Tensorial Methods and Renormalization in Group Field Theories

236 Pages·2014·3.76 MB·English

by  Sylvain Carrozza (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Sylvain Carrozza
Tensorial
Methods and
Renormalization
in Group Field
Theories
Springer Theses
Recognizing Outstanding Ph.D. Research
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Sylvain Carrozza
Tensorial Methods
and Renormalization
in Group Field Theories
Doctoral Thesis accepted by
the Université Paris-Sud, Orsay, France
123
Author Supervisors
Dr. SylvainCarrozza Prof.Vincent Rivasseau
LaboratoiredePhysiqueThéoriqued’Orsay LaboratoiredePhysiqueThéoriqued’Orsay
Université Paris-Sud Université Paris-Sud
Orsay Orsay
France France
Prof.Daniele Oriti
Max PlanckInstituteforGravitational
Physics(Albert EinsteinInstitute)
Potsdam-Golm
Germany
ISSN 2190-5053 ISSN 2190-5061 (electronic)
ISBN 978-3-319-05866-5 ISBN 978-3-319-05867-2 (eBook)
DOI 10.1007/978-3-319-05867-2
Springer ChamHeidelberg New YorkDordrecht London
LibraryofCongressControlNumber:2014935056
(cid:2)SpringerInternationalPublishingSwitzerland2014
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Wir sollen heiter Raum um Raum
durchschreiten,
An keinem wie an einer Heimat hängen,
Der Weltgeist will nicht fesseln uns
und engen,
Er will uns Stuf’ um Stufe heben, weiten.
Hermann Hesse, Stufen,
in Das Glasperlenspiel, 1943.
Supervisor’s Foreword
It is a pleasure to introduce Sylvain Carrozza’s thesis for publication in the
SpringerThesesBookSeries.Withsomanycurrentlycompetingapproachestothe
famous issue of quantizing gravity, it is possible to get confused. The physicists
who dare to engage in this deeply speculative domain can sometimes be viewed
from the outside as lost in byzantine quarrels. Nevertheless, the subject is so
fundamentalforourmostbasicunderstandingoftheworldthatitkeepsattracting
generation after generation of young and gifted scientists. They simply cannot be
stoppedintheirquest.Sinceitseemsunlikelythatadecisiveexperimentwilltruly
andunambiguouslyfavorasingletheoryofquantumgravityinthenearfuture,we
should be prepared to keep exploring the issue for a long time to come, using
simplicityandmathematicalconsistencyasrationalcriteriatoselectandstudyin-
depth the most promising formalisms.
It is in this spirit that Sylvain Carrozza’s thesis throws light at a particularly
interesting crossroad between several of the most prominent approaches to
quantum gravity. Together with co-authors D. Oriti and V. Rivasseau, Sylvain
Carrozzahasdefinedapromisingclassofnewmodels,calledtensorialgroupfield
theories, and explored in-depth their power counting and renormalization. Not
onlydoesthethesisgathersallcorrespondingresultsandtheorems,establishingin
this way solid mathematical foundations for this type of theories. It also contains
oneofthemostlucidandbestwrittenrecentintroductionstothesetopics.Starting
fromstandardnotionsofquantumfieldtheory,thethesispresentsindetailandina
pedagogicalmannertherelevantnotionsandresultsinloopquantumgravity,spin
foams, group field theories, and random tensor models, up to the level of present
day research.
Inthisway,thethesisachievesaremarkablesynthesisbetweendifferentsetsof
ideasandopensupfascinatingnewavenuesforresearch.Indeed,ontheonehand,
randomtensorsareafastdevelopingextensionofrandommatrices.Thelatterare
deeply connected to string theory, the leading formalism to quantize gravity, and
have been used successfully to understand quantum gravity and dynamical trian-
gulationsintwodimensionsaswellasquantumfieldtheoriesonnoncommutative
space-times. On the other hand, group field theories are the second quantized
version of loop quantum gravity, often considered the most important alternative
to string theory.
