Proc. Natl. Acad. Sci. USAVol. 88, pp. 994-998, February 1991Physics

Is the cygnet the quintessential baryon?(Cygnus X - 3/isospin/chronometric theory/elementary particle/cosmic ray)

I. E. SEGALMassachusetts Institute of Technology, Cambridge, MA 02139

Contributed by I. E. Segal, July 5, 1990

ABSTRACT The apparently new hadron-like particle("cygnet") indicated by cosmic ray observations on certainneutron stars is predicted to be a spin 1/2 fermion of magneticmoment and charge 0 and lifetime w. This derives from thenatural identification of the cygnet with the one hithertounobserved fundamental fermion of chronometric particletheory, the x or "exon", which plays the role of a quintessentialbaryon. The "partons" are represented by the other funda-mental fermions, consisting of e, ve, and v,,; e.g., n = x + e++ e-, p = x + e+ + Ve. With further empirical assignments,chronometric theory has a potential for explaining diversephenomena, such as mixing in the neutral kaon complex andthe nature of the higher electrons. Its fundamental fermion andboson fields transform indecomposably under its symmetrygroup, the conformal group G. Theoretical elementary parti-cles transforming irreducibly under G derive as successivequotients in a maximal chain of invariant subspaces. Massfixing by Mach's principle breaks the symmetry down tomicroscopically observed covariance with respect to the Poin-care group Po. The resulting representation is normally irre-ducible, but splits in the case of the K0 into two PO-irreduciblecomponents that are mixed by the excess of the chronometricover the relativistic energy ("gravity"), which provides a"superweak" force that may be explanatory of CP violation.

The isospin idea due to Heisenberg (1) has provided amathematical formulation of similarities between differentnuclear states. Further analysis by Breit, Condon, Wigner,and others led to general acceptance ofthe principle ofchargeindependence. Isospin analysis became central to the theo-retical treatment of the strong interactions; this inspired thederivation of the equation of Yang and Mills (2); this in turnbecame the prototype for modern gauge theory, which formsthe mathematical basis for the "standard model" representedby quantum chromodynamics in conjunction with elec-troweak theory.

Paradoxically, these last developments have not beenextremely successful in illuminating the basic phenomenafrom which the underlying theoretical formalism originated.Thus striking observations pertinent to the theory (e.g., theequality of the charges on the proton and the positron) areunexplained or appear rather accidental. The standard modelprovides a widely used and fairly coherent description ofparticle phenomenology but at the cost of hypothetical par-ticles of questionable ultimate reality, the involvement of alarge number of adjustable parameters, and the absence ofcorrelation with fundamental macroscopic issues such asgravity.The mathematically attractive string theory has in part

emerged from the standard formalism. Unfortunately, it ispresently lacking in empirical correlation and, on the theo-retical side, its consistency at a fundamental level withrelativistic causality and energy positivity appears uncertain.

Chronometric physics is an alternative fundamental theory(refs. 3 and 4 and references cited therein) that derives fromvery general considerations of causality, stability, and sym-metry. As such, it is naturally slightly abstract, and itsempirical implications require development, which like thoseof special relativity and quantum mechanics may initiallyappear contradictory ofaccepted doctrine. But its applicationto extragalactic astronomy (refs. 5 and 6 and references citedtherein) has shown that it is capable of precise and detailedpredictions regarding the cosmic redshift and other directlymeasured quantities in objective samples, notwithstanding itslack of adjustable cosmological parameters, as well as ofsimple explanations of otherwise obscure phenomena. Inparticular, the theory provides an explicit and statisticallyconsistent estimate of the cosmic distance scale, from directobservations on "superluminal" sources (7). This fundamen-tal length R, or "radius of the universe," determines inconjunction with h and c a physically complete set of naturalunits that are G-invariant. Relativistic quantum theory ap-pears as the limiting case of chronometric theory as R -X00in a sense similar to that in which classical physics appearsas the limit A-+ 0 of quantum theory, and nonrelativisticphysics appears as the limit c -Xoo of relativistic physics.Gravity is coherently and tenably representable as the excessof the chronometric over the relativistic energy (8, 9), whichin the case of free photons represents the cosmic redshift (3).At the same time, Friedman-Lemaitre cosmology (FLC)

appears seriously flawed by its inability to fit direct obser-vations on objective samples of galaxies and quasars, not-withstanding its two adjustable cosmological parameters.The hypothetical and unobservable perturbations that havebeen postulated to adjust this fit are contraindicated from anobjective scientific standpoint by the fact (e.g., refs. 5, 10, 11)that the pattern of deviations of FLC predictions is identicalto what is predicted by chronometric theory for the results ofanalyses predicated on FLC.

