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Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree.

There are eleven sections in this paper. The first six sections—not quite half of the paper—develop the subject model.

Two items need mention before we begin to validate the abstract. First an acknowledgement: due to the nature of this paper, it is necessary to describe in considerable detail the subject model whose development has been similarly described in references [

Next, a question, one that increasingly emerges in various guises at the frontiers of physics: “Just exactly what

Actually, the notion that there even are such things as elementary particles goes back (but apparently no further) to the fifth century BCE and the Greek philosopher Democritus for whom all matter was “composed of many different kinds of minute hard particles.” [

The notion of the atomicity of matter then languished for centuries. We are told that Isaac Newton in the seventeenth century also believed all matter to be somehow composed of infinitely small, infinitely hard atoms. Eventually, chemists found it expedient to postulate various kinds of elementary particles—atoms—each with a particular mass and electric charge in order to explain the variety of the substances of their experience. By the middle of the nineteenth century, a great variety of atoms were recognized and there was even an attempt to associate form and structure to them to explain the variety (see below). Nevertheless, it was not until the early twentieth century that most of the skepticism regarding the existence of atoms was dispelled. Albert Einstein’s explanation of Brownian motion was a big help in that regard. Nowadays, we have the technology that lets us “see” atoms, we know the mechanics of how they combine in the various forms of matter and we have a detailed taxonomy of atoms organized in terms of their internal structure; atoms are no longer elementary.

In fact, neither are all their immediate constituents. In the paradigmatic Standard Model (SM) of particle physics electrons are still elementary but the nucleons, that is the neutrons and protons, are endowed with an elaborate internal structure for which the ultimate elementarity resides in infinitesimal point particles, the quarks. For the time being anyway, the process of “reduction”—attributing behavior at one level to a further level of composition—seems to have ceased, or at least greatly diminished and it is the electrons (more generally the leptons) and the quarks that are considered to be “atomos”, structureless and without form.

Thus, although its elementary particles satisfy the Wigner criterion for elementarity, the SM simply takes the ontological default position for which those particles are still just featureless, infinitesimal points, a position fraught with mathematical complexity. While it is undeniable that no form or structure can yet be discerned for the electron, and that quarks evidently cannot be detected outside their nucleonic housing (ergo, additional theory to explain

Which brings us to our model, the subject of this paper. As we shall see, this is a model for which particle attributes and particle conformation in space are

But before we delve any further into the particulars of the model, we should mention one more bit of history, an early connection between knot theory and physics. This is the attempt in the mid nineteenth century by William Thompson, Lord Kelvin (in a sense, anticipating our model by something like a century and a half) to associate structure to the atoms known to exist at that time [

However, since we do, the model to be described therefore associates form and structure explicitly to those atomic

Going back to the quotation noted above, the knots in question are

where

A knot encircling a torus.

A clarification is appropriate here: the actual form and structure of the particles of the model are those of the

Which evokes another point of emphasis: it is this

A Moebius strip (MS) and its boundary.

We should point out that it is not only the MS border that is knotlike; we can also view an MS as a

An MS as a concatenation of torus knots.

Thus, concatenation is one way to “

The figure illustrates a number of relationships: the result for

Braids with closure and framing.

Also apparent is the requirement for two traversals to return to the origin in the case of all odd n but only one traversal for all even n, a requirement readily deducible from Equation 1-1. (e.g., The reference condition is

And finally, yet another quotation: “The braided representation makes manifest that all our “particles” belong to the same genus, namely the set of framed (2,

Another way to look at the rungs is as

Trivial and nontrivial vector bundles.

To return to the concatenation approach: a benefit is that it can be used to extend to an MS those results applicable to its constituent torus knots. For example, the speculation that an MS can exist as a soliton [

Which evokes a final historical note: In 1917 Albert Einstein wrote a paper [

However the ambiguity can be resolved by what might be termed “Einstein’s ansatz” in which incoming trajectories are mapped onto the

But consider the

To anticipate what we have to look forward to in what follows, here’s another quotation [

Composites will be addressed in the next section; here we introduce the kernel of our particle model:

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

The basic set of Flattened Moebius strips (FMS).

