Quantum Field Theory II: Introductions to Quantum Gravity, Supersymmetry and String Theory

374 Pages·2016·4.63 MB·English

Graduate Texts in Physics
Edouard B. Manoukian
Quantum
Field Theory II
Introductions to Quantum Gravity,
Supersymmetry and String Theory
Graduate Texts in Physics
Serieseditors
KurtH.Becker,PolytechnicSchoolofEngineering,Brooklyn,USA
SadriHassani,IllinoisStateUniversity,Normal,USA
BillMunro,NTTBasicResearchLaboratories,Atsugi,Japan
RichardNeeds,UniversityofCambridge,Cambridge,UK
Jean-MarcDiMeglio,UniversitéParisDiderot,Paris,France
WilliamT.Rhodes,FloridaAtlanticUniversity,BocaRaton,USA
SusanScott,AustralianNationalUniversity,Acton,Australia
H.EugeneStanley,BostonUniversity,Boston,USA
MartinStutzmann,TUMünchen,Garching,Germany
AndreasWipf,Friedrich-Schiller-UnivJena,Jena,Germany
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Edouard B. Manoukian
Quantum Field Theory II
Introductions to Quantum Gravity,
Supersymmetry and String Theory
123
EdouardB.Manoukian
TheInstituteforFundamentalStudy
NaresuanUniversity
Phitsanulok,Thailand
ISSN1868-4513 ISSN1868-4521 (electronic)
GraduateTextsinPhysics
ISBN978-3-319-33851-4 ISBN978-3-319-33852-1 (eBook)
DOI10.1007/978-3-319-33852-1
LibraryofCongressControlNumber:2016935720
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Preface to Volume II
My motivation in writing this second volume was to have a rather introductory
book on quantum gravity,1 supersymmetry,2 and string theory3 for a reader who
has had some training in conventional quantum field theory (QFT) dealing with
its foundations, with abelian and non-abelian gauge theories including grand
unification, and with the basics of renormalization theory as already covered in
Vol.IQuantumFieldTheoryI:FoundationsandAbelianandNon-AbelianGauge
Theories. This volume is partly based on lectures given to graduate students in
theoretical and experimental physics, at an introductory level, emphasizing those
parts which are reasonably well understood and for which satisfactory theoretical
descriptionshavebeengiven.
Quantumgravityisavastsubject,4andIobviouslyhavetomakeachoiceinthis
introductorytreatmentofthesubject.Asanintroduction,Irestrictthestudytotwo
differentapproachestoquantumgravity:theperturbativequantumgeneralrelativity
approach as the main focus and a non-perturbative background-independent one
referredtoas“loopquantumgravity”(LQG),wherespaceemergesfromthetheory
itselfandisquantized.InLQGweencounteraQFTinathree-dimensionalspace.
1FormoreadvancedbooksonquantumgravitythatIamfamiliarwith,seethefollowing:C.Kiefer
(2012): Quantum Gravity,by Oxford University Press, T.Thiemann (2007): Modern Canonical
Quantum Gravity, C.Rovelli (2007): Quantum Gravity, aswell as ofthecollection of research
investigationsinD.Oriti(2009):ApproachestoQuantumGravity,byCambrigeUniversityPress.
2Formoreadvanced books onsupersymmetry thatIamfamiliarwith,seethefollowingbooks:
H. Baer & X. Tata (2006): Weak scale supersymmetry: from superfields to scattering events,
M.Dine(2007):Supersymmetryandstringtheory-beyondthestadardmodel,S.Weinberg(2000):
TheQuantumtheoryoffieldsIII:Supersymmetry,byCambridgeUniversityPress,andP.Binetruy
(2006):Supersymmetry,experimentsandcosmologybyOxfordUniversityPress.
3For more advanced books on string theory that I am familiar with, see the following books:
K.Becker,M.Becker&J.H.Schwarz(2006):StringtheoryandM-theory-amodernapproach,
M.Dine(2007):Supersymmetryandstringtheory-beyondthestandardmodel,andJ.Polchinski
(2005):SuperstringtheoryI&IIbyCambridgeUniversityPress.
4Seethereferencesgivenaboveonquantumgravity.
v
vi PrefacetoVolumeII
Someuniquefeaturesofthetreatmentgivenare:
(cid:129) No previous knowledge of general relativity is required, and the necessary
geometricalaspectsneededarederivedafresh.
(cid:129) The derivation of field equations and of the expression for the propagator of
the graviton in the linearized theory is solved with a gauge constraint, and a
constraint necessarily implies that not all the components of the gravitational
fieldmaybevariedindependently—apointwhichismostoftenneglectedinthe
literature.
(cid:129) Anelementarytreatmentisgivenoftheso-calledSchwinger-DeWitttechnique.
(cid:129) Non-renormalizabilityaspectsofquantumgeneralrelativityarediscussedaswell
asoftherenormalizabilityofsomehigher-orderderivativegravitationaltheories.