This note represents a step toward the empirical correla-tion of chronometric theory at the opposite distance extreme:i.e., particle observations.

The Quintessential Baryon

The idea of charge independence can be formulated perhapsmost directly by postulating the existence of an underlyingelectromagnetically noninteracting particle x, of which n andp are composites, together with the corresponding electronsand neutrinos: n = x + e+ + e-, p = x + e+ + Ve. Inchronometric theory, there exists a fundamental bare particleof precisely the requisite character (12). It is a massive spinorthat transforms under the conformal group G with the weight(= conformal dimension) w = 5/2 that is required for dualitywith the electron, for which w = 3/2, and accordingly has noG-invariant interaction with the photon, for which w = 1 (thesum of the weights of fields having a G-invariant nonderiv-ative coupling must be 4). At the same time, cosmic rayobservations at varying but persistent levels of significance,including two recent independent observations (13, 14), have

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been indicative of a neutral extremely long-lived hadron-likeparticle coming from Cygnus X - 3 (the "cygnet;" refs. 15 and16) and from Hercules X - 1 (17). The chronometric exon isa logical theoretical counterpart for these particles. Its baremass is the same as that of the electron, but its physical massis presumptively fixed by Mach's principle and could be quitedifferent from that of e. If it is of the order of the neutronmass, confusion between x and n could be a factor in themany revisions in the estimated neutron lifetime in recentdecades and an anomaly in neutron scattering (18), butobservations on Hercules X - 1 suggest that it may be fairlylight, in which case it may be confused with a neutrino, ifproduced in high-energy collisions.

Notation and Background

Chronometric theory treats Minkowski space MO togetherwith its group of global causality-preserving transformationsP (the Poincare group PO extended by scaling, hence ofdimension 11) as a causally (= conformably) correct butmetrically distorted "flat" map of the true "curved" space-time M consisting of the Einstein universe E as a conformalmanifold, together with its group G of global conformaltransformations [i.e., the 15-dimensional group SU (2, 2) in amore global form]. The metric on E is dt2 ds2, where t istime and ds is the element of arc length on the sphere S3, c

and the radius R of space being taken as 1, which togetherwith h = 1 fix chronometric units, which will be used unlessotherwise specified. K denotes the Einstein isometry group,consisting of all translations in time and rotations of space;the chronometric or Einstein energy is the infinitesimalgenerator of time evolution in E, as represented quantummechanically. Relative to any point of observation in M, MOis imbedded P-covariantly, and the relativistic or Minkowskienergy is the infinitesimal generator of time evolution in MO,relative to the Lorentz frame in MO, that osculates maximally,at the point of observation, the frame defined by the space-time splitting in E. For any unitary positive-energy represen-tation of G, the corresponding Einstein energy exceeds theMinkowski energy by an amount that vanishes infinitesimallybut increases with the spatial support of the state in question.The excess in the case of the presumptive quantized field ofall elementary particles in the universe represents gravity.The inertial mass of a cosmologically long-lived particle isrepresented in accordance with Mach's principle as its inter-action energy with the cosmic background and is correspond-ingly only K-invariant (implying approximate local PO-invariance of its rest mass). (Additional background on

chronometric theory is given in refs. 4 and 19-24.)

