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PALEV STATISTICS AND THE CHRONON DavidRitzFinkelstein 2 1 0 2 n a J 8 Abstract A finite relativistic quantum space-time is constructed. Its unit cell has Palevstatistics definedbya spin representationof anorthogonalgroup.Whenthe ] StandardModelandgeneralrelativityarephysicallyregularizedbysuchspace-time h p quantization,theirgaugesarefixedbynature;thecellgroupsremain. - t n a u 1 Novus ordoseclorum q [ Thegoalisstillafinitephysicaltheorythatfitsourfinitephysicalexperiments.The 1 classical space-time continuumled to singular (divergent)quantum field theories: v Infinityin,infinityout.Inancienttimes,acontinuumwastheonlywaytounderstand 7 9 the translational and rotational invariance of Euclid’s geometry. Today there are 5 quantumspaceswithafinitenumberofquantumpointsthatstillhavethecontinuous 1 symmetriesofgravityandtheStandardModel,atleastwithinexperimentalerror.If . 1 electronspinwereanysimpleritwouldnotexist.Thestrategyisbuildthecosmos 0 fromsuchatoms. 2 Quantum spaces are represented by probability spaces that define the lowest- 1 order logic of their points. Modular architecture requires the higher-order logic, : v classically dealt with by set theory. Classical space-time and field theory are for- i mulated within classical set theory; perhaps quantum space-time and field theory X needaquantumsettheory.ClassicalsettheorywasinventedbyCantortorepresent r a themindoftheEternal.Quantumsettheoryisintendedtorepresentthesystemun- derstudyasquantumcomputer.Likeanyquantumtheoryitstatisticallyrepresents input/outtake(IO)beamsofsystemsbywhatHeisenbergcalledprobabilityvectors. Quantumlogicisasquarerootofclassicallogic:Transitionprobabilitiesinthe classicalsensearesquaresofcomponentsoftheprobabilityvector,transitionproba- bilityamplitudes.Callthequadraticspaceofprobabilityvectorsaprobabilityspace GeorgiaInstituteofTechnology,Atlanta,Georgia.e-mail:fi[email protected] 1 2 DavidRitzFinkelstein P. IngeneralP, like thequantumspace of Saller [11], includesbothinput(ket) andouttake(bra)vectors.Distinguishthesebythesignsoftheirnorms.P isnota Hilbertspace. Aquantumtheoryshoulddescribepopulationsaswellasindividuals.Enrichthe quadraticprobabilityspaceP toaprobabilityalgebraP whoseproductabrepre- sentssuccessiveapplicationofinput/outtake(IO)operators.Theone-systemproba- bilityvectorsformageneratingsubspaceP ⊂P.Pconsistsofpolynomialsover 1 P ,subjecttoconstraintsandidentificationssaidtodefinethestatistics. 1 A regular quantum theory is one with a finite-dimensional probability algebra [1]. TheprobabilityalgebraP− forfermionsisaCliffordalgebra,definedbyanti- commutationrelationsamongtheone-fermionvectors: ∀x∈P−: x2=kxk=x·x. (1) 1 ItsdimensionisDimP−=2DimP1,soFermistatisticsisregulariftheone-fermion probabilityspaceis. TheprobabilityalgebraP+forevenquantaiscommonlyassumedtobeaBose (Heisenberg,canonical)algebrawhosegeneratorsobey ∀x∈P : xy−yx=e (x,y) (2) 1 withagivenskew-symmetricbilinearforme onP ⊗P .Thisisnotexactlyright: 1 1 Bosestatisticsisalwayssingular.Pairsoffermions,however,obeyaregularstatis- tics whoseprobabilityalgebraenvelopsa Lie algebra,with Bose statistics andthe Heisenbergalgebraasasingularlimit.ThisisaspecialcaseofPalevstatistics[8,9], whichisreviewednext. 