AENEAS - A Custom-built Parallel Supercomputer for Quantum Gravity
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arXiv:gr-qc/9809090v1 30 Sep 1998UCI-98-31September1998
AENEASACustom-builtParallelSupercomputerforQuantumGravity
HerbertW.Hamber1DepartmentofPhysicsandAstronomyUniversityofCaliforniaIrvine,CA92171
ABSTRACTAccurateQuantumGravitycalculations,basedonthesimpliciallatticeformulation,arecomputationallyverydemandingandrequirevastamountsofcomputerresources.Acustom-made64-nodeparallelsuper-computercapableofperformingupto2×1010floatingpointoperationspersecondhasbeenassembledentirelyoutofcommoditycomponents,andhasbeenoperationalforthelasttenmonths.Itwillallowthenumericalcomputationofavarietyofquantitiesofphysicalinterestinquantumgravityandrelatedfieldtheories,includingtheestimateofthecriticalexponentsinthevicinityoftheultravioletfixedpointtoanaccuracyofafewpercent.1PhysicsGoalsOneoftheoutstandingproblemsintheoreticalphysicsisthedeterminationofthequantum-mechanicalpropertiesofEintein’srelativistictheoryofGravitation.Approachesbasedonlinearizedperturbationmeth-odshavenotbeensuccessfulsofar,astheunderlyingtheoryisknownnottobeperturbativelyrenormalizable.Duetothecomplexityofevensuchapproximatecalculations,afundamentalcouplingofthetheory,thebarecosmologicalconstanttermisusuallysettozero,thusfurtherreducingthepotentialphysicalrelevanceoftheresults.Furthermore,gravitationalfieldsarethemselvesthesourceforgravitation,whichleadstotheproblemofahighlynon-lineartheory,whereanysortofperturbativeresultsispossiblyofdoubtfulvalid-ity,especiallyinthequantumdomain,wherestrongfluctuationsinthegravitationalfieldsappearatshortdistances.Theabove-describedsituationbearssomeresemblancetothetheoryofstronginteractions,QuantumChromodynamics.Non-lineareffectsareknownheretoplayanimportantrole,andenduprestrictingthevalidityofperturbativecalculationstothehighenergy,shortdistanceregimewheretheeffectivecouplingcanbeconsideredweak.ForlowenergypropertiesWilson’sdiscretelatticeformulation,combinedwithcomputersimulations,hasprovidedsofartheonlyconvincingevidenceforquarkconfinementandchiralsymmetrybreaking,twophenomenawhicharecompletelyinvisibletoanyorderintheweakcoupling,perturbativeexpansion.Adiscretelatticeformulationcanbeappliedtotheproblemofquantizinggravityaswell.Insteadofcontinuousfields,onedealswithgravitationalfieldswhichliveonlyondiscretespace-timepointsandinteractlocallywitheachother.InRegge’ssimplicialformulationofgravity[1]oneapproximatesthefunctionalintegrationovercontinuousmetricsbyadiscretizedsumoverpiecewiselinearsimplicialgeometries[2,3].Insuchamodeltheroleofthecontinuummetricisplayedbytheedgelengthsofthesimplices,whilecurvatureisdescribedbyasetofdeficitangles,whichcanbecomputedviastandardformulaeasfunctionsofthegivenedgelengths.Thesimpliciallatticeformulationofgravityislocallygaugeinvariant[4],andisknowntocontainperturbativegravitonsinthelatticeweakfieldexpansion,makingitanattractivelatticeregularizationofthecontinuumtheory.Thestartingpointforanon-perturbativestudyofquantumgravityisasuitabledefinitionofthediscreteFeynmanpathintegral.InthesimpliciallatticeapproachonestartsfromthediscretizedEuclideanpathintegralforpuregravity,withthesquarededgelengthsasfundamentalvariables,
ZL=∞0s(Vd(s))σijdl2ijΘ[l2ij]exp−hλVh−kδhAh+aδ2hA2hquantumgravityZC=xgλ−k4RµνρσRµνρσ+···,(1.2)withk−1=8πG,andGNewton’sconstant,andreducestoitforsmoothenoughfieldconfigurations.Inthediscretecasetheintegrationovermetricsisreplacedbyintegralsovertheelementarylatticedegreesoffreedom,thesquarededgelengths.TheabovediscretegravitationalmeasureisthenthelatticeanalogoftheDeWittcontinuumfunctionalmeasure[5,6].TheδAterminthelatticeactionisthewell-knownReggeterm,andreducestotheEinstein-Hilbertactioninthelatticecontinuumlimit.Acosmologicalconstanttermisneededforconvergenceofthepathintegral,whilethecurvaturesquaredtermallowsonetocontrolthefluctuationsinthecurvature.Inpractice,andforphenomenologicalreasons,oneisonlyinterestedinthelimitwhenthehigherderivativetermsaresmall,a→0.Inthislimitthetheorydepends,intheabsenceofmatterandafterasuitablerescalingofthemetric,onlyononebareparameter,thedimensionlesscouplingk2/λ.Thediscretizedtheorycontainsonlyafinitesetofvariables,onceasetofsuitableboundaryconditionsareimposed,suchasopenorperiodic.Intheendtheoriginalcontinuumtheoryofgravityistoberecoveredasthespace-timevolumeismadelargeandthefundamentallatticespacingofthediscretetheoryissenttozero,possiblywithouthavingtorely,atleastinprinciple,onanyfurtherapproximationtotheoriginalcontinuumtheory.Quantumfluctuationsintheunderlyinggeometryarerepresentedinthediscretetheorybyfluctuationsintheedgelengths,whichcanbemodeledbyawell-definedMonte-Carlostochasticprocess.Inanalogywithotherfieldtheorymodelsstudiedbycomputer,allcalculationssofarhavebeenperformedintheEuclidean,imaginarytimeframework,whichistheonlyformulationamenabletoacontrollednumericalstudy,atleastfortheforeseeablefuture.TheMonte-Carlomethod,basedontheconceptofimportancesampling,iswellsuitedforevaluatingthediscretepathintegralforgravityandforcomputingtherequiredaveragesandcorrelationfunctions.Byacarefulanalysisofthelatticeresults,thecriticalexponentscanbeextracted,andthescalingpropertiesofinvariantcorrelationfunctionsdeterminedfromfirstprinciples.Studiesonsmalllatticessuggestarichscenarioforthegoundstateofquantumgravity[7].Thepresentevidencesuggeststhatsimplicialquantumgravityinfourdimensionsexhibitsaphasetransition(inG)betweentwophases:astrongcouplingphase,inwhichthegeometryissmoothatlargescalesandquantumfluctuationsinthegravitationalfieldarebounded,andaweakcouplingphase,inwhichthegeometryisdegenerateandspace-timecollapsesintoalower-dimensionalmanifold.Onlythesmooth,smallnegativecurvatureAdSphaseappearstobephysicallyacceptable.TheexistenceofaphasetransitionatfinitecouplingGimpliesnon-trivial,calculablenon-perturbativescalingpropertiesforthecorrelationsand