As per the quotation in the previous section, the boundaries of B and C are folded-over

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

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

which we plot in

Charge

Here

where _{3}_{3}

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

From another point of view, we recognize that the linear constraint imposed above between

where f(

where Δ_{i} is the change in the ribbon’s bearing in the plane at the ith quirk and _{q} Is the total number of quirks. The quantum condition becomes

where

ε_{i = }+1 for ccw rotation (an “up” quirk)

= - 1 for cw rotation (a “down” quirk).

and _{i} is the unit step function at _{i}.

Synthetic approach to flattening.

Up and down quirks; two views.

To discuss composites we need one more definition: associated with each of the four fermions in

Definition of an antiparticle.

As will be seen, this way to distinguish fermions from antifermions is

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

To demonstrate CPT invariance, consider for example the d quirk portrayed in

Reference “d” quirk.

1. It is reasonably clear by inspection that rotations around the

2. A rotation around the y axis produces

Rotated around “

We see that the attributes

and if we associate the value of −1 to a

3. A rotation about the

Rotation around ‘

We conclude that the

There is a

A “d*” quirk.

Now consider how that figure looks as seen from the

Another ramification is the relationship of quirks to

Four Instanton characteristics.

Now consider the same four diagrams

Quirk/antiquirk correspondence to instantons and reverse instantons.

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 [

where

Most of the variation in

Reference [

where

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

However, as shown in

Thus we see here not only a physical

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

“Flattened Moebius strips (FMS)—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.”

The all-important topic of degeneracy will be treated in detail later on but here we begin with a discussion of the

Fusion of

We note the fundamental dichotomy—connections are either horizontal or vertical—a situation reminiscent of the _{α}> and <_{α}>, respectively, where the value of ^{†} to be the disenabling and enabling operators

Also, we define the _{α}> to be (z_{α}) such that z_{d} = _{u} =

These functions turn out to find expression as coefficients in a tensor formalism [

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

Hopf Multiplication and comultiplication.

In the Hopf algebra these operations are representative of corresponding

The

As to those compositional products, two examples of what we shall designate as

Two mutually conjugate 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

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 ^{+ }is modeled herein, namely as a ^{-}

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” [

Once again, our model has been anticipated, although this time by only half a century; thus, our figures are graphic

The fact that traverse in our model does, indeed proceed

where

and summation over

Dirac imposed the SR compatibility constraint by demanding that

be equal to the Klein-Gordon equation,

which is the quantum mechanical equivalent of the Lorentz invariant

with

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

where _{α} = (_{1}, _{2})^{T}, _{β} = (_{3}, _{4})^{T} and

These equations have traveling wave solutions with exponents proportional to linear combinations of the time and space variables—

where

is essentially a spin matrix. Especially noteworthy is the _{α}, constitutes a _{β}, constitutes a

What is intriguing for present purposes is the _{α }and _{β}, the two-vector _{α }depends on _{β }(as modified by the spin matrix) and conversely, _{β }depends on _{α} in the same manner. It turns out that this dependence implies a

We can then demonstrate the

Dirac equation output; a fermion and antifermion bound pair.

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 “

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,

Of course, the actual,

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 [

which equals _{0} + _{1} for the case of first order fusion, that is , for s1 = s2 =1/2.

Specifically, the direct product,

of, respectively, the vector of four basic spin1/2 ^{T}^{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

Notice the two elements in _{0} consists of only two elements, namely, CB* and its complement, BC* the ones discussed above and shown in _{1} 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

where upon twist and charge become

indicating that twist and charge gradients are mutually orthogonal.

Twist loci and gradient; first-order fusion.

As indicated, twist loci are antisymetric about the principle diagonal, which, we note for future reference, also functions as the gradient of

First, the reason we are focusing on

Briefly, more than one

A given

A given

The possibilities for a given

Composition and contingency require a more detailed consideration and will be so discussed in _{2} = _{1}, we are led to the formal notion of symbolic

where _{μ}_{v}_{λ}

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

Operational diagram for convolution.

Note that the summation indicated in Equation (3-18) really denotes

Assembly of convolution products.

Especially noteworthy is that, with the replacement of letters A and B by D and C, respectively, the figure exhibits

As per the comment made above in connection with Equation (3-16), we can view

which equals _{1/2} + _{3/2} for s1 = 1 (or 0) and s2 = 1/2. The result can be viewed as a 4-vector,

whose elements are the matrices

For example, P_{1} is a

Geometrically, we can picture the vector _{k}

Twist loci; second-order fusion.

