§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 realizations 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 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 pairing 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 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. The tally is additive from those slots that contains a unit.

Thus the positron 1z000 has electric charge +1 and the electron 0z111 has charge −1, where the z is always neutral.

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.

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 00111 but may be operative on the left-handed 01111, for which a register flip in the doublet 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.

This link to section 3† takes us directly to the mass-generation model downscreen, thus bypassing the following section on the hierarchy of projective structures that the fermion code engenders.

§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. A fermion rendition of Desargues’ theorem is displayed.

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.

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.

The Massive Pairings

Inspection of the table suggests that we may identify a common factor Z for the z and collapse the rest of the combinatorics into a compact symbolic package for the four mass and four electric charge levels of a generation of fermions. The different color states are represented by the boldface notation Γ = {Γ_i}, i = 1,2,3, and Z ~ I₀+Γ₄:

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 Minkowski spacetime, namely that of a neutral massless scalar field in quartic self-interaction:

where the coupling parameter λ is dimensionless. This Lagrangian density differs from that of a basic Higgs by virtue of the reality of the field and the absence of a pseudo-mass term, which we shall introduce presently. The intrinsic sign of λ is left open for now.

The free hermitean 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 constants c and h have been rendered invisible as usual, here for typing (not aesthetic) ease, and will be reintroduced when it is helpful to know where they have gotten to. The free part of the associated Hamiltonian, normal ordered, is found to be the energy-weighted sum over the number operators for each momentum mode:

As to the quartic interaction, only the contributions that conserve particle number, i.e., those terms containing an equal number of creation and annihilation operators, are retained for the present. The time dependence is absorbed notationally through such abbreviations as

The following stew for the interaction Hamiltonian density is obtained from the quartic interaction term:

This expression, normal ordered, is now to be integrated over all space V to obtain the interaction Hamiltonian proper, and to which the free part is to be added. The latter was already secretly integrated above and its time dependence cancelled. One might just as well have bypassed the preliminaries and posited this final form directly:

The δ-functions encapsulate the various ways that momentum can be conserved at an interaction vertex. They allow contribution 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 a diparticle state of opposite movers:

This state is an eigenstate of the free Hamiltonian with the expected eigenvalue 2q:

To apply the interaction part of the Hamiltonian to |z(q)>, breakdown of δ-functions is needed. The first of the six terms is evaluated as follows; under normal ordering the others contribute equally by symbol renamings:

The result of applying the full Hamiltonian to |z(q)> is:

where the term (16k k' k'' k''')^½ in the interaction Hamiltonian has reduced to 4kq. Thus |z(q)> is not an eigenstate of the Hamiltonian when the interaction strength is nonzero. A way forward comes through entertaining a time-dependent superposition:

where the weighting w(q,t) is to be determined. Applying H to |z(t)> gives:

On account of the obliging form of the second nested sum, an interchange of k and q allows a common |z(q)> to be pulled out to the right:

The change of sign in the exponential alters nothing since both k-q and q-k take on all values, positive and negative, under the double summation. The second equality represents the goal, where m is the mass eigenvalue, which must be real and positive. The equality indeed holds provided the expression in square brackets is set to mw(q,t). Demanding such and solving for w(q,t) gives:

It is important that λ be nonzero. The functional form for w(q,t) is thus forced:

where C is a constant that can be used for normalization. On account of the spherical symmetry of w(q,t) the superposition |z(t)> has the same parity as |z(q)>, which is readily verified to be even. The oscillation at ‘double’ frequency is reminiscent of the zitterbewegung in Dirac theory, but the source of the interference is different. In the imagery described upscreen, we were indeed wishing the z to “tick.”

