§1  Coding Fermions

With absolute economy, the set of all 5-bit sequences (in a 2+3 split) encapsulates the repertoire of quantum numbers of a generation of fermions. Though different renditions of this startling regularity have been independently discovered for decades, the matter remains a curiosity in the literature. The table shows how the difference Δ in the fractional occupation by the unit ‘1’ in two segments of a code-string effectively monitors electroweak hypercharge Y = 2Δ.

1. Left-handed modes have an odd number of zeros, rights an even number. Particles interchange with antiparticles by replacing zeros by units and vice versa. We have labelled the fermions with first-generation notation, but “electron,” for example, might well stand for the congeries that is electron, muon, and tau.

2. The three entries under each quark heading in the table distinguish the strong color states. Since the coding of the triplet segment serves as efficient color labelling in itself, we usually dispense with the usual ‘reds’, ‘antireds’, etc.

3. Each 5-bit sequence, being of fixed handedness, corresponds to a massless mode that does not exist as an autonomous entity. The electron, for example, corresponds to a massive dynamical superposition of the two of massless modes 00111 and 01111, which we shall write as “0z111.” The charge assignments will be deemed pertinent only for such massive pairings.

4. A field theoretic model for the z (zbit, zeron, z, ziggs?) is entertained downscreen. It acts like a massive, neutral, real scalar boson. Analogies with the Higgs scheme, and departures from it, will become apparent. The main point is that the z is an ab intra affair of the fermion, in contrast to the received picture in which the fermion suffers an ad extra Yukawa-type coupling to the Higgs. The other 1+3 tetrad of charged slots will likewise be regarded as the result of a dynamic pairing that accompanies the formation of the z.

5. Leptons & antileptons are strong-color null, and one will note the complete filling and the complete absence of units within the triplet segment of their respective codings. Thus one may speak of whites (electrons and neutrinos) and blacks (positrons and antineutrinos) so that all the fermions are captured in a tetrad of reversible colors.

6. Electric charge in units e > 0 can be read off from a code string by using the following register assignment:

Thus the positron 1z000 has electric charge +1 and the electron 0z111 has charge −1, where the z is always neutral. The singlet-triplet structure of the charge tetrad itself is analogous—with a charge operator replacing a Hamiltonian—to the energy splitting caused by a spin-spin (magnetic) interaction. In the latter case, when an antisymmetric singlet state lies at energy +E above the noninteraction level, then a degenerate triplet of symmetric states lies at −E/3, that is, below the noninteraction level.

7. The difference Q − Δ is the third component of weak isospin, where Δ (half the weak hypercharge) was defined in code terms at the top of the screen.

8. A transition from 10••• to 01••• on the doublet side is associated with the emission of a W-boson with its positive unit of electric charge and weak isospin component, thus conserving weak hypercharge in the transition. By contrast, fully filled or empty configurations on the weak side of the form 11••• and 00••• represent weak isospin singlets. An interchange of identical elements does not change the coding, and the entity does not engage weakly.

9. Analogous interchanges on the triplet side correspond to the color-changing gluon processes. The fact that gluons do not carry electric charge is automatically built in to the code by the fact that such interchanges do not change electric charge. No register flipping occurs across the caesura in the code.

The code-string for a given entity thus provides us with an at-a-glance inventory of its menu of quantum numbers: (1) weak hypercharge, (2) electric charge, (3) strong color or anticolor, (4) weak isospin and third component, (5) handedness, and (6) baryon or lepton number. So much disparate information so economically encoded must invite speculation.

When we pursue geometrical imagery in which each code string represents a point in a 4-d projective space, the 00000 acts as origin, and lies outside the proceedings. In reality (all fermions are massive) this state, formally the left-handed anti-neutrino, finds itself in superposition with its right-handed counterpart 01000. The two modes engender one another dynamically, Pauli style, and thus yoked together yield the massive antineutrino 0z000, with its fully empty charge tetrad.

Ziggs boson anyone?