2 Palevstatistics ForanysemisimpleLiealgebrap,aPalevstatisticsofthepclassisonewhoseprob- abilityalgebraP isafinite-dimensionalenvelopingalgebraofp(haspascommu- tatorLiealgebra). FermiandBosestatisticshavegradedLiealgebrasf,bthatspecifytheircommu- tation relations. f and b as vectorspaces are also one-quantumprobabilityspaces. They have essentially unique irreducible unitary representations; these serve as many-quantum probability spaces. There are, however, an infinity of irreducible representationsofaPalevalgebrapthatmightserveasmany-quantumprobability space.EmpiricalchoicesmustbemadeforPalevstatisticsthatarealreadydecided forFermiandBosestatistics. Palevgivesarepresentationofsl(n+1)onaFockspaceW ofsymmetrictensors p ofdegreep.W isaHilbertspace,appropriateforhisapplicationsandnotforthese. p The probability space of a hypothetical quantum event must have enough di- mensionsto allow for the observedquantumsystems. Itis notclear that eventsin PALEVSTATISTICSANDTHECHRONON 3 space-timecan be experimentallylocated to within much less than a fermi, corre- spondingtoalocalizationintimeofabout10−25s.ThePlancklimitat10−43swas initially a conjecture based on pure quantumgravity. Our instruments, to be sure, donotseemtobemadeofgravitonsalone,buttakepartinalltheinteractions.The Plancktimeseemsatbestapoorlowerboundtothequantumoftime. Howevertheenergyatwhichalltherunningcouplingconstantsseemtoconverge is not very much greater than the Planck energy, so it may indeed have universal significance.Yetcrystalshavemanyscalesbesidescellsize,suchasDebyeshielding length,skindepth,coherencelength,andmeanfreepaths.ThePlancktimeandthe unification energy might correspond more closely to one of these than to the cell sizeX.Toavoidaprematurecommitment,callthenaturaltimeXthechrone. Howmanydimensionsmusttheeventprobabilityspacehave?Supposethelife- timeofthefour-dimensionaluniverseis1021s;anerrorbyafactorof100willnot mattermuch.IfX∼T ≈10−43 sthenthedimensionalityofthehistoryprobabil- P ity space of the cosmos—which we cannot observe maximally—is about 101024. The largest system that can be maximally observed by a co-system within such a cosmos—herewe renouncethe perspectiveof the Eternal—is much simpler. Its probabilityspacemighthavenomorethanlog 10256∼3000dimensions. 2 HerearetwoexamplesofPalevstatistics: 2.1 Rotatons Theso(3)LiealgebrawithcommutationrelationsrelationL×L=Ldefinesanag- gregateofpalevonsoftheso(3)kind,whosequantamustbecalledrotatons,since Landaupreemptedtheterm“roton”.RelativetoanarbitrarycomponentL asgen- 3 eratorofaCartansubalgebra,therootvectorsL =L ±iL representtheinputand ± 1 2 outtakeofspin-1rotatons.Anirreduciblerepresentationwithextremeeigenvalues ±il forL representsaPalevstatisticswithnomorethan2l+1rotatonspresentin 3 asingleaggregation. 2.2 Di-fermions Di-fermions are palevons. If the fermion probability space is 2NR then the di- fermionisapalevonoftypeso(N,N). This refers to the elementary fact that the Fermi commutation relations for a fermionwithNindependentprobabilityvectorsdefineaCliffordalgebraCliff(N,N), andthesecondgradeofCliff(N,N)isboththeprobabilityspaceforafermionpair, andtheLiealgebraspin(N,N)definingaPalevstatisticsofclassD . N 4 DavidRitzFinkelstein 2.3 Regular space-times Callaspacetimeregularifitscoordinatealgebraisregular.Allitscoordinatesthen havefinitespectra.Therearenotmanyregularspacetimesintheliterature. Singular(non-semisimple)Liealgebrascanberegularizedbyslightlychanging