Proceedings to the workshops 'What comes beyond the Standard model', 2000, 2001, 2002, Volume 2: Proceedings (Part I)

113 Pages·0.93 MB·English

by  N. Mankoc Borstnik

BLEJSKE DELAVNICE IZ FIZIKE LETNIK 3, SˇT. 4
BLED WORKSHOPS IN PHYSICS VOL. 3, NO. 4
ISSN1580–4992
Proceedings to the workshops
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What comes beyond the
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Standard model 2000, 2001,
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2002
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1 Volume 2
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Proceedings — PART I
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Edited by
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NormaMankocˇ Borsˇtnik1,2
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p- Holger Bech Nielsen3
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Colin D. Froggatt4
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v: Dragan Lukman2
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1UniversityofLjubljana,2PINT,3NielsBohrInstitute,4GlasgowUniversity
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DMFA – ZALOZˇNISˇTVO
LJUBLJANA, DECEMBER 2002
The 5th WorkshopWhat Comes Beyond theStandard Model
wasorganizedby
DepartmentofPhysics,FacultyofMathematicsandPhysics,UniversityofLjubljana
PrimorskaInstituteofNaturalSciencesandTechnology,Koper
andsponsoredby
MinistryofEducation,ScienceandSportofSlovenia
DepartmentofPhysics,FacultyofMathematicsandPhysics,UniversityofLjubljana
PrimorskaInstituteofNaturalSciencesandTechnology,Koper
SocietyofMathematicians,PhysicistsandAstronomersofSlovenia
OrganizingCommittee
NormaMankocˇBorsˇtnik
ColinD.Froggatt
HolgerBechNielsen
Contents
Preface............................................................. V
DerivationofLorentzInvarianceandThreeSpaceDimensionsinGenericFieldTheory
CD.FroggattandH.B.Nielsen........................................ 1
UnitaryRepresentations,NoncompactGroupsSO(q,d−q)andMoreThanOneTime
N.MankocˇBorsˇtnik,H.B.NielsenandD.Lukman....................... 20
WeylSpinorofSO(1,13),FamiliesofSpinorsoftheStandardModelandTheirMasses
A.BorsˇtnikBracˇicˇandN.MankocˇBorsˇtnik ............................. 27
ATightPackingProblem
A.Kleppe........................................................... 52
WhysoFewParticleSpecies?WhyOnlyFermions&Bosons?WhyU(1),SU(2)&SU(3)Symmetries?(ASpeculative
D.L.BennettandA.Kleppe ........................................... 71
AboutNumberofFamilies
D.Lukman,A.KleppeandN.S.MankocˇBorsˇtnik........................ 88
CouplingConstantUnificationinSpin-ChargeUnifyingModelAgreeingWithProtonDecayMeasurements
N.MankocˇBorsˇtnikandH.B.Nielsen.................................. 94