The Penrose picture of the worldline of a massive fermion in its rest frame is indeed presented as an alternation or superposition of massless modes in interaction. The drawing gives a schematic in terms of the 5-code for the electron. Only the second slot (colored red) can so ‘tick’ because the spectator charge tetrad is fixed by charge conservations. For an observer in the center of momentum frame, the zig and zag segments are equally likely, but for an observer moving relative to that frame, one or the other becomes dominant. The weak interaction is blind to the right-handed zigs (say) 00111 but may be operative on the left-handed zags 01111, where a register flip in the doublet segment produces the left neutrino mode 10111. With the spin direction thus fixed in space, the helicity reversals amount to maximally impulsive momentum reversals, which are assumed due to interaction with the Higgs field. Not so for us. In the sequel we entertain a scheme that proceeds in the spirit of the BCS Cooper-pair formalism in which entities of equal and opposite momenta engage in non-local pairing that leads to a mass gap. The figure is adapted from Penrose, p 630.

§2  Projective Structures from Fermions

“Circles of infinite radius give me a headache.”—Archimedes          

An exhaustive fermion systematics thus depends simply on all subsets of 5 items. More picturesque than code strings is the image of a 4-simplex with its five vertices “lit” (or not) in all possible patterns. These patterns break down into the familiar Pascal sequence of 1, 5, 10, 10, 5, 1 electrically and colorfully neutral families. Another image is provided by the Venn diagram built from five congruent ellipses projected onto a 2-sphere, where the configuration exhibits a 5-fold rotational symmetry as well as a polar symmetry. We will find in this section that the fermions also label cells in the Petersen graph or its dual, which arise from tilings of the projective plane. In fact the fermions produce a wonderful hierarchy of projective structures, including a riveting rendition of Desargues’ theorem.

The 5-bit coding introduced in the foregoing section provides a set of homogeneous coordinates for the points of a 4-d finite projective space known as PG(4,2). The essentials of the particle story can be seen in a smaller arena using the seven leptonic states alone, which is to say, on that particular Fano plane PG(2,2) within PG(4,2) for which the triplet part of a code string is strong-color null. Using the abbreviated boldface notation 1 = 111 and 0 = 000, inspect:

The Leptonic Fano Plane PG(2,2)

The last column in the table is formed as before from the difference in the fractional occupation by the ‘1’ in the two segments of each code string, and gives the hypercharges ½Y of the leptons. The line that is dual to each point abc is formed from all x, y, z that satisfy the condition ax + by + cz = 0. For example, code point abc = 111 requires that x + y + z = 0 in mod-2 arithmetic. That has solution set {110, 101, 011}, namely the left-handed states marked by the little open circles in the figures below. The ‘origin’, the left antineutrino, remains outside the pictures at the massless level.

Two renditions of the 7 points & 7 lines of the leptonic sector. The three left-handed modes (open dots) are assigned to the ideal line, while the four right-handed modes (filled dots) are assigned to the affine sector. Only five of the seven lines are reproduced in the second figure, but it is easy to see what must be connected and why we didn't try to draw them. The red symmetry axis contains the fixed-point states that do not engage in the weak interactions.

Now for the Quarks

Taking the story from 2-d to 4-d allows the quarks perfect accomodation. The seven leptonic states are augmented by 24 quark states to make the requisite 31 points of PG(4,2). From the vantage of any one of these points there are 15 rays (instead of only the 3 in the Fano case) harboring 30 points (not 6) in eclipsed pairs. The set of left-handed states can be assigned to a 15-point subspace PG(3,2), taken as the ideal sector, and all the right-handers are confined to the 16-point affine part. Who knew?

So let's look in more detail at the 15-point subspace that harbors the left-handed states alone. This is the 3-d counterpart to the circle in the first Fano figure drawn above. But here follows something truly striking: In the drawing below, the vertices of the two shaded triangles are occupied by the six color states of the left quark doublet, and it is noticed that these are in perspective from the positron at the top. Desargues’ theorem applies directly, and demands that the antiup quarks form a projective axis from which the other quark triangles are in dual perspective. They dutifully comply, as may be checked by doing the binary sums on the lines.