some vanishingcommutators,undoingthe flattening contractionthat led, presum- ably,fromtheregulartothesingular. Theprototypeofsuchregularizationbydecontractionis(special)relativization [13,6].ThisregularizestheGalileanLiealgebrag=g(L,K)ofEuclideanrotations LandGalileanboostsK,totheLorentzLiealgebraso(3,1).Writesuchrelationsas so(3,1)◦→ g or g←◦ so(3,1), (3) directedfromtheregularalgebratothesingular. The sole remaining culprit today is Bose statistics, the canonical Lie algebra. Its de-contraction requires adding new variables. This is the general case; special relativityandquantumtheorywereexceptionalinthisrespect. Somephysicalself-organizationmustthenfreezethese extravariablesoutnear thesingularlimit.Thiscanbetestedexperimentallyinprinciplebydisruptingthis organization.Hopefully,asuitableregularizationoftheremainingsingulartheories willonceagainimprovethefitwithexperiment. TheKillingformofclassicalobservablesisassingularascanbe:identically0. Itisnearlyregularizedbycanonicalquantization: a (x,p,i) ←◦ h(N), (4) comm wherethecanonical(Heisenberg)Liealgebrais h(N):[xn ,pn ′]=ih¯d nn ′, n ,n ′=1,...,N, (5) othercommutatorsvanishing.ThesolvableradicalCgeneratedbyisurvivescanon- icalquantization.Recallthathpdoesnotfitintoanysl(NR).(Theleft-handsidesof thecanonicalcommutationrelationswouldhavewell-definedtrace0,andtheright- hand side i would have non-zero trace.) Canonical quantization is a quantization interruptedbyprematurecanonization. 2.4 Feynman space-time Feynman[3]seemstohaveconstructedthefirstregularrelativisticquantumspace- timeF.Itspositionalcoordinatesarefinitespinsums: xm ←◦ xm =X[g m (1)+...+g m (N)], m =1,2,3,4. (6) b PALEVSTATISTICSANDTHECHRONON 5 Theg m haveunitmagnitudes,andXisthenaturalquantumunitoftime,orchrone.If thecommutators[g m ′(n′),g m (n)]vanishforn6=n′ thentheprobabilityvectorspace for F is a 16N-dimensionalCliffordalgebra.Quantificationtheory abbreviates(6 to xm =Xyg m y . (7) Each term in the sum represents a hypotheticalquantum elementof the space- b timeevent;callitachronon.TheFeynmanchrononhasspin0or1,becauseg m has bothascalarpartg 0 andavectorpart(g 1,g 2,g 3). Natureseemsto havea unitof space-timesize like X ateveryevent,fixingthe gaugeintheoriginalsenseofWeyl.Ifeacheventhasaspace-timemeasureX4,then the dimensionofthe one-eventprobabilityspace is proportionalto the space-time volume;asiftheeventstatisticsisextensiveinthesenseofHaldane[5,9]. 2.5 Yang space-time Yang [15] proposedthe first regularrelativistic space-time-momentum-energyLie algebra,leavingthesignaturesomewhatopen: y=(so(5,1)or so(3,3)) ◦→ hp(4). (8) y can represent the orbital variables of a spinless relativistic quantum. Yang re- stricted considerationto representationsin Hilbertspace, however,blockingregu- larity. Regularversionof the Yangtheory usesa finite-dimensionalrepresentation oftheYangLiealgebraso(3,3).Itsprobabilityalgebramustthenhaveanindefinite norm(§2.6). TheYangLiealgebrayisnottobeconfusedwiththeconformalso(3,3)Lieal- gebra.Theyareisomorphicbutactondifferentphysicalvariablesandhavedifferent physicaleffects.Wedealwithgroupsofphysicaloperations,notabstractgroups. 