An equivalent set of inclined planes (not shown) exists for charge and there is a corresponding set of inclined planar charge loci. The two sets of loci therefore intersect this time in a set of lines, discussed in more detail below. Also, for future reference, as was stated in the case of first order fusion, the simplified twist-charge (Gell-Mann/Nishijima) relationship that helped illustrated twist-charge orthogonality for first order fusion implies that

for second order fusion and therefore similar orthogonality for the twist and charge gradients as shown in

Orthogonal twist and charge gradients.

As was the case in first order fusion, the behavior shown in

Consequently, we can

for the summation of Equation (3-19) where, this time,_{μ}_{v}_{λ}

Operational diagram for second-order convolution.

Occupancy of the inclined planar twist loci of second-order fusion.

Convolution output twist assemblies; second-order fusion.

The linear intersections of twist and charge loci are readily visible as inclined groupings in each of the twist loci of this figure. For example, for

All the entries of this figure under the headings of NHT = ±1 and ±3 are

The results summarized in

Call the assembly, at each order of fusion, OF, of the permutations for a given value of twist, a twist assembly, TA. The permutations are organized in permutation groups, one to a combination. There are

(In this paper, we have used

which gives

For the rest of the range of duplication of letters, 0 <

which gives

In summary to this point, one way to

Twist-based quaternary number system; zeroth, first and second-order fusion.

We note that the quaternary system may be viewed as an extension in both directions of the basic fermions as displayed in

To this point we have ignored a major mismatch, namely that between our direct product/convolution approach and what we might describe as a ^{5} = 32

Availability of 32 binary choices in first-order fusion.

Second order fusion can take place in two distinct configurations as suggested by the stick figure representations in

Availability of 256 binary choices in second-order fusion.

To reconcile the discrepancy, the additional degeneracy due to the factors of

Additional degeneracy due to detailed composition; first-order fusion.

AA* ( |
CA* ( |

AB* (2 |
CB* (2 |

AC* ( |
CC* (4 |

AD* (0) | CD* (2 |

BA* (2 |
DA* (0) |

BB* (4 |
DB* ( |

BC* (2 |
DC* (2 |

BD* ( |
DD* ( |

For example, the three d quirks of letter A (d* for A*) count as a single point of first order fusion to form AA* because of the unbroken triangular symmetry of each component. On the other hand, each of the two d* antiquirks of letter B* counts as a potential point of fusion because the symmetry is broken by the direction of traverse and the existence of its u quirk. Hence the term 2

Similar considerations apply in the case of BC* and the two words CB* and CC* in the second column. In that regard, note that inverting, top to bottom, the second column and exchanging

Summing up the terms in each coefficient of the words then produces the total degeneracy for first order fusion as shown in

Composition-enhanced twist assemblies for first-order fusion

We can formally encode this ad hoc compilation [

and an

Here, _{μ}_{v}

For example, to compute the structural degeneracy associated with the fusion of a B and a C* we write

which we recognize as the relevant entry in

In [

The salient feature in

Half of contingency-enhanced permutaions; second-order fusion.

AA*A ( |
BA*A (2 |

AA*B ( |
BA*B (2 |

AA*C ( |
BA*C (2 |

[AA*D] | [BA*D] |

AB*A (2 |
BB*A {(4 |

AB*B (2 |
BB*B {(4 |

AB*C (2 |
BB*C {(4 |

AB*D (2 |
BB*D (4 |

[AC*A] | BC*A (2 |

AC*B ( |
BC*B {(2 |

AC*C ( |
BC*C {(2 |

AC*D ( |
BC*D {(2 |

[AD*A] | [BD*A] |

[AD*B] | BD*B ( |

[AD*C] | BD*C ( |

[AD*D] | BD*D ( |

As an example of the way in which contingency factors determine the entrees of this table, consider the word BB*B in terms of a fusion sequence that proceeds from left to right (although the result is independent of direction): the coefficient in this case is {(4

Half of contingency-enhanced twist assemblies; second-order fusion.

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

Ambiguities in the model-to-SM connection.

In practice, this kind of ambiguity is

Such ambiguity resolution will be illustrated presently in terms of well-known interactions, but in the meantime, we note that, in the

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 ^{0R} and ^{0L}, shown here of the neutral pion rather than the one, ^{0} usually shown in the SM.

Nucleons and excited state interactions mediated by pions.