(cid:129) A proof is given of the Euler-Poincaré Characteristic Theorem which is most
oftenomittedintextbooks.
(cid:129) A uniqueness property of the invariant product of three Riemann tensors is
provedwhichisalsomostoftenomittedintextbooks.
(cid:129) Anintroductorytreatmentisprovidedof“loopquantumgravity”withsufficient
details to get the main ideas across and prepare the reader for more advanced
studies.
Supersymmetry is admittedly a theory with mathematical beauty. It unites
particlesof integerand half-integerspins, i.e., with differentspins, butwith equal
masses in symmetry multiplets. Some important aspects in the treatment of the
subjectarethefollowing:
(cid:129) A fundamental property of supersymmetric theories is that the supersym-
metry charge (supercharge) operator responsible for interchanging bosonic
and fermionic degrees of freedom obviously does not commute with angular
momentum (spin) due to different spins arising in a given supermultiplet.
This commutation relation is explicitly derived which is most often omitted in
textbooks.
(cid:129) The concept of superspace is introduced, as a direct generalization of the
Minkowski one, and the basic theory of integration and differentiation in
superspaceisdeveloped.
(cid:129) A derivation is given of the so-called Super-Poincaré algebra satisfied by the
generators of supersymmetry and spacetime transformations, which involves
commutators and anti-commutators5 and generalizes the Poincaré algebra of
spacetimetransformationsderivedinVol.I.
(cid:129) Thesubjectofsupersymmetricinvarianceofintegrationtheoryinsuperspaceis
developed as it is a key ingredient in defining supersymmetric actions and in
constructingsupersymmetricextensionsofvariousfieldtheories.
(cid:129) A panoramaofsuperfieldsisgivenincludingthatofthe purevectorsuperfield,
andcompletederivationsareprovided.
5Suchanalgebraisreferredtoasagradedalgebra.
PrefacetoVolumeII vii
(cid:129) Once the theory of supersymmetric invariant integration is developed, and
superfieldsareintroduced,supersymmetricextensionsofbasicfieldtheoriesare
constructed,suchasthatofMaxwell’stheoryofelectrodynamics;aspin0–spin
1/2 field theory, referred to as the Wess-Zumino supersymmetric theory with
interactions;theYang-Millfieldtheory;andthestandardmodel.
(cid:129) There are several advantages of a supersymmetric version of a theory over
its non-supersymmetric one. For one thing, the ultraviolet divergence problem
is much improved in the former in the sense that divergences originating
from fermions loops tend, generally, to cancel those divergent contributions
originating from bosons due to their different statistics. The couplings in the
supersymmetric version of the standard model merge together more precisely
at a high energy. Moreover, this occurs at a higher energy than in the non-
supersymmetric theory, getting closer to the Planck one at which gravity is
expectedto be significant.Thisgivesthe hopeofunifyinggravitywith the rest
ofinteractionsinaquantumsetting.
(cid:129) Spontaneoussymmetrybreakingisdiscussedtoaccountforthemassdifferences
observedinnatureofparticlesofbosonicandfermionictypes.
(cid:129) Theunderlyinggeometrynecessaryforincorporatingspinorsingeneralrelativity
isdevelopedtofinallyandexplicitlyderivetheexpressionoftheactionofthefull
supergravitytheory.
In string theory, one encounters a QFT on two-dimensionalsurfaces traced by
stringsinspacetime,referredtoastheirworldsheets,withremarkableconsequences
in spacetime itself, albeit in higher dimensions. If conventional field theories are
low-energyeffective theories of string theory, then this alone justifies introducing
this subjectto the student.Some importantaspects of the treatmentof the subject
arethefollowing:
(cid:129) In string theory, particles that are needed in elementary particle physics arise
naturallyinthemassspectraofoscillatingstringsandarenot,apriori,assumed
toexistorputinbyhandintheunderlyingtheory.Oneofsuchparticlesemerging
fromclosedstringsistheevasivegraviton.
(cid:129) Withthestringsbeingoffiniteextensions,stringtheorymay,perhaps,providea
betterapproachthanconventionalfieldtheorysincethelatterinvolvesproducts
ofdistributionsatthesamespacetimepointswhicharegenerallyilldefined.
(cid:129) Details are given of all the massless fields in bosonic and superstring theories,
includingthedeterminationoftheirinheriteddegreesoffreedom.
(cid:129) The derived degrees of freedom associated with a massless field in D-
dimensional spacetime, together with the eigenvalue equation associated with
the mass squared operator associated with such a given massless field, are
consistently used to determine the underlying spacetime dimensions D of the
bosonicandsuperstringtheories.
(cid:129) Elementsofspacecompactificationsareintroduced.
(cid:129) The basics of the underlying theory of vertices, interactions, and scattering of
stringsaredeveloped.
(cid:129) Einstein’stheoryofgravitationisreadilyobtainedfromstringtheory.