Preface
Theseriesofworkshopson”WhatComesBeyondtheStandardModel?”started
in1998withtheideaoforganizingarealworkshop,inwhichparticipantswould
spendmostofthetimeindiscussions,confrontingdifferentapproachesandideas.
ThepicturesquetownofBledbythelakeofthesamename,surroundedbybeau-
tifulmountainsandofferingpleasantwalks,waschosentostimulatethediscus-
sions.
Theideawassuccessfulandhasdevelopedintoanannualworkshop. Thisyear
was a kind of small jubilee - the fifth workshop took place. Very open-minded
andfruitfuldiscussionshavebecomethetrade-markofourworkshop,producing
severalpublishedworks.IttakesplaceinthehouseofPlemelj,whichbelongsto
theSocietyofMathematicians,PhysicistsandAstronomersofSlovenia.
TheseworkshopshavealsoinspiredaseriesofEUROCONFERENCES,withthe
samenameastheworkshop(“WHATCOMESBEYONDTHESTANDARDMO-
DEL”).The first meeting inthe seriesentitled “EUROCONFERENCE ON SYM-
METRIESBEYONDTHESTANDARDMODEL”willtakeplacefrom12ofJulyto
17ofJuly2003athotelHistrioninPortorozˇ,Slovenia.Thisseriesofconferences
isalsomeanttoconfronttheideas,knowledgeandexperiencesderivedfromdif-
ferentapproachesto describing Nature beyond the Standardmodels of particle
physicsandcosmology.
Inthefifthworkshop,whichtookplacefrom13to24ofJuly2002atBled,Slove-
nia, we have tried to answer some of the open questions which the Standard
modelsleaveunanswered,like:
WhyhasNaturemadeachoiceoffour(noticeable)dimensionswhileallthe
•
others, if existing, arehidden? Andwhatarethe propertiesofspace-timein
thehiddendimensions?
HowcouldNaturemakethedecisionaboutthebreakingofsymmetriesdown
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tothenoticeableones,comingfromsomehigherdimensiond?
Whyisthemetricofspace-timeMinkowskianandhowisthechoiceofmetric
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connectedwiththeevolutionofouruniverse(s)?
Where does the observed asymmetry between matter and antimatter origi-
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natefrom?
Whydomasslessfieldsexistatall?Wheredoestheweakscalecomefrom?
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Why do only left-handed fermions carry the weak charge? Why does the
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weakchargebreakparity?
WhatistheoriginofHiggsfields?WheredoestheHiggsmasscomefrom?
•
Wheredoesthesmallhierarchycomefrom?(OrwhyaresomeYukawacou-
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plingssosmallandwheredotheycomefrom?)
VI Contents
DoMajorana-likeparticlesexist?
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Wheredothegenerationscomefrom?
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Canallknownelementaryparticlesbeunderstoodasdifferentstatesofonly
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oneparticle,withauniqueinternalspaceofspinsandcharges?
Howcanallgaugefields(includinggravity)beunifiedandquantized?
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Whydowehavemorematterthanantimatterinouruniverse?
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Whatisouruniversemadeoutof(besidesthebaryonicmatter)?
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WhatistheroleofsymmetriesinNature?
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Whatistheoriginofthefieldwhichcausedinflation?
•
Wehavediscussedthesequestionsfortendays.Someresultsofthiseffortappear
intheseProceedings,notonlyofthelastworkshop,butalsooftwoearlierwork-
shops.Thediscussionwillcontinuenextyear;attheEURESCONFERENCEand
intheWorkshop,whichwilltakeplaceaftertheconference,from17ofJulyto28
theofJuly,againatBled,againinthehouseofJosipPlemelj.
The organizers are grateful to all the participants for the lively discussions and
thegoodworkingatmosphere.
NormaMankocˇBorsˇtnik
HolgerBechNielsen
ColinFroggatt
DraganLukman Ljubljana,December2002
WorkshopsorganizedatBled
⊲ WhatComesbeyondtheStandardModel(June29–July9,1998)
⊲ HadronsasSolitons(July6-17,1999)
⊲ WhatComesbeyondtheStandardModel(July22–31,1999)
⊲ Few-QuarkProblems(July8-15,2000)
⊲ WhatComesBeyondtheStandardModel(July17–31,2000)
⊲ StatisticalMechanicsofComplexSystems(August27–September2,2000)
⊲ WhatComesbeyondtheStandardModel(July17–27,2001)
⊲ Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry
(August25–28,2001)
⊲ WhatComesBeyondtheStandardModel(July13–24,2002)