Indecomposable Elementary Particle Associations

The elementary particles in chronometric theory are closelyintegrated into coherent entities that will be called clans,consisting of all fields onM having designated transformationproperties under G; thus the particles represented are "offthemass shell" and have all theoretically possible masses for"bare" particles. Scalar, spinor, and vector elementary par-

ticles arise as subunits, and the fundamental interaction isbetween the fermion and boson clans as entities, the totalinteraction Lagrangian being representable as a sum of in-

teractions between individual elementary particles only in therelativistic limit. These features originate in the indecompos-ability of the fundamental clans as representations of G. Thiscontrasts with the complete decomposability built into con-

ventional particle families, as representations of Po andpossible internal symmetry groups. Since experimental data

are currently reduced and reported in terms of a fullydecomposable relativistic model, the empirical correlation ofchronometric theory requires an explicit correspondence

between clans and the corresponding families of individualparticles.To clarify this correspondence, which is basically of a

general group-theoretic character, I distinguish between areduced particle, a theoretical entity that is extracted froman ambient clan by the formation of subquotients, and anexact particle, which is represented by a vector in the clanand corresponds more precisely to a free physical state. Asubquotient of a representation R consists of the correspond-ing representation on the quotient space S/T between in-variant subspaces Sand T underR (where S C T). Thefactorsof R are those subquotients that are irreducible-i.e., forwhich T is a maximal invariant subspace of S. Any compo-sition series for R-i.e., maximal chain of invariant sub-spaces

0 = S0C S1C ... C Sn = R,

where R is the representation space for R-determines a setof factors for R, consisting of the representations on thesubquotients Sj+l/Sj(j = 0, 1, . . . , n - 1). The factors of therepresentation of G corresponding to a given clan define itselementary particle spectrum; the stable spectrum consists ofthe factors that are unitary and have a one-sided frequencyspectrum.Although there will in general be many inequivalent (non-

conjugate) composition series, the factors are unique asgroup representations. On the other hand, notwithstandingthis lack of unicity for composition series, in practice thereare nontrivial constraints on the order in which the factorsoccur, corresponding to the order of inclusion of the corre-sponding invariant subspaces. This contrasts greatly, ofcourse, with the entirely arbitrary order in which the factorsoccur in the case of a fully decomposable representation, asin conventional theory. Thus, in the chronometric fermionclan, the exon appears as a bottom invariant subspace, orfactor, and the electron as a top factor; in the middle are themuon and electron neutrino factors, in that order. In theboson clan, the photon appears as a bottom factor, abovewhich are bare versions of W and Z, in the subclan of allvectors. (A subclan is a subspace that is invariant undermultiplication by arbitrary smooth scalar functions; it corre-sponds to "subbundle.") The pion is included in a largersubclan, modulo which (i.e., in the top quotient clan) there isa factor identified with the neutral kaon, as discussed below.It is important for the formation of appropriate interactionsand the treatment of causality that the context is not purelygroup theoretic but involves the spatio-temporal labeling ofclan vectors and to a (necessarily) reduced extent of vectorsin the factors.Corresponding to any given chronometric clan is a rela-

tivistic free particle family consisting of the direct sum of thestable factors, restricted to P0 and fixed in mass. Because ofthe indecomposability, the action of P0 on the clan effectivelymixes up the factors, and so is quite different from therelativistic action ofPO on the direct sum ofthe stable factors.The mixing implies that chronometric free temporal evolutiongives rise to apparent particle production within the frame ofthe relativistic limit. I call this indecomposable production, todistinguish it from Lagrangian production of the conven-tional type; both are causal and covariant. Since indecom-posable production is absent in the relativistic limit of chro-nometric theory, it appears as a weak interaction in conven-tional terms, descriptive of weak decays. But there are alsoLagrangian interactions between neutrinos and other parti-cles, which would be classified as weak in relativistic theory,as discussed below. Analytic treatment ofthe mixing dependson the analysis of the matrices whose components areoperators from one reduced particle space to another, cor-responding to the given representation of G.

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The Chronometric Clans and Their Interaction

Most of the foregoing applies to general G-covariant clansover M but certain specific clans appear especially naturaland close to well-established parts of relativistic theory.Briefly, I specialize in accordance with the assumptions:

(i) The fermion clan is based on the half-spin representationof G, as locally the orthogonal group SO (2, 4). Equivalently,it is induced to G from the simplest faithful finite-dimensionalrepresentation of P, which in the relativistic limit is just theusual spin representation, as extended to P. (The correspond-ing reduced particles are massive and massless spinors.)