When the explicit form for w(q,t) is substituted back into its defining equation, the dynamical constraint on the mass eigenvalue m is obtained:

where the constant C and the time dependence have both cancelled, as well they must. It is convenient to pass to integral form through the usual replacements:

So obtain

The angular integrations simply produced 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 universal constants and numerical factors that do not matter for the present. 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 solve for the 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 simple form:

The vertical axis is 4π²/3λ (“inverse temperature”) and the x-axis carries the positive mass ratio 2M/m. We’ve sketched the two parts of the logarithm to show how singlet and effective doublets of mass eigenvalues can be accomodated depending on the sign of λ. Below the x-axis is the overly-excited negative-temperature regime for which all mass eigenvalues exceed the Planck mass. Above the x-axis is the positive temperature regime in which the observed particle masses can be set. The point off the graph at the upper right represents m=0, while the point off the graph at the bottom represents a degenerate pair at m=2M. In terms of λ, these unattainable points are really the same, corresponding to λ=0 approached from opposite sides; in fact their unattainability is a mimicry of the third law of thermodynamics.

The eigenvalue condition as it pertains to the right-hand branch of the graph, namely for m(λ) < 2M, is given by:

For λ fractional and positive, +1 in the denominator is insignificant, and we can view m(λ) as if it were a Boltzmann-like suppression, comely if gargantuan, of 2M:

Some numbers. In a coincidence of minor merriment, λ = 1/3 yields a mass value in the vicinity of the top quark mass:

where M = (hbar c/G)^½ ~ 1.22 × 10⁺¹⁹ GeV/c². In the 5-bit coding imagery given before, we would say that the top quark mass is “mostly z,” with little correction due from the charge tetrad. The corresponding statement in the Higgs picture is that the top quark couples “most strongly” to the Higgs field.

If perturbation expansions in λ must be supported, the highest mass that the model can formally accomodate corresponds to λ = 1:

Values at this level are sometimes associated with the threshold of the unification scale. The advance from here to the Planck mass would witness λ passing from 1 (Planck temperature) to infinity, and may be an unphysical domain.

For λ fractional and negative, it is the exponential that is negligible compared with the +1 in the denominator of the mass eigenvalue condition, and all mass eigenvalues associated with the lower right branch of the graph cluster just under 2M, while those associated with the lower left branch are just over 2M.

While we are usually impressed with the great range over which the observable masses are spread (some dozen or so orders of magnitude) that range is really but a narrow sliver of values relative to what could be a priori. That this sliver should be picked out with an amenable range of λ on the “good side of order unity,” is encouraging. If λ took the same numerical value as low as the electromagnetic 1/137, for example, the mass obtained is bogglingly lower than anything conceivable (hand calculators don’t even try.) Even the root of 1/137 ~ 0.085 (that is, in terms of charge instead of charge-squared) yields a mass some fifty orders of magnitude lower than that of the top-quark mass m(1/3), even though the strengths differ only by a factor of about 4 numerically:

The mass eigenvalue condition cannot be realized through perturbation methods launched from the non-interacting situation. Pre-BCS perturbative approaches to superconductivity were blocked in the same way. In fact the BCS approach was the proximate inspiration for the scheme presented here. The diparticle state machinations are analogs to the Cooper-pair formalism. The Planck cutoff finds a counterpart in the Debye energy cutoff of lattice phonons, and the binding energy of the electron pairs in the lattice exhibits the same nonanalytic functional dependence on the coupling strength as found here. In both schemes, bound states emerge for arbitrarily weak coupling, and the dependence at vanishing strength is that of an essential singularity. Despite the starkly different material bases of the two models—charged massive fermions in a metallic lattice vs. neutral massless bosons in vacuum—the assumption of interacting pairs of oppositely directed momenta is enough to provide significant parallels.

