On Generalised Statistical Equilibrium and Discrete Quantum Gravity (Springer Theses)

194 Pages·2022·3.42 MB·English

by  Isha Kotecha

Springer Theses
Recognizing Outstanding Ph.D. Research
Isha Kotecha
On Generalised
Statistical
Equilibrium
and Discrete
Quantum Gravity
Springer Theses
Recognizing Outstanding Ph.D. Research
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Isha Kotecha
On Generalised Statistical
Equilibrium and Discrete
Quantum Gravity
Doctoral Thesis accepted by
Humboldt-Universität zu Berlin, Berlin, Germany
Author Supervisor
Dr.IshaKotecha Dr.DanieleOriti
QubitsandSpacetimeUnit ArnoldSommerfeldCenterforTheoretical
OkinawaInstituteofScience Physics
andTechnology Ludwig-Maximilians-UniversitätMünchen
Okinawa,Japan Munich,Germany
ISSN2190-5053 ISSN2190-5061 (electronic)
SpringerTheses
ISBN978-3-030-90968-0 ISBN978-3-030-90969-7 (eBook)
https://doi.org/10.1007/978-3-030-90969-7
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SwitzerlandAG2022
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Wherethevoiceofthewindcallsour
wanderingfeet,
Throughechoingforestandechoingstreet,
Withlutesinourhandsever-singingweroam,
Allmenareourkindred,theworldisour
home.
—Sarojini Naidu,inWanderingSingers
Formygrandparents,myguidingstars
andmyparents,myguidinglights
Supervisor’s Foreword
ItisapleasuretointroducetoyouthisworkbyDr.Kotecha,resultingfromherPh.D.
thesisattheMaxPlanckInstituteforGravitationalPhysicsinPotsdam,Germany.It
is a top-quality presentation of a difficult and fascinating area of research, dealing
withtrulyfoundationaltopicsintheoreticalphysics,combiningtechnicalaswellas
conceptualchallenges.Itincludesthedetailsoftheoriginalresearchcarriedoutby
the author and a lucid and insightful discussion of the context and implications of
thisresearch,providingausefulintroductiontothefield.
Thegeneraltopicofthisworkisthedefinitionofafullycovariantquantumstatis-
ticalmechanicsframeworkforquantumgravitydegreesoffreedom.Theproblemof
definingstatisticalmechanicsinabackgroundindependentcontextwhenthegravi-
tationalfieldandgeometryarethemselvesdynamicalisanopenissueintheoretical
physicsandnotoriouslychallengingatbothconceptualandtechnicallevels.When
thegravitationalfieldandspacetimearethemselvesquantized,thechallengeisway
bigger.Further,inthecontextofquantumgravityapproacheswherespacetimeand
thegravitationalfield(classicalorquantum)donotfeatureamongthebasicentities
buthavetobereconstructedfromthecollectivebehaviourofdifferent,morefunda-
mentalquantumdegreesoffreedom,thechallengeisevengreater.Yet,thiswasthe
challengethatDr.KotechasetforherselfandthecontextofherPh.D.thesis.
The motivation for tackling this challenge is as big as its difficulties. A proper
understandingoftheinterplaybetweengravitational,quantum,andthermaleffectsis
ofparamountimportance,infact,wellbeyonditsfoundationalsignificanceandthe
formaldefinitionofatheoryofquantumgravity.Itisalsonecessary,forexample,for
understandingthephysicsofquantumblackholesandcosmologicalhorizons,even
inasemi-classicalregime,andfortacklingseveralissuesconcerningthephysicsof
theprimordialuniverse.
Thespecificformalcontextofthethesisisthatofdiscretequantumgravityandof
tensorialgroupfieldtheorymodels,buttheresearchquestionstackledbyDr.Kotecha,
thetechnicalstrategiesadoptedtoanswerthem,andthepotentialimpactofherresults
go way beyond this specific context. On the one hand, the tensorial group field
theoryformalismhasbeenchosensince,duetoitstechnicalpeculiarities,itallows
fortherigorousimplementationofconceptualandmathematicalstrategiestowarda
ix
x Supervisor’sForeword