(ii) The boson clan in the minimal form used here issimilarly based on the adjoint representation of G. Equiva-lently, it is induced from a certain 15-dimensional indecom-posable representation of P. (The corresponding reducedparticles are scalar and vector, as representations of P0.)

(iii) The interaction Lagrangian LI is the essentially uniqueG-invariant bilinear coupling of the boson clan with the localbilinear fermion clan current. (In the relativistic limit, LIrepresents three distinct types of interactions. One is similarto the electroweak Lagrangian without the Higgs boson. Thetwo others are roughly semihadronic and purely hadronic.)

(iv) Charged particles are either electrons or compositeswith electrons.These assumptions determine chronometric theory prac-

tically uniquely, apart from a possible enlargement of theboson clan (22) and specification of the role of tachyons (ifany). I mention also that, as earlier indicated (4, 22, 24), it isnatural and useful for the treatment of discrete symmetries tobegin with real clans and use positive-energy and symmetryconsiderations to derive the action of the complex unit i. Thederived complex clans largely suffice for present consider-ations, and only these are treated here.The fermion clan F is the direct sum of a stable subspace

FS and a tachyonic subspace F0. FS is the direct sum of apositive-frequency representation F+ with its complex con-jugate F -. F + is indecomposable and has 4 factors that occurin its composition series. In the relativistic limit, all 10 factorsoccur as direct summands in the space of all square-integrable spinor fields on Mo. Physical assignments for thereduced particles in F+ are given in Table 1; those for theantiparticles F- are obtained by interchanging the left andright spins.

In this table, column I gives a physical particle assignment,within the limitation of approximation by a reduced particle.The assignment is determined virtually uniquely by themassive/massless character of the particle and the vanish-ing/nonvanishing of its interaction with the photon.The reduced exon is identical to the exact exon, which may

tend to enhance its stability. Only the lightest electron isexpected to be well approximated by its reduced form; theheavier electrons (muon, tau, . . .) are expected to be rep-resented by exact particles with significant components in thelower invariant subspaces modulo the exon subspace-i.e.,in terms of the relativistic limit, be linear combinations withcomponents in the neutrino spaces.

Table 1. The stable elementary fermion spectrumBare Left Right Principal G-K

Particle mass spin spin weight Height dimension(I) (II) (III) (IV) (V (VI) (VII)

Exon, x 5/2 1/2 0 /2 1 4(cygnet)

3/2 /2 0 /2 2 3Ve 3/2 1'/2 0 3/2 3 3Electron 5/2 '/2 0 3/2 4 4

G-K, Gelfand-Kirillov.

The higher electrons are distinguished experimentally bythe sharpness and stability of their relativistic masses, inaddition to their interaction with the photon, which is deter-mined by their top residue class. In chronometric terms, thestability means that if M2 denotes the usual relativistic massoperator p2 - p2 - p2 - p2 and H, the chronometrichamiltonian in their actions on F, then e-iHM2eiH has thehigher electron state in question as an approximate eigenstateof the same narrow-width eigenvalue, for a nongenericallylong time interval. Thus in particular, [M2, H] should haveexpectation value 0. Although the relativistic and chrono-metric hamiltonians have no simultaneous eigenvectors, vec-tors in F whose relativistic masses exceed that of e satisfy thestability and narrow-mass-width constraints just indicated,and have nonvanishing top residue from which they areidentifiable as electrons, appear likely to exist and be com-putable from these constraints. They would be identical toelectrons in charge and magnetic moment apart from thesmall perturbation deriving from their different mass andotherwise represent higher forms of the electron, as regardstheir interaction with w = 1 bosons. On the other hand, theirinteractions with w = 0 and w = -1 bosons may be quitedifferent.Column II of Table 1 presents the intrinsic (or bare) mass