Ten ‘Desargues States’ as Section of a Complete 5-point

In the figure above, the vertices of the complete 5-point (all of code weight 4) are labelled 1 to 5 and are indicated by the open black circles. Taking those in pairs we can find the third point on each of those lines, which are all of code weight 2. Coxeter’s Projective Geometry (2nd ed. p 19) gives a crafty proof of (the converse of) Desargues’ theorem by making use of this auxiliary construction, though not in the context of a finite field.

Coxeter suggests coloring the Desargues configuration of ten edges and ten vertices in such a way as to reveal a pair of dual subconfigurations. A complete quadrilateral (4 edges & 6 vertices) is so situated that its vertices lie on the edges of a complete quadrangle (4 vertices & 6 edges.) The former has 2 lines passing through each point with 3 points to a line, while the latter has 3 lines passing through each point with 2 points to a line, i.e., a tetrahedron. In left-handed fermion terms, the positron & down triplet compose the quadrangle, while the up and antiup quark triplets compose the quadrilateral. Done this way, these dual structures are separately charge neutral. One can see this construction easily in the preceding figure by noting that the up and antiup quarks are coplanar and form the vertices of the quadrilateral in question.

Another suggested coloring reveals the Desargues structure as a pair of pentagons “mutually inscribed” in the sense that consecutive sides of each pass through alternate vertices of the other, a result discovered by Graves in 1839.

It is a pity that those authors could not have seen these pretty systematics mirrored by the basic ingredients of matter.

Above: a different view the ten ‘Desargues states’ assigned more symmetrically to the 4 vertices and 6 edges of a tetrahedron. The five-point has not been included in the figure, but they are easily imagined, for its vertices consist of the centroids (call it) of the four triangular faces, together with the centroid of the large tetrahedron itself.

States that constitute a Desargues configuration are limited to left-handers, which is a curious manifestation of the maximal violation of parity. In fact, had we not known about parity asymmetry before now, we would have good reasons to be disturbed by the potential physical consequences of this feature of binary arithmetic. A parallel case will arise presently that shows a fundamental distinction between particles & antiparticles, between color & anticolor.

The Affine Sector of PG(4,2)

Shown in this next figure is the complement space to the ideal space of left-handed states dealt with above. This tesseract contains all the right-handed states as a 16-point (4-d) affine space. The set of all ideal third points on each edge are the left-handers shown before, and can be found as always by addition mod-2.

Different separations of the particles can be assigned to the affine and ideal parts. While the figure preceding is with the set of right-handed, it can also be done with the the weak doublets alone, or just matter states in contrast to antimatter states. When strong color charge is not distinguished, the tessaract reduces simply to a square (the Fano quadrangle pictured up-screen.) There are other interesting divisions that put sixteen of them on a Boolean logic lattice of two propostions. This can be done in five ways, one of which is shown below:

Above: An affine space isomorphic to a 2-proposition logic lattice. The right neutrino is the tautology at top and (any) one of the five basis vectors can serve as the contradiction at bottom from which all else in the family is consequent. The states in each corresponding ideal sector of the full space have, in effect, some particular color suppressed because they are all confined to this affine part.

Tilings of the Projective Plane

The full 31 pointset of PG(4,2) finds a natural labelling through the minimal triangulation of the real projective plane. It needs ten triangles sharing 6 vertices. The points on the outside edge of the figure are identified with their antipodes, i.e., the point in the top right corner is the same as that in the lower left, and so all the way around.

Above: The minimal triangulations of the real projective plane (in blue) is superimposed on a Desargues configuration (red) so that each point of the latter occupies exactly one of the ten trianglar faces.

Quarks & the Petersen Graph

Above: There is a dual tiling (red in the figure) with six pentagonal faces known as a Petersen graph. Its 10 vertices and 15 edges can be labelled in terms of the six numbers used in the figure preceding. In contrast with the Desargues configuration, three of the edges (α, β, γ) of the Petersen configuration pass through points at infinity.

The table below shows all 31 massless modes mapped onto such labelled cells. These consist of 6 vertices, 15 edges, and 10 triangles. The icosahedron has exactly double this repertoire, which are then identified in pairs by the antipodal map to obtain the tiling in question. For convenience, the cell labelling has been made consonant with the 5-bit particle coding that we have been using.