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

Now, recalling the previous discussion of ^{+} and ^{-}, respectively, note that if

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

In terms of the nomenclature that we found in ^{+} and Δ^{-} are two members of the last triplet in the NHT = −1 column of that figure and Δ^{++} and Δ^{0} are two members of the last triplet in the NHT = +1 column. However we can also express two of the delta particles as

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

We note for future reference that

Delta particles with pion constituents.

As per [

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

which “is the accepted SM composition (also viewed as a ^{0}.” In the same vein, we note that the two pion versions can also form the superposition for the

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

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

where, as defined previously,

First stage of neutron decay.

On the basis of the quirk structure, C and C* translate unambiguously to _{e} (for its lepton number).

All this can be accomplished if we just ^{- }

Second stage of neutron decay.

Muon decay is a similar process; we begin again with an excited state, this time in the NHT= −3 column of

First stage of muon decay.

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 _{e} in order to cancel the electron’s lepton number. And finally, the B of the second fermion must translate to a muon neutrino _{μ}^{-}v_{μ}

Second stage of muon decay.

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

and here (

The bosonic operator matrix in SM nomenclature.

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 ^{-}^{+}^{0}, are in the lower left quadrant and in the (diagonally) next, overlapping quadrant are the pions involved in “strong” interactions: ^{-}, ^{+}, ^{0L} and ^{0R}. In the SM these two groups of operators enact parallel roles in terms of weak and strong

Model of Deuteron stability mediated by pions.

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

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 ^{+} splitting in analogy with the ^{-} and so on to perpetuate the electron/antineutrino pair. That is, we begin by making the correspondences:

which imply the additional correspondences

Thus the above interactions

In other words, the vector bosons of the model mediate lepton pair exchanges in “

Finally, we recall the discussion of the Dirac theory in ^{0}.

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

The basic fermions in order of twist.

These four objects can also be viewed as four points on a line in what we define as “

Basic fermions in “label space”.

The idea here is to initiate an organizing

Three-label scaffold (More to come).

Note that the triangular display accommodates seven appropriately placed intermediate points for a total of

Note also the

The key feature here is that of

Four-label scaffold.

However, accommodating the addition of another flavor (for a total of five) by trying to fit to another regular three-dimensional structure is

(where

In

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

Six-label scaffold; that’s all!

Furthermore, as noted in [

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

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

Sample “duad”.

Following Baez [

The duads can be assembled into groups of three called

Of particular interest because of its correspondence with the three generations of the SM is the

The three orthogonal duads of the first syntheme.

And their normals.

These three mutually orthogonal duadic planes and the associated set of orthogonal axes define the geometry of the icosahedron as a

We are not yet done with the “speculative vein” adopted above: given the duadic planes, the role if any of the associated orthogonal

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

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

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

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

The Spin ½ Baryon Octet.

Nominally we would be left with the intermediate ^{0} and Λ^{0} particles, with the overall result that we realize, as per [

Returning to the

The Spin 3/2 baryon decuplet.

As a final example, we show in

The Spin 0 Octet.

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

Since this paper is not concerned with particle kinematics, position/momentum relationships are not an issue. However, there is clearly a statistical element in the way FMS are combined in the fusion operation and this should rightly be treated in a quantum mechanical manner. We begin by recalling

In detail, the operator role is actually implemented by the individual elements (in other words, the bosons) of matrix M. In this sense, these elements implicate the

where ^{-} = BC*, ^{0L} = BB*, ^{0R} = CC*, ^{+} = CB*.

On another level, we note that a salient feature of quantum mechanics is its formulation in terms of complex algebra. Coincidentally, a salient feature of matrix M, in addition to its role as an operator, is the

and each basic antifemion, say y*, with a corresponding conjugate state vector

In formulating the fusion process, we should like to combine states in a

such that what appears in each exponent is in the nature of the

From a little different point of view, we note that each of these exponential functions is actually the

Some bosonic twist-charge state vectors.