definitionofquantumstatisticalmechanics,whichhavebeenproposedforquantum
gravity more generally. On the other hand, other quantum gravity formalisms like
spinfoammodelsandloopquantumgravityareverycloselyrelatedtotensorialgroup
fieldtheoriesandsharethesametypeofdiscretefundamentaldegreesoffreedom.All
thisiseloquentlyexplainedinthereviewpartofthiswork,whichclarifiesconceptual
issues,previousresearchonthistopic,andtherequiredtechnicalbackground.
The original research at the basis of the present book is the result of four main
researchprojects.Thefirsttwotackledtheissueofdefiningstatisticalequilibrium
infullquantumgravity.Thisisanabsolutelynon-trivialproblemsincefundamental
quantum gravity lacks any preferred notion of time, which is usually needed to
definewhatequilibriummeans.Thefirstprojectprovidedafewcandidateconstruc-
tionsofequilibriumstatesingroupfieldtheory:onebasedonKMSconditionsand
rootedinthealgebraicdefinitionofthetheory(inthespiritoftheso-called‘thermal
timehypothesis’)andassociatedtovariousinternalsymmetriesofthecorresponding
algebraofobservables,interpretedaspossible‘temporal’evolutionflowsassociated
to internal degrees of freedom of the models used as ‘relational clocks’; another
restingonthedeparametrizationofthesametheory,withanysuchrelationalclock
acquiringthestandardroleasexternalevolutionparameter,insomeapproximation;
andyetanotheronebasedonJaynes’entropymaximizationprincipleatfixedaverage
volume.Thesecondprojectexpandsandgeneralizesthislastapproach,anditdefines
statisticalequilibriumbasedonJaynes’entropymaximizationprinciple,subjectto
generalmacroscopicconstraints,providingthenanumberofconcreteexamples.The
procedurewasappliedtosystemsofclassicalandquantumtetrahedra,thedegreesof
freedomsharedbydiscretequantumgravity,spinfoammodels,andgroupfieldtheo-
ries.Infact,itcanalsobeseenasafundamentaldefinitionofthegroupfieldtheory
formalism, as a background independent statistical mechanics system of quantum
simplices. This second work also rests on a geometric formulation of statistical
mechanics applied to quantum gravity, which is of great interest in itself. The last
twoprojectsconcernedtheconstructionofathermofielddynamicalframeworkfor
group field theories, and associated thermal representations, and studied coherent
states associated to such thermal representations. Next, they combined the ther-
mofieldformalismandtheassociatedcoherentstatestostudytheemergentcosmo-
logicaldynamicsfromgroupfieldtheoriesandtheroleofthermalfluctuationsinit.
Theequilibriumstatesforquantumgravitysystemsthathadbeendevelopedinthe
previousprojectsfoundthenafirstphysicalapplication.Thewholeconstructionis
highlynon-trivialatthemathematical level,anditisasolidbasisforanumber of
currentandfuturephysicalapplications.
Butthebookisfarfromasimplecollectionofresearchresults.Itisacoherent
andorganicexpositionofthisfoundationalresearcharea,startingfromtheconcep-
tual motivations, developing the mathematical foundations, and then presenting in
a logically crystal-clear manner the various developments. The author provides us
withaveryclearvisionofthefieldandpresentsitinaninsightfulanddeepmanner.
Moreover, it is, quite simply, a pleasure to read: the ideas are clearly exposed and
logically connected to one another, the more formal aspects presented with rigour
butnounnecessaryorpedanticdetail,thephysicalaspectsdulyemphasizedandwell
Supervisor’sForeword xi
placedinabroadercontext,andthewholepresentationisfullofinterestinginsights.
IamcertainthatthereaderwillenjoyitasmuchasIdid.
Munich,Germany Dr.DanieleOriti
March2021 Seniorresearcherandgroupleader,
HeisenbergGrant—DFG