Lorentz covariant statistical mechanics and thermo

arXiv:0712.2725v2[hep-th]18Dec2007LorentzcovariantstatisticalmechanicsandthermodynamicsoftherelativisticidealgasandpreferredframeK.Kowalski,J.Rembieli´nskiandK.A.Smoli´nskiDepartmentofTheoreticalPhysics,UniversityofL´od´z,ul.Pomorska149/153,90-236L´od´z,PolandAbstractTheLorentzcovariantclassicalandquantumstatisticalmechanicsandthermodynamicsofanidealrelativisticgasofbradyons(particlesslowerthanlight),luxons(particlesmovingwiththespeedoflight)andtachyons(hypotheticalparticlesfasterthanlight)isdiscussed.TheLorentzcovariantformulationisbasedonthepreferredframeapproachwhichamongothersenablesconsis-tent,freeofparadoxesdescriptionoftachyons.Thethermodynamicfunctionswithinthecovariantapproachareobtainedbothinclassicalandquantumcase.PACSnumbers:03.30.+p,05.20.-y,05.30.-d,05.70.-a,05.70.Ce1I.INTRODUCTIONIdealgasesareoneofthemostimportantmodelsystemsofthenonrelativisticstatisticalmechanicsandthermodynamics.Examplesrangefromequationsofstateforclassicalgasestodescriptionofelectronsinmetalsandsuperconductors.Inspiteofthefactthatthestudiesofarelativisticgasofmassiveparticles(bradyons,alsocalledtardyons)gobackto1911[1],therelativisticstatisticalmechanicsandthermodynamicsarefarfromcomplete.Ontheonehand,thereasonareapparentlimitedapplicationsoftherelativisticthermodynamics.Forexample,inopinionofTerHaarandWergeland[2]:“Atextremelyhightemperaturesrelativisticeffectsmayofcoursebeimportant.Then,however,matterbehavesasmixtureofidealgasesandthislimitingcaseposesnoproblem.Byandlarge,arelativistictheoryofheatseems,therefore,tobeoflittlepracticalimportance.”Nevertheless,theargumentsofTerHaarandWergelandevidentlyfailforluxonsandtachyonswhicharerelativisticparticlesregardlessoftheconcretevalueofthetemperature.Furthermore,aspointedoutbyArag˜aodeCarvalhoandGoulartRosa[3],afullyrelativistictreatmentisrequiredbyastrophysicalsystemssuchaswhitedwarfsandneutronstars.Ontheotherhand,thedevelopmentoftherelativisticstatisticalmechanicsandthermodynamicswassloweddownbythelackofthecovariantformulation.Inparticular,wepointoutdifferenttransformationrulesoftherelativistictemperaturesuggestedbyEinstein,PlanckandvonLaue,byOttandbyLandsberg[2].Theonlyformulationworkinginthecaseoftachyons,basedonthenonstandard(absolute)synchronizationscheme,wasintroducedinaveryrecentpaper[4].InthisworkwestudytheLorentzcovariantstatisticalmechanicsandthermodynamicsoftherelativisticidealgasofbradyons,luxonsandtachyons.Insection2werecalltheformulationofspecialrelativitybasedontheabsolutesynchronization.Section3isdevotedtotheclassicalrelativisticidealgas.Inparticular,wederivethecovariantformofthermodynamicfunctions.Insection4wediscussthequantumstatisticalmechanicsandthermodynamicsoftherelativisticidealgas.Besidesderivationofthecovariantformsofthermodynamicfunctionswealsodiscusstheclassicallimit.2II.ABSOLUTESYNCHRONIZATIONSCHEMEInthissectionwerecallthebasicfactsabouttheformulationofspecialrelativitywiththehelpoftheabsolutesynchronizationofclocks[5].Amongothersthisapproachprovidesacorrectdescriptionoftachyons.Tachyonsarehypotheticalfasterthanlightparticles.Besidestheirintriguingtheoreticallypredictedproperties[6],tachyonstakeattentionofphysicistsbecausetheyarecandidatesforthedarkmatter[7]anddarkenergy[8].Moreovertheyappearinbranetheoriessuchasthebraneexcitationsaswellasincosmologicalmodels(socalledrollingtachyonmodels)[9].Forthisreasonitisinterestingtoinvestigatestatisticalandthermodynamicalpropertiesoftachyonicgas.Unfortunately,standarddescriptionoftachyonsisplaguedbynumberofinconsistencies.Typicalexamplesofsuchdifficultiesarethecausalparadoxes(tachyonanti-telephone[10]),theproblemofsocalledtranscendentaltachyon(thespaceoftachyonvelocitiesisnotaLorentzgroupcarrierspace[5,11])andvacuuminstabilityonthequantumlevel[12].AswasstatedmanyyearsagobySudarshan[11]aconsistentdescriptionoftachyonsdemandsapreferredreferenceframeonthefundamentallevel.However,thismeansthattherelativityprincipleisnecessarilybrokeninsuchacase.ThiscausesanapparentconflictwiththestandardLorentzgrouptransformationsintheMinkowskispace-time.Toovercomethisdifficultyletusnoticethatintroductionoftheinertialpreferredframe(PF)meansthatweshouldrealizetheLorentzgroupnotonlyonthespace-timecoordinatesbutalsoonthefour-velocityofthePFasseenbyinertialobservers.ThisgivesusthenecessaryfreedomtoreconcilebreakingoftherelativityprincipleandsimultaneouslytopreserveLorentzco-variance.SucharealizationoftheLorentzgroupwasgivenin[5]andithasanelegantexplanationintermsofthebundleofframesaswellasthephysicalinterpretationintermsoftheabsolutesynchronizationschemeforclocks[5,13,14].Inparticularin[5]aconsistentclassicalandquantumdescriptionoftachyonswasbuiltinthisframework,withoutoftheabovementionedinconsistencies.Itisimportanttostressthatformasslessandmassivesub-luminalparticles(luxonsandbradyonsrespectively)thisschemeiscompletelyequivalenttotheEinsteinsynchronizationscheme(socalledconventionofsynchronization[5])whereasitprovidesaconsistentdescriptionoftachyons.Asanimportantapplicationoftheabsolutesynchronizationinquantummechanicsthecovariantrelativ