The Higgs Lagrangian

We should now like to retrace our massless pairing approach using a Higgs Lagrangian. Everything perks along just as before except that the conformally invariant quartic self-coupling of the massless field φ is now replaced by the shifted device:

where ½μ² = 2λ<φ>² is a constant that introduces a length scale into the proceedings via the vacuum expectation value <φ>. The crossterm has merely the appearance of a mass term, and that of anomalous sign when λ<0. Just as with the non-adulterated quartic interaction, we persist in using an expansion in massless field modes for φ. Only the first term of the Hamiltonian is in need of adjustment:

where again we have retained only the contribution that conserves particle number, and all constants have been dropped. When this new Hamiltonian is applied to the paired state |z(q)> the original energy eigenvalue 2q becomes:

Nothing else is modified, and all proceeds as before. The resulting eigenvalue condition for the composite’s mass m is now given by a more complicated integral having a quadratic expression in the denominator:

where a+b = m and ab = 2μ². By way of check, when b, say, vanishes, then the integral reduces to the original μ=0 form, which is now an extremum case. We seek values for the eigenvalue m when b is non-vanishing. Explicitly, these non-negative real zeros of the quadratic are:

where for convenience the square-root is called m' = a − b. When λ > 0 (so that μ is real) then μ possesses a maximum. The integral can be easily separated into two parts by the method of partial fractions. The result is:

When the square-root vanishes, so that a = b = ½m, the integral becomes

A ½m has replaced the erstwhile m. A μ-dependent range for the mass eigenvalues is thus provided, the extremal values of which are separated by a factor of 2:

where it is the lower bound that occurs for μ=0, and the upper bound for μ maximized. All this is pointedly evocative of black holes that possess some μ-attribute in addition to mass. For example, the charged Reissner-Nordström black hole has event-horizon radii r+ and r− that stand as analogs to the quadratic roots found above:

where μ²=Q²/G. When Q is the Planck charge (hbar c)^½, then m = μ = M is the Planck mass. This case minimizes the entropy, and a factor of 2 separates it from the maximum entropy of the neutral μ=0 situation. Similar antics occur in the case of Kerr geometry, where μ instead represents specific angular momentum. For a Kerr hole of Planck mass, the minimum entropy occurs for angular momentum hbar, the smallest non-zero value permited for a boson. Again, a factor of 2 separates this minimum entropy from the maximum entropy that corresponds to the spinless μ=0 situation. For both geometries, intermediate entropies must be absent on account of the respective quantizations. If the analogy held for the particle model, we would conclude that if we cannot achieve the theoretical maximum for the vacuum expectation value, then it must be zero. I suppose we should take a look what that maximum would be. Using as before the value λ = 1/3 obtain:

This value is half the oft-cited value for the Higgs vacuum expectation value that signals the electroweak unification scale. And our value for the vacuum expectation value at the same strength is:

We hasten to affirm that we are not construing the state z as a black hole when λ is fractional and positive, for such strengths confine z’s mass to the observed particle range where the Minkowski approximation, and hence ordinary field theory, is held to be valid. Nor is λ permitted to breach the high-temperature transition to negative values. For with μ thus rendered imaginary, b becomes negative, which is impossibile. This extended model confines us to the cis-planckian world. We take this result as a hint that we should be barred from the trans-planckian regime in the original model as well. We can be sanguine about μ and <φ> vanishing, for unlike the Higgs case, we do not need them for generating mass.

Intermezzo, With Photons and Trumpet

If one were to embark on a photon model along the same lines, there is a way, courtesy John Wheeler, that the Schwartzschild radius R for the Planck mass M establishes a gravitational length scale from the beginning. As in the scalar case, the free field photon operator A(x,t) can be expanded in non-interacting normal modes, but now there is an additional polarization sum over two states σ = 1 and 2. Opting for the integral form in Gaussian units, write:

The two polarization vectors are distinguished through the contrasting cross-products

Wheeler now multiplies A(x,t) by the crafty geometrical scaling factor G^½/c², which yields the “reduced” dimensionless operator:

where R = 2GM/c², M = Q/G^½, and Q = (hbar c)^½ is a notional Planck charge. With spherical symmetry, which we shall enjoy for the same reasons as before, even that 4π in the denominator goes away. These are attractive units indeed.