2.6 Interpretationoftheindefinitenorm InspecialrelativitythesignoftheMinkowskimetricformgdistinguishesallowed (timelike)directionsfromforbidden(space-like)ones. InarelativisticquantumtheoryoftheDirackind,theprobability-amplitudeform b isneutral(ofsignature0). bYY =Y b ◦Y (9) givesthemeanfluxofsystemsfromtheexperiment(nottheabsoluteflux).Thesign distinguishesinputprobabilityvectorsfromouttakeprobabilityvectors[2]. 6 DavidRitzFinkelstein Let source kets bras have positive norm and sink bras negative. This changes nophysicsin theusualquantumtheory,whichdoesnotaddbrasandkets. Hereit enlargesthegroupandmustbetestedbyexperiment. Aregulartheorymightassociateanelementaryparticlewithanirreduciblefinite- dimensionalisometricrepresentationofasimpleLiealgebraythatapproximatesthe Poincare´Liealgebra: y ◦→ hp. (10) Thenallone-particleobservableshavefinitespectra. Theconstantih¯ of the usualquantumphysicsis thenanothernon-zerovacuum expectationvalue,afrozenvariableliketheMinkowskimetricgm ′m andtheHiggs field. Centralizing a hypercomplex number by a condensation gives mass to any gauge boson that transports that number; this was shown for the quaternion case, for example. Such a frozen i must be assumed in the Yang and Segal space-time quantizations[15,13]basedontheLiealgebra y=so(3,3)∼=spin(3,3)∼=sl(4R). (11) Infinitesimalgeneratorsofso(3,3)∼=spin(3,3)∼=sl(4R)makeupatensor[Ly′y] (y,y′=1,...,6)with15independentcomponents,representingorbitalvariablesof theYangscalarquantum.ThisyisalsoacandidateforthescalarparticleLiealgebra y of (10).The Feynmanquantumspace-timeand the Penrose quantumspace [10] canberegardedasspinrepresentationsoftheYangLiealgebra. ı, the quantized i, is a normalized2×2 sector [Lz′z] (z,z′ =5,6) of [Ly′y] in an adaptedframe.Thetensor[Ly′y]thenbreaksupaccordingto b Lm ′m ixm ′ ipm ′ 4×44×2 [Ly′y]=ixm 0 L56 ∼(cid:20)2×42×2(cid:21), y,y′=1,...,6, (12) ipm L65 0   m whichincludestheLorentzgeneratorLm ′m asa4×4block,positionx andmomen- tum pm (m ,m ′=1,2,3,4)as4×1blocks,andLz′z (z,z′=1,2)asa2×2block. Posit a self-organization,akin to ferromagnetism,that causes the extra compo- nentL toassumeitsmaximummagnitudeinthevacuum.Smallfirst-orderdepar- 65 turesfromperfectorganizationofimakesecond-ordererrorsin|i|. 2.7 Locality OnemorelimitstandsbetweentheregularYangLiealgebrayandsingularcanon- icalfield theories.Thespecial-relativistic kinematicsandy have a canonicalsym- m m metrybetweenthex andp .Yettherearegreatphysicaldifferencesbetweenthese m variables. Under the composition of systems, pm is extensive and x is intensive. ThefundamentalgaugeinteractionsoftheStandardModelandgravityarelocalin PALEVSTATISTICSANDTHECHRONON 7 m x and not in pm ; unless asymptotic freedom can be regarded as a weak form of localityin pm . Thissuggeststhatthereisaricherclassofregularquantumstructuresthathave classicaldifferentialgeometryandgaugefieldtheoriesasorganizedsingularlimits, withatleastthreequantificationlevels:thechronon,theevent,andthefield. Wignerproposedthatan elementaryparticle correspondsto an irreducibleuni- tary representation of the Poincare´ group. Up-dates in this concept are called for byhislaterwork.TheWignerconceptofelementaryparticlegivesnoinformation aboutinteractionsbetweenparticles.Gaugetheoryrequiresanelementaryparticle to have a location in space-time where it interacts with a gaugeon. It would then seem useful to associate an elementary particle with an irreducible representation oftheHeisenberg-Poincare´Liealgebrahp(xm ,pm ,Lm ′m ,i)instead,whichfusesthe Poincare´ and the Heisenberg(canonical)Lie algebras.Thisis the Lie algebrathat Yangregularized. 