Clearly, fixing twist fixes the real component of each of the associated complex constituent vectors, which therefore differ only in their

The transform can also be used to illustrate another connection to the quantum mechanical formalism, namely the algebra of commutativity. Consider the commutator (actually expressing the fusion of x and y* or y and x* to create bosons).

where, as before, _{1} + _{2}. Clearly, we have commutation only for _{1} = _{2} (or x = y) which translates to bosons located along the twist gradient of matrix M (or

We also need to address the formulation of state functions that incorporate the

where

and the values of the

For example, for

so that Eqation (9-5) evaluates as

Second order fusion is understandably more complicated: we write

where

For example, for the case of

so that Eqation (9-7) evaluates as

In summary, the foregoing demonstrates a formulation that combines complex algebraic precepts of Quantum Mechanics with the degeneracies and primitive combinatorics inherent to the FMS genus.

The model described herein is based on quite a small set of basic notions:

1. The particles of the model are localized

2. There is an underlying

3. To create a basic set of particles, each of the four MS is

4. A counterclockwise

5. Time is identified as the odd dimension in the 2 + 1 dimensions occupied by each FMS

6. A corresponding basic set of four conjugate particles (antiparticles) is defined by a

7. All other particles are constructed as composites by fusing basic particles and antiparticles.

From the point of view of

Flattening and traverse really need to be discussed together. Flattening gives us the two dimensional FMS planform and the small basic set of FMS. However, it is in

Identifying the out-of plane dimension with time gives us CPT invariance and the notion of antiparticles as particles moving backward in time.

And finally, the notion of fusion makes possible the realization of all values of twist by the composition of basic FMS and the notion of

Once, what may be termed the “axiomatic” basis of the features listed above is in place, there follows, almost inexorably, the development of a set of interaction models and a taxonomy organized in terms of the product of SU(2) vector spin spaces, followed by the combinatorial analysis of composite degeneracy in terms of twist and charge categories. That analysis is formalized as a process of symbolic convolution the results of which are displayed in geometric fashion. The analysis of

A ubiquitous,

Finally, we note that this model is a work in progress. For one thing, as it stands the model’s ability to discuss interactions is hampered by a major shortcoming, namely the lack of a dynamics. On the plus side, connections to other aspects and areas of physics and mathematics, some of which are discussed below, have emerged and continue to do so. These need to be pursued.

While this paper has emphasized the role of Moebius strips as the basis of elementary particle ontology, clearly, the interdependence of MS and torus knots is fundamental, whether we describe the latter as the boundary of the former or, conversely, the former as the concatenation of the latter. All of which makes the knot connection to physics important to our model and this paper, a connection that, as we saw goes back to the valiant mid-19th century efforts of

The Fermi-Yang paper has another interesting connection to our model, one presented as a highlight of a paper submitted to another journal [

Nowadays, there is a large and growing literature [

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” [

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 [

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

An approach to validating the model as a Topological Quantum field Theory (TQFT).

With regard to the lower branch of the figure, the equivalence between the

Although that logic leaves out the demonstration of a common

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

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 ^{+} meson (

Regarding the meson we note that all bosons in our model consist of two quirks and two antiquirks. As to the pentaquark, the report (^{+} meson. As pointed out in [^{+} into a ^{+} by the change of a d* antiquirk into an s* antiquirk as portrayed in

Our model of the pentaquark.

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 ^{+} which tallies with the figure. Also found were a proton and a ^{-},which as in [^{-} that has similarly decayed into a ^{-}.

Source of additional decay products.

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

One approach to answering that question is to turn the

One possibility is to invoke the notion, introduced in

This Appendix is essentially a summary of [

where the

On the basis of the knot’s characterization as noted above, plus the implied differentiations

where

where ^{2} + ^{2}^{2})]^{1/2}, which, is

This is the

Most of the variation in

The second approach involves a

where ^{4}/4^{2}/m, c is the velocity of light and ^{-11} m^{3}/kg⋅s^{2} is the gravitational constant.

To specify

Here ^{μk}R_{μk}

is the Energy Scalar, contracted from _{μv}

Actually, what we are really interested in here is expressing the EMT in terms of the CvS, which we can

where we note, the denominator of the RHS embodies both toroidal radii, as is to be expected and the magnitude of curvature is a maximum on the two toroidal equators, vanishing on the polar circles where curvature changes from one sign to another.

Combining equations A-7, 8 and 9 then gives the desired relationship,

which we set equal to

where the integrand has the dimensions of energy volume density. The corresponding

which is Eqation (A-4) to first order in

The paper [

where

Then by considering an MS to be a

We note, however, a major

in order to emphasize that it applies only to traverse to the

but

Thus we see here not only a

In [

Then we differentiate both sides with respect to traverse variable (

which leads to the corresponding dynamic equation, Equation (A-14).