Corresponding reduced versions of the electric & magnetic fields, which are derivatives of the a(x,t) in the usual way, then take on dimensions of inverse length. Quartic combinations can then be treated much as in the scalar case under the safety of a dimensionless coupling constant. The derivatives are

The transverse gauge has been chosen, so that e(x,t) does not involve the gradient of a reduced scalar potential. The polarization vectors change sign with momentum, and for oppositely directed pairings, we have three possibilities (+1,0,−1) for the dot product:

Photon states composed of equal and opposite momenta will serve our purposes only if they are eigenstates of parity, specifically even parity. That is a three-step process: Since the states of linear polarization are not eigenstates of spin, we must first introduce circular polarization. These can be obtained from the linear polarization vectors by rotating their reference base by π/4, thus achieving maximal mixing. Abbreviate:

Step two, construct four different helicity states from the two circular polarizations:

Step three: Under parity |RR> becomes |LL> and vice-versa, so the sum |RR>+|LL> is an eigenstate of even parity, and has therefore the quantum numbers appropriate to a neutral scalar in the center-of-momentum. This state, call it |z(k>, is the one we pay attention to. The mixed states |RL> and |LR> are also of even parity, but for spin 2. The only odd-parity state is the combination |LR>−|RL>, which corresponds to a neutral pseudoscalar. If the scalar exists, perhaps they all must? That is not perhaps such a pleasant prospect in these circumstances.

The quartic interaction, analog to λφ⁴, that Misner and Wheeler (1957) experimented with in a now-classic paper is given by the following expression:

where φ=e+ib. Henceforth write e² for e•e, etc. According to the received discipline, the reduced momentum density for this interaction part is formed from

from which the (non-invariant) interaction Hamiltonian density is constructed:

A feature of this RMW interaction (the R is for George Rainich) is the invariance of the Lagrangian under a rotation of the components of the field pair through the same arbitrary angle ω:

The π/2 case is familiar from the free Maxwell equations. The magnetic field B can replace the electric field E, provided E become −B. A general “duality rotation,” as it is known, is not always visualizable as a rotation about some axis in the 3-space of the fields themselves, for it is a curious affair of a 6-space.

For plane-waves, the six real degrees of freedom in φ reduce to four through a self-orthogonality (null) constraint φ•φ=0, which insists that the field vectors are of equal magnitude |e|=|b| and are mutually orthogonal, e•b = 0. Mass generation depends on departure from these null conditions, a feature already suggested at the kinematic level (discussed in the appendix downscreen.) A free-Wheelerism has it that such non-null fields have acquired a definite “complexion.”

The scalar and pseudoscalar combinations that make up the Lagrangian constitute a duality pair of their own, with doubled angle, as substitutions from above will reveal:

The duality invariance of the Lagrangian follows by squaring and summing these lines. In null conditions, ω is undetermined. Just as the duality pair (e,b) was combined to form the complex φ = e + ib, so this higher-order duality pair can be combined to form the higher-order complex Φ:

The interaction density thus casts itself into the simple form ¼λΦ*Φ, which vanishes in null conditions. Further manipulation reveals it to be the Lorentz-invariant composed from the energy density of the field:

and the Poynting flux density, analog to momentum:

Individually, ε and p are not invariant operators, but the difference of their squares is, hence the identification of a mass-squared density (operator) in the final expression:

where ε²−p•pc²=M²c⁴. The resultant combination 2GM/c² is precisely the form of the Schwartzschild radius R for mass M. Thus identify

Despite the superficial appearance, we cannot forget that R is an operator density here. When e and b are not orthogonal, there exists a local Lorentz frame S' in which they are observed parallel (e'•b'=e'b') and the Poynting flux density vanishes. This is the frame in which an observer can be said to be locally at rest with respect to the em field, and it is the natural vantage for spying on proper mass.

... to be continued, um, maybe. This is starting to feel a lot like work.