The 31+1 Massless Particle Modes Labelled with n-Cells

The ten left-handed Desargues states that we delineated upscreen are all edges in this representation (highlighted in red in the table.) Their corresponding right-handed antiparticles are represented by the ten triangles listed at the bottom of the table, but these do not constitute a Desargues configuration. Just as the code automatically builds in an asymmetry in parity, the code also builds in another sort of maximal difference between particle and antiparticle.

The Massive Pairings

By using the combination Z = I₀ + Γ₄ (under some meaning of the + sign) we can split off a common factor for the neutral z and collapse the rest of the combinatorics into a compact symbolic package for the four mass and four electric charge levels of the first generation of fermions. The boldface notation Γ = {Γ_i}, i = 1,2,3 is used to represent the different color states together.

In terms of the pentagonal tiling, the same troupe of 31 massless states can be described in terms of 6 faces, 15 edges, and 10 vertices of the Petersen configuration. The dodecahedron has exactly double this repertoire, which are again identified in pairs by the antipodal map to obtain the dual tiling. The Euler number is the signed sum χ(projective plane) = 6 − 15 + 10 = +1, which being positive reveals the surface to be elliptic. The genus g = 2 − χ is also +1, consistent with one’s understanding that the real projective plane is obtained by grafting a single Möbius strip onto a sphere.

Summary

The 31 + 1 massless states of one generation organize themselves in a hierarchy of finite projective spaces, each electrically and colorfully neutral:

• PG(−1,2). The empty set as “origin” 00000. With all slots empty it does not engage in standard model interactions. It corresponds to the massless left-handed antineutrino mode, and lies outside the scheme until mass is introduced. In that case it comes into an observer-dependent superposition with the 01000, the massless right-handed antineutrino mode. The massive superposition 0z000 (“the antineutrino”) does engage weakly via its right-handed condition.

• PG(0,2). The single point 11111 is a neutral singlet that corresponds to the massless right-handed neutrino mode, which with all slots filled also does not engage in standard model interactions. When mass is brought into the picture it goes into superposition with the weakly interactive massless left-handed neutrino mode 10111 to produce the observed massive neutrino 1z111.

• PG(1,2). Cyclic triplets. The full space of 31 massless modes contains 155 such lines. All of the 15 lines that contain a state-antistate pair also contains the 11111, which thus symbolically represents their fusion. Or conversely, the 11111 produces all such pairs through its symbolic fission in all possible ways. For example, 11111 = 10000 + 01111 is a separation into the massless right electron and the massless left positron modes.

• PG(2,2). The smallest officially accepted projective space. There are 7 massless states in each of these planes, of which there are 155 in all in PG(4,2). One of these harbors the 7 leptonic states together in a neutral grouping, which includes the 11111. All leptons have a coding of the form ••111 or ••000, and are thus characterized by the absence of strong color.

• PG(3,2). There are 15 massless states in each of these volumes. An important case for our purposes contains all of the left-handed states together in a neutral grouping, and this forms the ideal subspace of the full PG(4,2). A feature of this grouping is that ten of these points constitute a Desargues configuration, while their antistate counterparts do not do so. The 10 Desargues points result from taking a section of the complementary 5-point.

• PG(4,2). There are 31 massless states in the full neutral hypervolume. While the most important division for us is the separation into the 15 left- and 16 right-handed species, there are by duality 31 such divisions in all. A division of a different sort breaks the 4-d hypervolume into the leptonic sector of 7, a Fano plane, and the quark sector of 24, which is not itself a projective space (merely a matroid.)

The numbers 1, 3, 7, 15, 31 are the first few q-analog integers

for the special case of q = 2. Such 2-analogs and their higher counterparts also arise outside projective geometry in diverse algebraic and combinatorial contexts, such as in the enumeration of lattice walks, partitions, etc. They were introduced first by Gauss in his powerful generalization of the binomial coefficients. They are known as Gaussian coefficients or q-Gaussian coefficients. It is a delight that they should show up so overtly amongst the particles of nature. By coincidence or not, the particle information encoded in the

points of PG(4,2) with its three ‘particles’ to the line can be reconfigured as the two dimensional finite plane PG(2,5). The lower dimension is paid for with the use of a higher 5-analog integer:

where now 6 points compose each line. This pair of geometries is special in that it provides the unique example of finite projective spaces of different dimension that harbor the same number of points.

