [1] “points to the existence of heretofore unobserved particles”. The particles of interest are a four-quark meson (http://xxx.lanl.gov/abs/hep.lat/0001023) and a “pentaquark”, a five-quark baryon which decayed into a neutron and a K⁺ meson (http://www.phy.ohiou.edu/~hicks/thplus/thplus3.html).

Regarding the meson we note that all bosons in our model consist of two quirks and two antiquirks. As to the pentaquark, the report (http://www.phy.ohiou.edu/~hicks/thplus/thplus3.html) of the experimental result states that “The valence quark configuration of the exotic state contains two up quarks, two down quarks, and an anti strange quark”8 (See Figure 57 below) which was found to decay into a neutron and a K⁺ meson. As pointed out in [1] this sounds very much like our delta particle model (specifically a delta plus) if we also recall the transformation of a π⁺ into a K⁺ by the change of a d* antiquirk into an s* antiquirk as portrayed in Figure 54 of Section 8.

The experiments of reference were conducted in the early-and-mid years of the last decade at the Jefferson Lab in Newport News, Virginia. They involved high energy gamma rays impinging upon a deuterium target. The pentaquirk decayed into a neutron and a K⁺ which tallies with the figure. Also found were a proton and a K^-,which as in [1], suggests the delta minus-like particle shown below but with a π^- that has similarly decayed into a K^-.

The existence of a pentaquark was originally suggested by researchers at the Petersburg Institute in Russia in 1997. There was supporting evidence in data taken in 2002 at Osaka University in Japan using carbon targets bombarded by X-rays and in a review in Germany of accelerator data taken in the years 1997/98. However subsequent disappointing experimentation has since led to a consensus of doubt as to the actual existence of a pentaquirk; perhaps the subject ought to be revisited.

As is well known, there are numerous candidates for the constituents of dark matter (and dark energy as well). Here is our model’s candidate: Always lurking at the perimeters of disquisition but never really contributing is the enigmatic fourth elementary fermion, the one labeled “D” with an NHT of +3 and a charge of +2e. Clearly, this FMS and its combinations have absolutely nothing to do with the alternative model’s taxonomy or interactions. Nevertheless, they must appear by virtue of symmetry and so they do, at each level of fusion as a completely dual organization, a kind of particle-for-particle super (anti) symmetry. That being the case, the question naturally arises as to whether they might actually have some relation to reality.

One approach to answering that question is to turn the problem of nonparticipation into a virtue. In other words, we would assume that our dark particles actually exist; they consist of the D and its combinations but we can’t “see” them simply because they do not interact with normal matter. However, even though they are “dark”, they still have mass, possibly very heavy mass, which brings up the matter of symmetry breaking. Of course, the original, basic MS mirror symmetry is broken by the choice of traverse direction, but what we need here is something over and above that, something in the nature of a spontaneously broken symmetry, the kind of thing provided in the SM by a Higgs field (which we have so far managed to avoid!).

One possibility is to invoke the notion, introduced in Section 8 of the triplication of time. Although, such triplication is visualized as existing at every point of space, there appears to be no a priori reason for the “ordinary time of a dark particle to be the same as it is for our ordinary taxonomical world. It could instead be one of the other orthogonal time coordinates in the triplication, perhaps the one associated with our heaviest generation in which case finding a “dark” particle might be very difficult; as appears to be the case! Again, further discussion is postponed for future publication.

Quaternions are useful for manipulating quantities in four dimensions as are complex variables in two. In this section we investigate first how they relate to the 1 + 3 dimensional spacetime introduced in Section 8 to help explain the triplication of generations on the particle family tree. There are also taxonomical implications which follow.

A quaternion can be expressed as

(E-1)

where and .

While the p's and q's provide the information explicit to diverse situations, the λ's define the quaternion algebra common to all. Using their definition we find the outer product of two such entities to be expressible as

(E-2)

where and .

The outer product can of course also be expressed as a 4 × 4 matrix, the direct product of a column and a row vector (as in Section 3 to form boson matrix M) as

= (E-3)

However, we can also subdivide this matrix so as to summarize and highlight its intrinsic organization as a 2-dimensional manifestation of the content of Equation E-2. Thus, in Figure E1, we see a scalar, an inner product, two vectors and a cross product, the latter split between two pieces of matrix real estate, as shown in more detail together with the principal diagonal in Figure E2.