For _{L}

This time, proceeding as in the above produces

which leads to

instead of Equation (A-14), and, as per the previous definitions, but with modifications:

That is, _{R}_{L}_{L}_{L}_{L}_{R}

Finally, it is instructive to demonstrate the extension of the solitonic behavior into the

where

The expression in brackets will be recognized as the

As mentioned in

Half of contingency-enhanced permutaions; second-order fusion.

AA*A ( |
BA*A (2 |

AA*B ( |
BA*B (2 |

AA*C ( |
BA*C (2 |

[AA*D] | [BA*D] |

AB*A (2 |
BB*A {(4 |

AB*B (2 |
BB*B {(4 |

AB*C (2 |
BB*C {(4 |

AB*D (2 |
BB*D (4 |

[AC*A] | BC*A (2 |

AC*B ( |
BC*B {(2 |

AC*C ( |
BC*C {(2 |

AC*D ( |
BC*D {(2 |

[AD*A] | [BD*A] |

[AD*B] | BD*B ( |

[AD*C] | BD*C ( |

[AD*D] | BD*D ( |

The situation here is similar to the case of first order fusion except that the coefficients associated with the constituents of the input columns are the

Junctions and Available antiquirks, First Order fusion

Available Junctions | Available Antiquirks | Available Junctions | Available Antiquirks |
---|---|---|---|

AA*(x) | 2d* | CA*(x) | 2d* |

AB* (2x) | 2(d* + u*) | CB*(2x + 2y) | 4d* + 2(d* + u*) |

AC* (x) | 2u* | CC*(4y + x) | 4(d* + u*) + 2u* |

AD* (0) | 0 | CD*(2y) | 2u* |

BA* (2x) | 2d* | DA*(0) | 0 |

BB* (4x +y) | 4(d* +u*) + 2d* | DB*(y) | 2d* |

BC* (2x + 2y) | 2(d* + u*) + 4u* | DC*(2y) | 2d* + 2u* |

BD* (y) | 2u* | DD*(y) | 2u* |

As an example of the impact of the availability coefficients, recall the general convolutional format for second order fusion. As portrayed at the stage portrayed below in

Second-order Convolution for

However, if we take into account the enhancement indicated in the Available Antiquirks columns in

(Note that we have written d!A instead of 3dA in order to indicate that the second order fusion term with three quirks operates as a single point of fusion in this situation).

The logic of the example used to illustrate how the availability information comes about translates into a formalism as follows: first we note that the enhanced products of first order fusion (in the Available Junctions” columns of _{μ}β_{v}

Here (

which we interpret as subtracting (

in agreement with

The objective here is to summarize a demonstration of the relationship between our model and a

Free and Fused FMS Junctions.

And for comparison, here is the Hopf algebra’s diagrammatical way to picture the two operations of

Hopf Multiplication and Comultiplication.

Diagrammatically, the two sets of operations basically 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. However, there’s more to it than that: the operations of multiplication and comultiplication in the Hopf algebra are representative of corresponding

From the alternative model point of view, the inputs

where

Implementing fusion is then expressed by equating a subscript of

where, with certain restrictions, m can be either

Note, however, that something has been lost, namely, the nature of the crossover junction and knowledge of which FMS are involved. To rectify that deficiency, we invoke the fusion state function introduced in

treating it as an operator and evaluating it as per the delta function—that is, we write

where

We still need to implement the proper association between operator and junction and, to do that we employ, again, the Kronecker delta function but this time operating to the left as well. In short, this leads to the operator

which, when applied to the RHS of 10.9 produces

the representation (as per Eqation C-1) of a free pair in coexistence. For example, for the case, again, of

(Note that the order of the subscripts and superscripts in this result is immaterial here).

We can go on to characterize the Hopf algebraic concept of an

is assumed to commute with the elements of

where ^{2}. For the special case of

Note that only the connected entrees in the LHS of the bracket diagram commute; all of the others anticommute. Finally, the associated

which, for the special case, again, of

The approach here is therefore to associate matrix

Rotating this diagram 90 deg. CW produces the geography of matrix

Clearly, the inverse matrix implicates an FMS ^{-} and ^{+}) and the pair (BC* and CB*) which correspond to the charged pions. In summary, given the identification the model’s fusion and fission operations with the algebra’s multiplication and comultiplication operations plus the identification of bound particle conjugation in the model with the algebraic concept of an antipode, we conclude that our model qualifies as a Hopf algebra, a conclusion that awaits formal documentation.