§3  Generating Mass

The idea that interacting massless quanta can lead to bound states of nonzero proper mass is now sketched in the context of the simplest possible field model in spacetime, namely that of a neutral massless scalar field in quartic self-interaction:

where the coupling parameter λ is dimensionless. We make no assumptions about its intrinsic sign at this stage. The sign will in fact determines whether singlet or doublet mass states are generated. This Lagrangian density differs from that of the Higgs by virtue of the reality of the field and the absence of an undetermined pseudo-mass term.

The free hermitian field is represented in the usual way as noninteracting normal modes confined to a volume V:

where kx = −k•x+|k|t and |k| = k is the energy corresponding to 3-momentum k. The operators satisfy the familiar commutation relations and act on the appropriate Fock states. The free part of the Hamiltonian, normal-ordered, is the sum over all number operators weighted by energy:

The constants c and hbar are rendered invisible as usual, here for typing (not aesthetic) ease here in html-land. They will be reintroduced whenever it is not obvious where they have gotten to, or when we want to know whether a term is physically dominant or negligible compared with others.

As to the quartic interaction, only the contributions to the energy that conserve particle number are pertinent to the purpose. Thus any term that does not contain an equal number of creation and annihilation operators is ignored. The time dependence is absorbed notationally through such abbreviations as

The following stew for the truncated interaction Hamiltonian density is obtained from the quartic interaction term in the Lagrangian density given above:

This expression, normal ordered, is now to be integrated over all space V to obtain the interaction Hamiltonian proper. Adding the free part, already secretly integrated above, we obtain the following expression for the total Hamiltonian H. Actually one might just as well have bypassed the preliminaries and posited this form directly:

The δ-functions encapsulate the various ways that momentum can be conserved at an interaction vertex. They allow contribution when and only when each indicated vector sum vanishes.

Eigenstates of H

An attempt is made now to construct eigenstates of the preceding Hamiltonian in terms of states consisting of two massless scalar quanta having equal and opposite momenta q and −q. Introduce the tensor product (diparticle) state of opposite movers:

This state is an eigenstate of the free Hamiltonian with the expected eigenvalue:

To apply the interaction part of the Hamiltonian to |z(q)>, breakdown of δ-functions is needed. Here is one of the three done in detail:

There are thus 12 terms that contribute in equal measure by symmetry. The result of applying the full Hamiltonian to |z(q)> is:

Thus |z(q)> is not an eigenstate of the Hamiltonian when the interaction is non-zero. A way forward comes through entertaining a superposition over the modes q:

where the unknown weighting f(q) is begging to be determined. Applying H to |z> gives:

On account of the obliging symmetry of the second nested sum, interchange the k and q to obtain

The second equality represents the goal, where m is the mass eigenvalue. The equality indeed holds provided the expression in square brackets is set to mf(q). Demanding such and solving for f(q) gives:

where λ is nonzero. The explicit functional form for f(q) is thus forced:

where C is a constant that cancels. On account of the spherical symmetry of f(q) the superposition |z> has the same parity as |z(q)>, which is readily verified to be even. When this explicit form for f is substituted back, the dynamical constraint takes the entrancing form

where λ is nonzero. It is convenient at this stage to pass from the summation to integral form through the usual replacements:

So finally:

The angular integrations simply produce a factor of 4π in this spherically symmetric situation, leaving only the divergent k-integration. The essential physics is governed by the singularity at k = ½m. Let us see what comes numerically if we introduce a Planck regime cutoff at k = M, which depends only on the three universal constants c, G, h and possibly a numerical factor that does not matter, for now. The principal part and pole contribution can be evaluated [e.g. Weinberg p 112] between the limits k = 0 and k = M:

by which device an infinitesimal interval around 2k=m has been excised for orderly proceedings, and the delta function integrated to ½. And so obtain