ofthe particle, defined as the minimum ofthe Einstein energy(the minimum of the Minkowski energy vanishes). Thisintrinsic mass is far below the level of physical observability,since mp -= 104, but higher intrinsic masses have an apparenttendency to be associated with higher physical masses, forboth bosons and fermions.Columns III and IV specify the spatial transformation

properties of the ground state of the particle via the spins ofthe corresponding representations of the spatial isometrygroup SU(2)L X SU(2)R of the Einstein universe. Space 53can be identified with SU(2) by an isometry, and the groupaction is then' U x V: W -- UWV-1. The "particle quantumnumbers" (columns II-IV) determine the irreducible repre-sentation ofG in question (25). The angular momentum spinis the absolute difference between columns III and IV, thus1/2 here.Columns V and VI present properties of the spatio-

temporal description of the particle as a clan member that areimportant for the specification of its interactions. They areentirely independent of the reduced particle quantum num-bers in columns II to IV. As yet it is not formally proved thatcolumn VI applies to all composition series for F+, but noothers are known. (The composition series shown on p. 34 ofref. 22 is inexact and should be replaced by that indicatedhere.) Having w = 5/2 is a G-invariant property, but having w= 3/2 is invariant only in the relativistic limit or may beregarded as a property of the quotient clan modulo the w =5/2 subclan. The purely electromagnetic interactions of thefermions (as well as those with the bare WandZ) are uniquelydetermined by this residue class. The exact higher electronsare expected to include substantial w = 5/2 components thatwill differentiate them from e independently of their massesbut which are important for the determination of thesemasses. In principle an infinite number of higher electronsmay exist, but their stability and hence observability may belimited by phase-space considerations. Such considerationsbased on the theoretically more fundamental but only indi-rectly observable K-covariant quantum numbers depend onthe Gelfand-Kirillov quantum number given in column VII.The chronometric interaction is specified by an interaction

Lagrangian that is uniquely determined by the followingfeatures, apart from its overall scale: (i) it is bilinear in thefermion clan (field), linear in the boson clan (field), and local;(ii) it is G-invariant; and (iii) (almost automatic) it is U(1)gauge-invariant. Specifically, if , denotes the fermion and A,

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the boson field, where A is represented canonically by amatrix on the fermion spin space, then

L1(4f,A) = f (( A+, i )) d4x,

where the inner product is the invariant one in the fermionspin space at x. Kinetic terms are implicit in the clantransformation properties. The bosons have weights dual tothose of the fermion currents; since these are of weights 3/2 +3/2, 3/2 + 5/2, and 5/2 + 5/2, the boson weights are 1, 0, and -1.The weights 0 and -1 are well-defined only in the relativisticlimit; w = -1 states "leak" (by virtue of indecomposability)under the action of G into w = 0 states, and w = 0 states leaksimilarly into the w = 1 subspace, which is G-invariant.Corresponding to the three different types of currents justindicated, there are essentially three different types of inter-actions, in terms of the relativistic limit.

(i) Two w = 3/2 fermions and a w = 1 boson: The two w =

3/2 fermions are electrons (of which only the top residueparticipates, since it determines the corresponding constitu-ent of the Lagrangian) and neutrinos (for which the same istrue). The w = 1 bosons include the photon, at the bottom ofthe subspace, and distinct candidates for the bare W = WO andthe Z, the former in a neutral form (the physical W+ beingcomposites of WO with electrons and other particles). Allthree reduced particles have distinct quantum numbers thatplay a role comparable to the gauge degrees of freedom in thestandard model. Charges of the w = 3/2 particles are includedautomatically in the form of the Lagrangian; e.g., the neu-trino-photon integrated interaction vanishes, as a conse-

quence of the transformation laws (or equivalently, the Diracand Maxwell equations). The neutrino interactions with theWO and the Z are nonvanishing and parallel those of e with thelatter, providing a form of weak isospin.