If we delete all the terms containing p₀ or q₀ in Figure E1 we are left with a 3 × 3 matrix, the direct product of the 3-vectors or alternatively, the direct sum of the inner product and cross product of the vectors shown in Figure E2. We also note that, from the quaternion point of view, vectors are also known as pure quaternions and the 3 × 3 matrix is a 2-dimensional manifestation of their outer product. In any event, whether in full or reduced form, the matrix of Equation (E-3) constitutes an operational representation of the outer product of two quaternions, and can be used to operate on additional quaternions.

In Section 8 it was suggested that quaternions might prove useful in characterizing the 1 + 3 dimensional “personal spacetime” experienced in the traversal of an FMS in the case of time triplication. With that in mind, we consider the case of a 4-vector and its conjugate in such a spacetime, defining

(E-4)

where with s associated with space, the t's with time and with the unit vectors isomorphic to the λs of Eqation (E-1), we have

(E-5)

and similarly for the conjugate. Instead of Figure E1 we now have Equation E6.

(E-6)

The scalar and vector terms involve space only and space-time interaction, respectively; deleting them leaves us with the inner and outer products that occupy the following 3 × 3 matrix, the direct product of the triplicated time vector and its conjugate:

(E-7)

At the same time, we recall that the Standard model employs the Kobayashi-Maskawa (KM) matrix [27] to systemize intergenerational interactions.

V = (E-8)

Now, if we identify indices i = 1, 2 and 3 of the t_i with u, c, and t, respectively and j = 1, 2, 3 of the t_j^* with d, s, and b, respectively, we see that the two matrices are isomorphic. Also, as noted in [17] a useful approximation to the KM matrix is

(E-9)

where λ = s₁₂, and s_ij = sinθ_ij, the angle being the “mixing angle” between the ith and the jth generations. Thus our t₁t₂* and t₂t₁* terms correspond to the KM terms λ and -λ, respectively, of the first and second generations, as they should, and our t₂t₃* and t₃t₂* terms correspond to the KM terms λ² and –λ² of the second and third generations, as they should.

We conclude that there is a demonstrable correspondence between the time triplication approach and the KM matrix approach to systematizing the triplication of generations on the family tree.

As discussed in the foregoing, the four basic FMS give rise to a “super(anti)symmetric” kind of duplicate taxonomy: there is a half that involves the fermion triplet A, B and C and their conjugates and combinations and which largely replicates the taxonomony of the Standard Model; and there is another half involving D, C and B, which has no such connection but which exists by reasons of symmetry. We recall, that with the replacement of A, B and C by D, C and B, respectively, the two halves are manifestly isomorphic.

In Section 9 we have seen the development of a complex algebra as a result of the orthogonality of charge and twist in taxonomical development. Since the taxonomy of our model develops as per the dictates of the gauge group SU(2), it would appear that either of the above triplets ought to be amenable to treatment in terms of a quaternionic algebra, given the close association of that algebra and that group. Thus suppose, as in the preceding we regard the three basic fermions A, B and C as a vector—that is as a pure quaternion—then add an identity element and define a conjugate. Thus we consider as quaternions the 4-vectors

(E-10)

where

and similarly for the conjugate with the 's being isomorphic to the λs. We remark that the unit quaternion is isomorphic to the gauge group SU(2) with the unit vectors being isomorphic to the Pauli matrices.

In the case of first order fusion, instead of Figure E1 we then have Equation E-11, the matrix version of the outer product

(E-11)

where

(E-12)

Again, if we delete the scalar and vector terms (which involve identity elements f and f*), we end up with a 3 × 3 matrix as in Figure E3

which is matrix M rotated 90 deg. CW and without the row and column associated with enigmatic label D. It is organized according to the taxonomical analogs to Figure E2 for cross product and principal diagonal as shown in Figure E4.

Using Standard Model nomenclature for our boson model, Equations (E-12) then become

(E-13)

or, as translated into Standard Model particle nomenclature,

(E-14)

In the case of second order fusion, we operate on a third fermion, say

(E-15)

with matrix Ф⊗Ф* of Figure 59—i.e., we form

(E-16)

However, since we are really interested only in the case where f = f* = g = 0, we find that the RHS of Equation E-16 reduces to the direct sum of the following two outer products,

and

which evaluates as the direct sum of six matrices, listed below. Together these contain a total of 27 three letter terms, each of which coincides uniquely with one of the 27 three letter terms that remain in Figure 28 and Figure 29 of Section 4 after all terms containing letters D or D* are deleted.

,.

Collecting terms then reduces the direct sum of these six terms to the outer product of two vectors

where σ* is the direct sum of three terms,

(E-17)

Although this expression still encompasses all of the 27 terms, it is a little more succinct.