Appendix A: The Mass of Massless Systems

Massive systems can be constructed from massless constituents even at the kinematic level. Not much construction is needed; the difficulty would lie in not producing mass. Indeed, the mass “hiding in the light” of an ordinary flashlight beam that exhibits the slightest spatial divergence or convergence has a proper mass that can be estimated. The simplest case suffices to tell the tale. For a single free photon of energy E and momentum p consider the relativistic invariant

Its vanishing asserts masslessness of the photon. The extension to two and more free photons considered as a single system immediately gives us something new because the total momentum can vanish as a vector sum even though the contributing momenta are not zero. Write p + p' and E + E' for the total momentum and energy, and consider the counterpart expression.

This quantity, also a relativistic invariant, does not vanish in general on account of the pesky cross-terms. In the center-of-momentum frame, in which p = −p', it falls to the square of the total energy to carry the invariant value, which we can call (mc²)². Any inertial observer then identifies the fixed number

with the proper mass of the 2-photon system. Upon substituting the single-photon relationships E = c|p| and E' = c|p'|, and expanding the squares, obtain:

where θ is the angle between p and p'. In the extension to n massless particles of total momentum p + p' + p" + ..., a sum of n(n − 1)/2 such terms ensues, one for each pair i = 1,2,3, ... of photons:

Differently moving observers see a different set of angles, but the energies conspire to keep m a fixed constant for all. Only under the idealization of precisely parallel motion (the limiting plane wave) is the total proper mass of a photon system additive, that is, equal to zero. Parallel motion is also distinguished in general relativity [e.g. Zee], for while divergently directed photons engage gravitationally, those in parallel motion do not. We shall be concerned with the antiparallel case that allows us zero net momentum pairings and maximal system mass.

For a single monochromatic point source of photons of energy E propagating outward in spherically-symmetric fashion in the rest frame of the source, the proper mass of the radiation field is well approximated by

which is negligibly different from nE/c² when n is large enough. This quantity must be distinguished conceptually (and numerically) from the “mass equivalent” of the energy nE held in total by the individual massless quanta.

With some thousand million photons per extant baryon in the universe, there are a surfeit of contributing divergent pairs to build a certain amount of system mass into the microwave background. Paired trajectories need not be in mutual proximity to make this contribution.

Massive particles. The general nonadditivity of proper mass is not confined to the massless case. Given an isolated system of N massive particles, the sum of the proper masses of the constituents is in general less than the proper mass of the system taken as a whole. And as before, additivity holds only in the special case of all velocities being identical, which is to say, in both direction and magnitude. This demand on magnitude was simply automatic in the massless case.

Take two neutrinos, each of rest mass m, and respective energy-momentum (E, p) and (E', p'). Consider as before the invariant mass M of the system taken as a unit:

With E = γmc², p = γmv, γ = (1 − β²)^−½ and β = |v|/c, and likewise for the primed quantities, we can express this invariant in terms of velocities. Substitution yields

where k(θ) is the kinematical factor in brackets and θ is the angle between p and p'.

(i) The additive case: In the special case of equal velocities, β = β' and cosθ = +1, then k = k(0) takes on its minimum value of +2. In that case we have M = 2m for all β.

(ii) The maximally nonadditive case: In the case of equal speeds but opposite direction, i.e., cosθ = −1, the kinematical factor assumes the form

Now it is only for vanishing β that M = 2m. For speeds near the speed of light, such as is characteristic of cosmological neutrinos, the proper mass of each oppositely directed pair is well approximated by M ~ 2mγ, where γ is arbitrarily large. That looks harrowingly like additivity of relativistic mass, but proper mass it is for the two-particle system in this situation.

Neutrinos thus contribute to the system mass of the universe, not so much on account of their large numbers, but rather on the number of their pairings. And again, like photons of the microwave background, the paired trajectories need not be in mutual proximity to make their contribution.

§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.

• Lightman, A., Press, W., Price, R., Teukolksky, S. Problem Book in Relativity & Gravitation, problems 4.2-4.6. Princeton U Press (1975).

• Misner, C. & Wheeler, J. Ann.Phys. 2 525 (1957).

• 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.

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 von der Lin & Val von der Lin, are hobbyists, retired in Calgary Canada. Last update: June 2008.
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