2.8 Quantizationand quantification Quantificationand quantizationare related like archeologyand architecture.They concernsimilarstructures,butquantificationsynthesizesthemfromthebottomup, whichlikelytheorderofformation,whilequantizationanalyzesthemfromthetop down,theorderofdiscovery. Quantizationre-introducesaquantumconstantthattheclassicallimiteliminates. Quantificationdoesnotintroduceonebecausethequantumindividualalreadypro- videsit. In particular, space-time quantization introduces a new quantum entity, the chronon, carrying a time unit, the chrone X, and an energy unit, the erge E. The chrononisnoparticleintheusualsensebutaleastpartofthehistoryofaparticle. Canonicalquantizationcanalso beinterpretedasa quantificationwitha singu- larstatistics. Whatissometimescalled“secondquantization”ismoreaccuratelya secondquantification. 2.9 Gauge A gauge is an arbitrarily fixed movable standard used in measurements. It is part of the co-system, the complementof the system in the cosmos. As partof the co- system, a gauge is normally studied under low resolution and treated classically. A field theory may postulate a replica of the gauge at every event in space-time, formingan infinite field of infinitesimal gauges.Weyl’s originalgaugewas an in- finitesimalanalogueofacarpenter’sgaugeoramachinist’sgaugeblock,amovable standardoflength;hencethename. 8 DavidRitzFinkelstein Agaugetransformationchangesthegaugesbutfixthesystem.TheyformaLie group,thegaugegroup,whichindicatesthearbitrarinessofthegauges.tiscustom- ary to assume that the relevant dimensions of the gauge field are fixed during an experimental run, so that the experimental results can be compared meaningfully witheachother.Thismeansthatthegaugemustbestiff.Forexample,machinist’s gaugeblocksare oftenmadeof tungstencarbide.Suchrigidconstraintsbecomea sourceofinfinities. In a simple quantum theory, however, all variables have discrete spectra. All eigenvaluescanbedefinedbycountinginsteadofbymeasuring.Thereisnoneedfor arbitraryunits,externalgauges,orgaugegroup;Natureprovidesthegaugewithin thesystem.Thusagaugegroupisanothersignofinterruptedquantization. Onewell-knownwaytobreakagaugegroupisbyself-organization.Itisoften supposedthattheHiggsfield,whichbreaksanisospingroup,issuchacondensate. A gauge group is also broken, however, when further quantization discovers a natural quantum unit, fixing a gauge. The quantized i that breaks su(2) and im- partsmassinquaternionquantumgaugetheoryisofthatkind.Soisthequantized imaginaryıthat breaksYang so(3,3). The Higgsh thatbreaksisospin so(3), and thegravitonicgm ′m thatbreakssl(4R)∼=so(3,3)mayalsobesuchnaturalquantum gauges,tobberecoveredbyregularizingthekinemabticalLiealgebraoftheStandard Modelthroughbfurtherquantization. Notation:Theone-quantumtotalmomentum-energyvectoris, uptoaconstant, thedifferentiator[¶ m ],canonicallyconjugatetothespace-timepositionvector[xm ]. [¶ m ] reducesto agauge-invariantdifferentiator[Dm ], also canonicallyconjugateto xm ,relatedtokineticenergy,andavectorG m (x)thatcommuteswithposition,related topotentialenergy: ¶ m = Dm + G m . Total= Kinetic+Potential (13) Thegaugecommutatoralgebraa(xm ,Dm (x),Fm ′m ,...)isgeneratedbythespace- m timecoordinatesx andthekineticdifferentiatorDm (x),andincludesthefieldvari- ablesFm ′m =[Dm ′,Dm ]andtheirhighercovariantderivatives.Itsradicalincludesall m functionsofthex .Thismakesitsingulartoo. Gaugingsemi-quantizes.Itconvertsthecommutativeoperators¶ m intothenon- commutativeonesDm ,andforindividualquantatheseareobservables.Itscontrac- tionparameteristhecouplingconstant.Landauquantizationinamagneticfieldis ofthatkind. Gaugingalsoquantifies:Itconvertsonefinite-dimensionalglobalgaugegroupG intomanyisomorphsofG,oneateachspace-timepoint. Gauging introduces infinities because the number of gauges is assumed to be infinite.Thusquantumgaugephysicscanbe regularizedbyregularizingitsquasi- Liealgebras.Thiseliminatesgaugegroupsaswellastheorysingularities.Thiswill betakenupelsewhere. PALEVSTATISTICSANDTHECHRONON 9 3 Higher-order quantum settheory Classical set theory iterates the power-set functor to form the space of all “regular”(ancestrallyfinite,hereditariliyfinite)sets.Aregularsettheorymighttherefore iteratetheFermiquantificationfunctor[4],asfollows. ThePeanoi,with {a,b,c,...}:={a}{b}{c}...=iaibic ..., (14) definesmembershipa∈b: a∈b :≡ ia⊂b (15) Let S designate the classical algebra of finite sets finitely generated from the emptyset1bybracingix={x}andthedisjointunionx∨y(agroupproductwith identity1,theemptyset).SetsofSareherecalledperfinite(elsewhere,ancestrally orhereditarilyfinite).Theyarefinite,andsoaretheirelements,andtheirelements, andsoforth,allthewaydowntotheemptyset.Let sbethesetoffinitesubsets ofs.Then W :S→S= S. (16) _ _ An elementof S is a set or simplex whose verticesmay be sets or simplices. S is supposedlycomplexenoughtorepresentanyfiniteclassicalstructure. AquantumanalogueSisakindoflinearizationofS: ForanyquadraticspaceS,let SdesignatetheCliffordalgebraoffinite-degree b polynomialsoverS,modulotheeFxclusionprinciple ∀s∈S:s⊔s=0. (17) S and ⊔ correspond to the classical power set and the symmetric union (XOR). FIf P1 is a one-fermionprobabilityspace then P =CliffP1 is the many-fermion probabilityalgebra. EachquantumsubclassofasystemisassociatedwithasubspaceC ⊂P inthe probabilityspaceofthesystem,andsowithaCliffordprobabilityvectoreC,atop vectoroftheCliffordalgebraCliffC ⊂CliffP. Thendefinei :P→CliffP asaCantorbrace,modulolinearity: ∀p∈P :i p:={p}, mod i(ax+by)≡aix+biy. (18) Take S (as a first trial) to be the least Clifford algebra that is its own Clifford algebra: b :S→ S=S. (19) G G CallthequantumstructureswithprobbabilityvbectobrsinSquantumsets.Thequantum setissupposedlycomplexenoughtorepresentanyfinitequantumstructure. b Table1arrangesbasicprobabilityvectors1 ofSbyrankrandserialnumbern. n b 10 DavidRitzFinkelstein Table1 Quantumandclassicalsets1 byrankrandserialnumbern n 6 1 11 11 111 11 111 111 1111 111 1111 1111 11111 ... ... ... ... hexp6 5 1 11 11 111 11 111 111 1111 111 1111 1111 11111 ... ... ... ... hexp5 4 1 11 11 111 11 111 111 1111 111 1111 1111 11111 ... 16 17 18 19 20 21 22 23 24 25 26 27 ... 3 1 11 11 111 11 111 111 1111 111 1111 1111 11111 4 5 6 7 8 9 10 11 12 13 14 15 2 1 11 2 3 1 1 1 0 1 0 r 1 n n 3.1 SpinstructureofS b Foreachrankr,S[r]isnaturallyaspinorspace: • D[r]=hexprbisitsdimension. • S[r−1]isitsCartansemivectorspace. • W[r−1]:=S[r−1]⊕DualS[r−1]isitsunderlyingquadraticspace. b • SO(D[r−1],D[r−1])isitsorthogonalgroup. b b • ThereisaneutralsymmetricPauliformb [r]:S[r]→DualS[r]forwhichthefirst gradeg w∈Cliff[r]arehermitiansymmetric. b b