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viii Supervisor’sForeword
The publication of this Ph.D. as a Springer Thesis will hopefully further
acceleratethediffusionanddiscussionofthisexcellentsynthesis.ReadingSylvain
Carrozza’sthesiswillbebothenjoyableandinstructiveforanyscientistinterested
in understanding some of the most up-to-date current research on the quantum
aspects of space, time, and gravity.
Orsay, February 2014 Prof. Vincent Rivasseau
Abstract
In this thesis, we study the structure of Group Field Theories (GFTs) from the
pointofview ofrenormalizationtheory.Suchquantumfield theoriesarefound in
approaches to quantum gravity related to Loop Quantum Gravity (LQG) on the
onehand,andtomatrixmodelsandtensormodelsontheotherhand.Theymodel
quantum space-time, in the sense that their Feynman amplitudes label
triangulations, which can be understood as transition amplitudes between LQG
spin network states. The question of renormalizability is crucial if one wants to
establish interesting GFTs as well-defined (perturbative) quantum field theories,
andinasecondstepconnectthemtoknowninfraredgravitationalphysics.Relying
on recently developed tensorial tools, this thesis explores the GFT formalism in
twocomplementarydirections.First,newresultsonthelargecut-offexpansionof
the colored Boulatov-Ooguri models allow to explore further a nonperturbative
regimeinwhich infinitely manydegrees offreedomcontribute. Thesecondsetof
results provide a new rigorous framework for the renormalization of so-called
Tensorial GFTs (TGFTs) with gauge invariance condition. In particular, a
nontrivial 3d TGFT with gauge group SU(2) is proven just-renormalizable at the
perturbative level, hence opening the way to applications of the formalism to (3d
Euclidean) quantum gravity.
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Acknowledgments
First of all, I would like to thank my two supervisors, Daniele Oriti and Vincent
Rivasseau. Obviously, the results exposed in this thesis could not be achieved
without their constant implication, guidance, and help. They introduced me to
numerousphysicalconceptsandmathematicaltools,withpedagogyandpatience.
Remarkably,theirteachingsandadviceswerealwayscomplementarytoeachother,
somethingIattributetotheiropen-mindednessandwhichIgreatlybenefitedfrom.
I particularly appreciated the trusting relationship we had from the beginning.
It was thrilling, and tome the rightbalance between supervisionand freedom.
I feel honored by the presence of Bianca Dittrich, Razvan Gurau, Renaud
Parentani,andCarloRovelliinthejury,whokindlyacceptedtoexaminemywork.
Many thanks to Bianca and Razvan especially, for their careful reading of this
manuscript and their comments.
I would like to thank the people I met at the AEI and at the LPT, who
contributed to making these 3 years very enjoyable. The Berlin quantum gravity
groupbeingalmostuncountable,IwillonlymentionthepeopleIhadthechanceto
directlycollaboratewith:AristideBaratin,FrancescoCaravelli,JamesRyan,Matti
Raasakka, and Matteo Smerlak.
It is quite difficult to keep track of all the events which, one way or another,
conspired to pushing me into physics and writing this thesis. It is easier to
remember and thank the people who triggered these long-forgotten events.
Firstandforemost,myparents,whoraisedmewithdedicationandlove,turning
the ignorant toddler I once was into a curious young adult. Most of what I am
today takes its roots at home, and has been profoundly influenced by myyounger
siblings: Manon, Julia, Pauline, and Thomas. My family at large, going under the
name of Carrozza, Dislaire, Fontès, Mécréant, Minden, Ravoux, or Ticchi, has
always been very present and supportive, which I want to acknowledge here.
The good old chaps, Sylvain Aubry, Vincent Bonnin, and Florian Gaudin-
Delrieu,deeplyinfluencedmyhighschoolyears,andhencethewayIthinktoday.
Meeting them in different corners of Europe during the 3 years of this Ph.D. was
very precious and refreshing.
MyfriendsfromtheENStimesplayedamajorroleintherecentyears,bothat
the scientific and human levels. In this respect, I would especially like to thank
Antonin Coutant, Marc Geiller, and Baptiste Darbois-Texier: Antonin and Marc,
for endless discussions about theoretical physics and quantum gravity, which
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