(ii) A w = 3/2 fermion, a w = 5/2 fermion, and a w = 0 boson:This is not readily characterized in relativistic terms butseems to underlie low-energy-electron and top-neutrino in-teractions with baryons and light mesons. The large nucleonto physical electron mass ratio appears to give this interactiona strong appearance in relativistic terms, although in barechronometric terms it appears formally as approximatelysymmetric between the e and the x. The w = 0 sector includesa natural candidate for the neutral pion, whose decay into twophotons may derive primarily from the leakage of the w = 0

bosons into the w = 1 subspace. The decay into neutral pionsof the KO may be of similar character.

(iii) Two w = 5/2 fermions and a w = -1 boson: Thisinteraction appears as purely strong in relativistic terms. Thestable reduced elementary boson in this sector shows mixingof two relativistically invariant components and would beexpected to leak into w = 0 bosons, among other possibledecays. This suggests identification with the K°, but themixing shown by the BO and the DO, together with their decayproducts, suggests they may be higher forms ofthe K° via themechanism proposed above to apply to the higher electrons.The top positions of the e and the K° in their respective clansshould facilitate this mechanism. The conformal weight sumconstraint suppresses decay of the K° into ,+,- but allowsK° -- xX.

The U(1) gauge-invariance ofthe interaction in conjunctionwith the weight structure implies an overall number conser-

vation law, which in the relativistic limit breaks into separatelepton and baryon number conservation laws. Separate elec-tron and muon number conservation is intelligible as a

corollary to this in conjunction with the w = 5/2 character ofcomponents of the muon and the muon neutrino and theabsence of significant such components for the electron andthe electron neutrino. Relativistically, the half-spinors ofweight 5/2 in terms of relativistic scaling are the same, withappropriate boundary value specification, as the w = 5/2

subspace of the fermion clan, and the half-spinors of weight3/2 similarly are the same as the quotient of the fermion clanmodulo the w = 5/2 subclan.

The Cygnet/Herculon and the Exon

Estimated distances of the order of - 104 light years forCygnus X - 3 and Hercules X - 1 largely rule out thepossibility that the cygnet ( or "herculon," here identifiedwith the cygnet) could be a neutron. Relativistic models haveinvolved many ad hoc adjusted features (see, e.g., refs. 15,26, and 27). The reluctance to accept the cygnet as a newparticle (28) is understandable in view of the thoroughexploration of the hadron spectrum beyond the expectedmass level and its dubious impact on the standard model, butits recent reappearance (13, 14) confirms its reality.The observations on Cygnus X - 3 and HerculesX - 1 are

indicative of a free physical particle whose parameters,insofar as they are observable, are precisely those of thetheoretical exon. Further cosmic ray observations areneeded, but conclusive identification of the cygnet with theexon will depend on observation of the latter in acceleratorexperiments. It should be possible to produce it in energeticelectron-nucleon or nucleon-nucleon collisions.The mass of the physical exon is not determined by the

cosmic ray observations; it may be of the order of, or greaterthan, the nucleon mass or considerably less. A rough timinganalysis (29, 30) suggests that the mass of the hadron-likeparticle indicated by observations on HerculesX - 1 (17, 31),is S 60 MeV (1 eV = 1.602 x 10-19 J). If m, << mp, thedisintegration of nucleons into their elementary constituentscould result in a chain reaction playing a basic role in theenergetics of galactic nuclei, supernovae, and stars such asCygnus X - 3. Irrespective of its mass, the exon may be adynamically significant component of possible dark matter.

Mixing and Internal Quantum Numbers

The unicity of CP violation in the kaon complex (32) hasappeared almost as puzzling as the fact of CP violation itself.The superweak force postulated in ref. 33 provides a phe-nomenological explanation, without however designatingany specific force. Both CP violation and its unicity in thekaons admit simple explanations if the chronometric assign-ment for the kaon is valid. However, because of the com-posite character of the K+ in chronometric theory, it is to beexpected that 71+- and 100 will differ slightly.