Now substitute m = α + iβ and equate real and imaginary parts. The latter condition leads to the periodicity relation:

for n any odd integer, positive or negative. Conclude that β vanishes for all n, which is to say that all states have zero decay width. Our ground state is stable as must be, and there are no unstable excited states labelled by n. The eigenvalue constraint thus reduces to a single relation between the groundstate mass m(λ) of a scalar composite and the Planck mass M. With x = 2M/m, the eigenvalue condition takes a very simple form:

We've sketched above the two pieces of the logarithm to show how stable doublet or singlet mass eigenstates can be accomodated depending on the sign of λ. For λ < 0, there is a pair of (“negative temperature”) solutions with both masses exceeding the Planck mass. So far we have no use for such solutions. The unallowed limit λ = 0 corresponds to the zero temperature limit approached from both the positive and negative sides; the former is the unrealized massless limit (off the graph on the upper right) while the latter is the unrealized limit of a degenerate doublet at 2M (off the graph at the bottom.)

A Black Art

For masses of observable interest, m/M < 1, the mass eigenvalue condition has the functional dependence of Bose-Einstein character:

where λ > 0 is acting as the analog to positive temperature. We are surprised that this pleasantry should emerge from a quartic interaction, but by Jove we’ll take it. The −1 in the denominator is wildly insignificant for λ of order unity, so we can view m(λ) as if it were a Boltzmann-like suppression, comely if gargantuan, of twice the Planck mass:

where M = ε/G^½ and ε = (hbar c)^½ is a notional Planck charge. The question is often asked, “With the Planck mass so large why are the observed particle masses so tiny?” Perhaps a dark red exponential may happen along to dampen things down. The peculiar (I’ll say attractive) feature is that the dynamics that brings Planck masses down to the observed particle range is the very same that conspires to bring a system of massless particles up to the observed particle range. Some actual numbers will accrue presently.

With the functional form of m(λ)/2M so pointedly evocative of Bose-Einstein statistics, let us pursue the notion that the exponential is indeed some kind of a state-counting apparatus:

where k is Boltzmann's entropy quantum and S is the entropy of a suitable information micro-surface at the Planck scale. We hasten to add that this scheme does not not oblige us to construe the low-mass scalar z itself as a black hole buried within our fermions.

Experiment: So let us see what happens if we model the information surface by that of a rotating black hole of mass M, zero charge, and specific angular momentum a = Jc/GM (expressed thus in mass units.) The horizon area A is given by the Kerr form:

In the case that a² = M² = hbar c/G, the implied angular momentum J is simply hbar, a circumstance that is calculated to warm hearts everywhere. The horizon area is thus minimized at

We imagine a smallest “physical length” to be the circumference C of some Planck-scale sphere or loop; such a length after all has no meaningful beginning or end point. The abstract radius ρ of such a loop, which effectively defines a smallest rotation radius, can still be used as a basic length unit, and that we avail ourselves of. Set

With the obliging 4 already in place, the Bekenstein-Hawking entropy then assumes the form

That the h (and any multiplicative factors that might come with it) has safely cancelled away in the entropy indicates that the classical-quantum interface has been negotiated without offense to either side. The jiggery-pokery with the units was clearly done to obtain an expression that invites cordial comparison with the argument of our damping exponential (which is the thing we do trust here.) In this way, massless field theory is providing the anchor for establishing the most appropriate units. In any case, with the choice thus set, identify:

thereby fixing the value λ = 1/3, in units of Planck charge per Planck charge. In a coincidence of minor merriment, this coupling strength yields a mass value in the vicinity of the top quark mass:

where 2M = 2ε/G^½ ~ 2.441 × 10⁺¹⁹ GeV/c². In the imagery described upscreen, the top quark mass is thus “mostly z.” The counterpart statement in the Higgs picture is that the Higgs field couples most strongly to the top quark. Full Planck charge λ = 1 formally yields

Values at this level are sometimes associated with the threshold of the unification scale. But this coupling strength takes us well under the minimum permissible entropy, and the z should not exist there. When further high-mass fermions are discovered we shall erase these words with no little alacrity and some hilarity.