BLEDWORKSHOPS Proceedingstothe5thWorkshop
INPHYSICS Whatcomesbeyond... (p.1)
VOL.3,NO.4 Bled,Slovenia,July13-24,2002
Derivation of Lorentz Invariance and Three Space
Dimensions in Generic Field Theory
CD.Froggatt†⋆ andH.B.Nielsen‡⋆⋆
†DepartmentofPhysicsandAstronomy,GlasgowUniversity,GlasgowG128QQ,
Scotland
‡DeutschesElektronen-SynchrotronDESY,Notkestraße85,D-22603Hamburg,Germany
andTheNielsBohrInstitute,Blegdamsvej17,CopenhagenØ,Denmark
Abstract. Averygeneralquantumfieldtheory,whichisnotevenassumedtobeLorentz
invariant,isstudiedinthelimitofverylowenergyexcitations.FermionandBosonfield
theoriesareconsideredinparallel.Remarkably,inbothcasesitisarguedthat,inthefree
andlowestenergyapproximation,arelativistictheorywithjustthreespaceandonetime
dimensionemergesforeachparticletypeseparately.InthecaseofFermionfieldsitisin
theformoftheWeylequation,whileinthecaseoftheBosonsitisessentiallyintheform
oftheMaxwellequations.
1 Introduction
Since many years ago [1], we have worked on the project of “deriving” all the
knownlawsofnature,especiallythesymmetrylaws[2],fromtheassumptionof
the existence of exceedinglycomplicatedfundamentallawsof nature.However
thederivationsaresuchthatitpracticallydoesnotmatterwhattheseexceedinglycom-
plicatedlawsareindetail,justprovidedweonlystudytheminsomelimits suchasthe
low energy limit. This is the projectwhich we have baptized“Random dynam-
ics”, in order to make explicit the idea that we are thinking of the fundamental
laws of nature asbeing given by a particularmodel pulled out at randomfrom
a very large class of models. In this way, one can overcome the immediate re-
proachtotheprojectthatitiseasytoinventmodel-proposalswhich,indeed,do
not deliverthe laws of natureaswe know them today.We only makethe claim
that sufficiently complicated and generic models should work, not very special
ones that could potentially be constructed so as not to work. Also it should be
stressed that there is a lot of interpretation involved, as to which elements in
the “random” model are to be identified phenomenologically with what. As a
consequence,theprojecttendstobesomewhatphenomenologicalitself,honestly
speaking.However,inprinciple,weshouldonlyusethephenomenologytofind
outwhichquantitiesinthe “randomdynamics”modelaretobe identifiedwith
whichphysicallydefinedquantities(concepts).
⋆
E-mail:[email protected]
⋆⋆
E-mail:[email protected];[email protected]
2 CD.FroggattandH.B.Nielsen
Oneofthemostpromisingsteps,indevelopingthisrandomdynamicspro-
ject, was [1,2] to start without assuming Lorentz invariance but to assume that
wealreadyhaveseveralknownlawssuchasquantummechanics,quantumfield
theoryandmomentumconservation.Lorentzinvariancewasthen“derived”,at
leastforasinglespeciesofWeylparticleswhichemergedatlowenergy.However
this“derivation”ofLorentzinvariancemightnotactuallybethemostinteresting
resultfromthisstep inrandomdynamics;itisafterallnotsuchanoverwhelm-
ingsuccess,sinceitonlyworksforoneparticlespeciesonitsownanddoesnot,
immediatelyatleast,leadtoLorentzinvarianceifseveralparticlespeciesarein-
volved.Itmayratherbethepredictionofthenumberofspacedimensionswhich
ismoresignificant.Actuallythefundamentalmodelisassumedtohaveanarbi-
trarynumberofdimensionsandhasmomentumdegreesoffreedominallthese
dimensions, but the velocity components in all but three dimensions turn out
to be zero. In this way the extra dimensions are supposedly not accessible. So
thepredictioniseffectivelythattherearejustthreespatialdimensions(plusone
time)!
In these early studies only a fermionic field theory (without Lorentz sym-
metry) was considered, while Bosons were left out of consideration; we then
sometimes speculated that the Bosons could at least be partly composed from
Fermions and thus inherit their Lorentz symmetry. Indeed, evenin more recent
work, it is the Fermions that play the main role [3,4]. For a summary of other
recenttheoreticalmodelsand experimentaltestsof Lorentzinvariance breaking
see,forexample,reference[5].
It is the purpose of the present paper to review the work with Fermions
stressing a new feature aimed at solving a certain technical problem—the use
of the “Homolumo-gap-effect” to be explained below—and to extend the work
tothecaseofbosonicfields,whichisahighlynon-trivialextension.
In the following section we shall putforwardour verygeneralfield theory
modelandthen,insection3,weshallwritedowninparalleltheequationsofmo-
tionforBosonsandFermionsrespectively.Itturnsoutthatweobtainacommon
equationofmotionforthe“fields”in“momentum”representation—momentum
here being reallythought of asa rather general parameterisationof the degrees
offreedom,onwhichtheHamiltonianandcommutationrulesdependsmoothly.
Thisequationofmotioninvolvesanantisymmetricmatrixwhichdependsonthe
“momenta”.Thebehaviouroftheeigenvaluespectrumofsuchanantisymmetric
realmatrixisstudiedinsection4,withthehelpofsomeargumentsbasedonthe
Homolumo-gap-effectwhicharepostponedtillsection5.Theconclusionsareput
intosection6.
2 A randomdynamicsmodel
Since it is our main purpose to derive Lorentz symmetry together with 3 + 1
dimensions, we must start from a model that does not assume Lorentz invari-
ancenor theprecisenumberofspacedimensionsfromthe outset.Wewould,of
course, eventuallyhope to avoid having to assume momentum conservation or
eventheexistenceoftheconceptofmomentum.Howeverthisassumptionisless