All of the chronometric elementary particles remain irre-ducible on restriction to P and decompose into the continuumof all masses on restriction to P0, with two exceptions. Oneof these is identified with the W0, which occurs only in acomposite form having a large mass width, making it difficultto observe possible mixing in the underlying W0. The other isthe K°, but its quantum numbers (5, 1/2, 1/2) indicate that itsplits discretely into two P0-invariant subspaces of nearlyidentical relativistic mass, which appear as natural candi-dates for the Ks and the KL. However, the currently acceptedvalues for the spins of these particles (34) are both 0, ratherthan the 1 and 0 predicted by the present theory. Thisapparent discrepancy may result from the extrapolation of astandard but optimistic approach to kinematic analysis be-yond the point of approximate validity.More specifically, the spins of short-lived particles have

been inferred by using theoretical relativistic kinematics inconjunction with the assumption that the particles are instates ofexact energy momentum and angular momentum intheir center of mass frames. Generic relativistic particles arenot of this character, since a truly physical state must havefinite energy, which implies that a physical particle must berepresented by a normalizable vector in the (Hilbert) space

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corresponding to its spin and mass. In the Lorentz frame inwhich the particle is maximally at rest-i.e., has minimumexpected energy-the expected momentum vanishes but theexpected orbital angular momentum generically does notvanish. The argument for the spin ofthe Ks is sensitive to thisquantum effect, since the two pions into which it decays willbe distinct if their relative orbital angular momentum isnonzero.The designation of the P0-irreducible component to which

a physical kaon state belongs represents an internal quantumnumber equivalent to strangeness and explains its conserva-tion in strong, and nonconservation in weak interactions.Strangeness in the higher baryons and mesons may bephenomenologically representable in a similar way, by theassociation of the multiplet in question with a discrete seriesrepresentation ofG in the stable spectrum of the local tensorproduct ofthe sub- or quotient clans ofits elementary particleconstituents. More generally, all relativistically "internal"symmetries may originate in the interplay between the quan-tum numbers associated with the maximal subgroups K andP ofG in the chronometric clans and thus be of an ultimatelygeometrical character.

Discussion

Detailed applications of the proposed theory depend onquantum field and harmonic analysis computations regardingthe fundamental clans and their interaction. In the presentunresolved foundational state of nonlinear quantum fieldtheory, renormalizability in perturbation theory becomes anissue. In practice, renormalizability has been substantiallyequivalent to conformal invariance within the addition ofmass terms. This suggests that the success ofrenormalizationmay originate in its ready correlation with chronometrictheory in conjunction with mass-fixing via Mach's principle,on the basis ofeffectively similar symmetry groups and fields.In any event, nonlinear K-invariant quantized theory, whichis theoretically fundamental, is less divergent than the cor-responding relativistic theory. In particular, it provides afirst-order approximation to the perturbative S matrix forconformally invariant interactions that is rigorously a self-adjoint operator in Hilbert space (35).

In principle, the higher baryons are determined from theinteracting quantized theory as bound states of the exon withother elementary particles. Much simpler approximate de-scriptions by higher discrete series representations ofG (36,37), whose discrete splitting on restriction to P has beencorrelated with relativistic theory (38, 39), may provideuseful phenomenological models of mass multiplets.The leakage of the top fermion space into that just below

implies the process e -- e + ve + Me, leading to the corollaryprocess

n =x + e+ + e-

(x +e + +re) +e- + we = P+ e + e

i.e., beta decay. If 1L is a higher electron whose exact clanrepresentative differs from that of e by its inclusion of a largev. component (in addition to the mass difference), leakagefrom the top into the two lower spaces should result in theobserved form of muon decay. The similar process ve ve +v,, + v in these two lower spaces may contribute to the solarneutrino deficiency, by the attrition in flight of lve numbersdue to conversion into v, pairs that are unable to revert to ve.On the other hand, the "inverse" process defined by revers-ing the incoming and outgoing particles can proceed on acomparable scale only by Lagrangian rather than indecom-posable production.

I thank J. Bernstein, I. M. Gelfand, and D. A. Vogan, Jr., forhelpful discussions. I thank also U. J. Becker, G. F. Chew, R. H.Dalitz, H. Feshbach, J. I. Friedman, P. Morrison, L. Rosenson,M. A. Ruderman, and L. Wolfenstein for informative comment. Theresearch reported here includes earlier work that was supported inpart by the National Science Foundation.

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