The maximum entropy case, namely the static Schwarzschild limit of the Kerr black hole, corresponds to zero angular momentum, a = 0. The horizon area is simply double that of the preceding minimum entropy case, and so leads to a halving of the coupling to λ = 1/6. The answering mass value is below the threshold of current observation:

We would be forced to say that this what the lowest neutrino mass would be before reckoning its fermionic nature, i.e., before a dynamic accounting is taken somehow of the attendant tetrad of slots in the fermion coding. In any case, we have found that the extremal entropies do indeed provide mass values that bracket those of the observed particles, if perhaps generous to a fault at the lower bound. The observed range of particle masses, though covering many orders of magnitude, is still but a sliver of the a priori range, and we can be moderately encouraged that this sliver has been essentially identified dynamically in the entropy approach. If, for example, the dimensionless scalar coupling were only modestly weaker, say matching that of the dimensionless electromagnetic vector coupling strength, then:

The exponential exp(4π²/3λ) emerged naturally in the simplest consistent interacting massless field model in spacetime. As it happens, precisely the same exponential appears in the Higgs formalism [e.g. Novaes 2000, p 86] where it serves to establish an upper bound on the Higgs mass. Though arrived at in a different way (a renormalization group technique was referred to in the Novaes lectures) the form is owing to the shared assumption of a quartic self-coupling. This agreement somehow gives us confidence in the technical correctness of the diparticle state construction. The trenchant difference is that the Higgs first shows up for duty with an undetermined imaginary mass, while the “non-local constituents” of the z are pristinely massless. Another encouraging feature is that the coupling parameter runs nicely on the “good side of order unity,” as well it must in a perfect world.

The BCS model of superconductivity was the proximate inspiration for the scheme presented here. Despite the starkly different physical bases of the two models, namely charged massive fermions in a metallic lattice versus neutral massless bosons in vacuo, the assumption of interacting pairs of oppositely directed momenta is evidently enough in itself to provide significant parallels. The Planck cutoff finds a counterpart in the Debye energy cutoff of lattice phonons. The binding energy of the electron pairs in a superconductor exhibits the same nonanalytic functional dependence on the coupling strength as found here. In both situations bound states emerge for arbitrarily weak coupling, and the dependence at vanishing strength is that of an essential singularity.

This page sporadically emends itself. The basic ideas behind this approach to mass generation were first introduced in unpublished lecture notes of Prof. F.A. Kaempffer, University of British Columbia 1971. The authors, Carol & Val von der Lin, are cranks who are retired in Calgary Canada. Last update: May 2008.
      If you happen to have an abiding thirst for 19th-century postal markings, do check out our other hobby page at Kashmir Stamps.

§4  References

• Coxeter, H. Projective Geometry, 2nd ed. Springer-Verlag New York, Inc. (1987) The colorings of the Desargues configuration are set up on p 27.

• Georgi, H. & Glashow, S. Phys. Rev. Lett. 32, 438 (1974). The fountainhead.

• Novaes, S. Standard Model: An Introduction (p.86) hep-ph/0001283 v1 27 Jan 2000.

• Penrose, R. The Road to Reality. The ‘zigzag’ electron. §§25.2-25.3. Jonathan Cape. London 2004.

• Polster, B. “Pretty Pictures of Geometries”. Bull. Belg. Math. Soc 5 pp 417-25 (1998).

• Thurston, W. Three-Dimensional Geometry and Topology. Volume I, ed. Silvio Levy. Princeton University Press, New Jersey, 1997.

• Weinberg, S. The Quantum Theory of Fields, Vol I, p 112. Cambridge U Press 2005.

• Wilczek, F. “Beyond the Standard Model: An Answer and Twenty Questions.”23 Feb 1998, hep-ph / 9802400v1

• Zee, A. Quantum Field Theory in a Nutshell, Princeton U Press 2003. (a) A Binary Code for the World, p 410; (b) The gravity of light, p 427.