(cid:129) TheYang-Millsfieldtheoryisreadilyobtainedfromstringtheory.
viii PrefacetoVolumeII
Thisvolumeisorganizedasfollows.InChap.1,thereaderisintroducedtoquan-
tum gravity, where no previous knowledge of general relativity (GR) is required.
Allthenecessarygeometricalaspectsarederivedafreshleadingtoexplicitgeneral
Lagrangiansforgravity,includingthatofGR.Thequantumaspectofgravitation,as
describedbythegraviton,isintroduced,andperturbativequantumGRisdiscussed.
The so-called Schwinger-DeWitt formalism is developed to compute the one-
loop contribution to the theory, and renormalizability aspects of the perturbative
theory are also discussed. This follows by introducing the very basics of a non-
perturbative,background-independentformulationof quantum gravity, referred to
as “loop quantumgravity” which gives rise to a quantization of space and should
be interestingto the reader.In Chap.2, we introducethe readerto supersymmetry
and its consequences.In particular,quite a detailed representationis givenfor the
generationofsuperfields,andtheunderlyingsectionshouldprovideausefulsource
ofinformationonsuperfields.SupersymmetricextensionsofMaxwell’stheory,as
well as of Yang-Mills field theory, and of the standard model are worked out,
as mentioned earlier. Spontaneous symmetry breaking, and improvement of the
divergenceprobleminsupersymmetricfieldtheoryarealsocovered.Theunification
of the fundamentalcouplingsin a supersymmetricversionof the standardmodel6
isthenstudied.Geometricalaspectsnecessarytostudysupergravityareestablished
culminating in the derivation of the full action of the theory. In the final chapter,
the reader is introduced to string theory, involvingboth bosonic and superstrings,
andtotheanalysisofthespectraofthemass(squared)operatorassociatedwiththe
oscillatingstrings.Thepropertiesoftheunderlyingfields,associatedwithmassless
particles, encountered in string theory are studied in some detail. Elements of
compactification, duality, and D-branes are given, as well as of the generation of
verticesandinteractionsofstrings.Inthefinalsectionsonstringtheory,wewillsee
howonemayrecovergeneralrelativityandtheYang-Millsfieldtheoryfromstring
theory.Wehavealsoincludedtwoappendicesattheendofthisvolumecontaining
useful informationrelevant to the rest of this volume and should be consulted by
the reader. The problemsgiven at the end of the chaptersform an integralpartof
the books, and many developments in the text depend on the problems and may
include, in turn, additional material. They should be attempted by every serious
student. Solutions to all the problems are given right at the end of the book for
theconvenienceofthereader.Wemakeitapointpedagogicallytoderivethingsin
detail,andsomeofsuchdetailsaresometimesrelegatedtoappendicesattheendof
therespectivechapters,orworkedoutintheproblems,withthemainresultsgiven
inthechaptersinquestion.TheverydetailedintroductiontoQFTsinceitsbirthin
1926inVol.I,7aswellastheintroductionstothechapters,providethemotivations
6The standard model consists of the electroweak and QCD theories combined, with a priori
underlyingsymmetryrepresentedbythegroupproductsSU.2/(cid:2)U.1/(cid:2)SU.3/.
7QuantumFieldTheoryI:FoundationsandAbelianandNon-AbelianGaugeTheories.Istrongly
suggestthatthereadergoesthroughtheintroductorychapterofVol.Itoobtainanoverallviewof
QFT.
PrefacetoVolumeII ix
and the pedagogical means to handle the technicalities that follow them in these
studies.
This volume is suitable as a textbook. Its content may be covered in a 1 year
(two semesters) course. Short introductoryseminar courses may be also given on
quantumgravity,supersymmetry,andstringtheory.
I often meet students who have a background in conventional quantum field
theorymentionedearlierandwanttolearnaboutquantumgravity,supersymmetry
and string theory but have difficulty in reading more advanced books on these
subjects.Ithusfeltapedagogicalbookisneededwhichputsthesetopicstogether
anddevelopstheminacoherentintroductoryandunifiedmannerwithaconsistent
notationwhichshouldbeusefulforthestudentwhowantstolearntheunderlying
different approaches in a more efficient way. He or she may then consult more
advancedspecializedbooks,alsomentionedearlier,foradditionaldetailsandfurther
developments,hopefully,withnotmuchdifficulty.
Ifirmlybelievethatdifferentapproachestakenindescribingfundamentalphysics
atveryhighenergiesoratverysmalldistancesshouldbeencouragedandconsidered
asfutureexperimentsmayconfirmdirectly,orevenindirectly,theirrelevancetothe
realworld.
I hope this book will be useful for a wide range of readers. In particular, I
hope that physics graduate students, not only in quantum field theory and high-
energy physics but also in other areas of specializations, will also benefit from it
as,accordingtomyexperience,theyseemtohavebeenleftoutofthisfundamental
areaofphysics,aswellasinstructorsandresearchersintheoreticalphysics.
EdouardB.Manoukian