The object here is to demonstrate a knot theoretic demonstration of how the switching of quirks in beta decay comes about. We begin with

Two Different Intersections.

Suppose we view an intersection as a short duration loop out of the plane, an overpass so to speak, as in

Intersections as Overpasses.

The Difference between the Two Intersections.

This kind of demonstration can also be used to illuminate what takes place in the Beta switch. A simple approach is to begin with

Before and After Antiquirks in the Beta Switch.

The Difference Between the Two Antiquirks.

A more complete analysis carried out in three dimensions is found to lead to the same result.

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

A quaternion can be expressed as

where

While the

where

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

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

Quaternionic matrix subdivision.

Cross product and scalar components of the reduced quaternionic matrix.

If we delete all the terms containing _{0} or _{0} in

In

where with s associated with space, the

and similarly for the conjugate. Instead of

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:

At the same time, we recall that the Standard model employs the Kobayashi-Maskawa (KM) matrix [

Now, if we identify indices _{i}_{j}^{*}

where _{12}, and _{ij} = sin_{ij}, the angle being the “mixing angle” between the _{1}_{2}* and _{2}_{1}* terms correspond to the KM terms _{2}_{3}* and _{3}_{2}* terms correspond to the KM terms ^{2} and –^{2} 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

where

and similarly for the conjugate with the

In the case of first order fusion, instead of

where

Again, if we delete the scalar and vector terms (which involve identity elements

“Pure” quaternionic matrix for the reduced basic fermion vector.

which is matrix

Cross product and scalar components of

Using Standard Model

or, as translated into Standard Model

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

with matrix Ф⊗Ф* of

However, since we are really interested only in the case where

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

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

where

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

Both the Kauffman bracket polynomial [

We begin with the recursion formula of the Kauffman bracket polynomial [

The associated closed form version is readily shown to be in the form of a convolution

where ^{-3}(which is equal to _{-1}) and _{0 }= -(A^{2} + A^{-2}).

Note that with this notation Equation F-1 becomes

The form of these equations suggests an isomorphic relationship to the field of sampled data feedback systems as summarized in the following diagrammatic portrayal. The open switch in the figure signifies that the _{0} input exists for only an initial sample). Due to the circulation around the closed loop, the system response at time ^{u-m}^{n-m}_{0}, switched on at ^{n}_{0}. Note that the output shown in the figure is just Equation (F-3).

Operational model of Eqation F-2.

The simplest example of what is meant herein by a (2, _{1} + _{2}. We find that

Interchanging subscripts 1 and 2 in this equation and collecting the four terms that result into two convolution groups gives the more symmetrical formulation

or, more concisely,

where

The case for three components, _{1} + _{2} + _{3}, proceeds in essentially the same way and in analogy with Equation (F-5) we find

where the coefficients are _{i}^{ni}_{i}

Cyclic permutation of the subscripts gives two more equations and collecting terms associated with each of the three convolutions produces the final, symmetrical form,

where _{1} = _{2}_{3} + _{2}_{3} + _{2}_{3}, _{2} = _{3}_{1} + _{3}_{1} + _{3}_{1}, _{3} = _{1}_{2} + _{1}_{2} + _{1}_{2}.

We can readily generalize to the case of an arbitrary number, _{F}

For any particular value of _{F}_{F}

1. All subscripts, including that of the

2. The number of _{F}_{F}

3. The highest y subscript value is _{F}_{F}

For example, we can write down, ab initio, the equation analogous to Equation (F-8), for the case of _{F}

Not that the meaning of the word "is" is frivolous!.

The search for the Higgs goes on.

More on flattening below.

* Note the analogy to the annihilation and creation operators of quantum field theory

which may have relevance to the subject of particulate genesis (but not in this paper).

Note the analogy to the SM's (enigmatic) switch of the neutronâ€™s down quark to an up quark, by means of which the neutron is converted into a proton plus a W particle.

We note the correspondence to the six-dimensional vector space (three electric and three magnetic} of electromagnetism.

Emphasis added

Not a new result but presented as emerging from a unique basis.

Computational details are provided in [