BRST Symmetry and De Rham Cohomology

205 Pages·2015·1.126 MB·English

by  Soon-Tae Hong

Soon-Tae Hong
BRST
Symmetry
and de Rham
Cohomology
BRST Symmetry and de Rham Cohomology

Soon-Tae Hong
BRST Symmetry and de
Rham Cohomology
123
Soon-TaeHong
ScienceEducation
EwhaWomansUniversity
Seoul,RepublicofKorea
ISBN978-94-017-9749-8 ISBN978-94-017-9750-4 (eBook)
DOI10.1007/978-94-017-9750-4
LibraryofCongressControlNumber:2015935190
SpringerDordrechtHeidelbergNewYorkLondon
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Preface
Since Dirac proposed quantization for systems with constraints, there have been
considerableprogressesinHamiltonianquantizationmethodassociatedwithBecci-
Rouet-Stora-Tyutin(BRST)charge.EspeciallyintopologicalsolitonssuchasO(3)
nonlinearsigmamodel,CP(N)model,Skyrmionmodelandchiralbagmodel,there
appeargeometricalconstraintswhichcanberigorouslytreatedintheHamiltonian
quantizationschemebyexploitingStückelbergfieldsin extendedphase spaces.In
this book,by includingghostand antighostfields in these extendedphase spaces,
we constructthe BRST invarianteffectiveLagrangianforthese solitonsandsome
othermodelsdescribedbelow.
Moreover, exploiting the Hamiltonian quantization method, there have been
attempts to quantize the geometrically constrained systems such as free particles
on a sphere and on a torus to investigate the BRST symmetries involved in the
systems. Both symplectic embedding and Hamilton-Jacobi quantization schemes
have been also developed to analyze Proca model, self-dual master Lagrangian
and nonholonomic system. The BRST symmetries in SU(3) linear sigma model,
fractional spin statistics of CP(1) model with Hopf term and gauge symmetry
enhancementinenlargedCP(N)modelcoupledwithU(2)Chern-Simonstermhave
beenstudiedintheHamiltonianquantizationmethod.
Phenomenologically, flavor symmetry breaking effect on SU(3) Skyrmion has
beeninvestigatedtoyieldrelevantmassspectraincludingWeylorderingcorrections
associatedwiththeconstraints.StrangenessinSU(3)chiralbagmodel,whichisa
hybridoftheSkyrmionandMITbagmodel,hasbeenevaluatedintermsofbaryon
octet and decuplet magnetic moments to predict data of SAMPLE and HAPPEX
experimentsonprotonstrangeformfactor.Mostofphysicalsystemsaresupposedto
possessconstraintsandthussignificanceoftheHamiltonianquantizationforthese
systemsisbeingemphasizedincreasingly.
Finally, in this book, BRST charge, de Rham cohomology and closed algebra
ofquantumfieldoperatorshavebeeninvestigatedin’tHooft-Polyakovmonopole,
which is classified as second class system in the Dirac quantization formalism.
To this end, the first class Hamiltonian of the monopole has been constructed
to define a monopole charge in U(1) subgroup of U(2) gauge group in the first
v
vi Preface
classconfigurationandtoinvestigateBogomol’nyiboundonextendedinternaltwo-
sphere. The explicit form of the BRST invariant Hamiltonian has been studied to
discussgeometricaspectsofthecorrespondingdeRhamcohomology.
Seoul,RepublicofKorea Soon-TaeHong
November2014
Acknowledgments
The author would like to thank G.E. Brown, Y.M. Cho, D.K. Hong, W.T. Kim,
Y.W. Kim, K. Kubodera, B.H. Lee, J. Lee, S.H. Lee, T.H. Lee, C.M. Maekawa,
R.D. McKeown,D.P. Min, F. Myhrer,A.J. Niemi, P. Oh, B.Y. Park, Y.J. Park, M.
Ramsey-Musolf,M.RhoandK.D.Rotheforhelpfuldiscussions.
vii

Contents
1 Introduction .................................................................. 1
2 HamiltonianQuantizationwithConstraints.............................. 5
2.1 HamiltonianQuantizationofFreeParticleonSphere................. 5
2.2 HamiltonianQuantizationofFreeParticleonTorus.................. 11
3 BRSTSymmetryinConstrainedSystems................................. 15
3.1 BRSTSymmetryinFreeParticleSystemonSphere.................. 15
3.2 BRSTSymmetryinFreeParticleSystemonTorus................... 19
4 SymplecticEmbeddingandHamilton-JacobiQuantization............ 25
4.1 SymplecticEmbeddingofFreeParticleonTorus..................... 25
4.2 Hamilton-JacobiQuantizationofNonholonomicSystem............. 28
4.3 SymplecticEmbeddingandHamilton-JacobiAnalysisof
ProcaModel............................................................. 34
5 HamiltonianQuantizationandBRSTSymmetryofSolitonModels... 51
5.1 HamiltonianandSemi-classicalQuantizationofO(3)
NonlinearSigmaModel ................................................ 51
5.2 SchrödingerRepresentationofO(3)NonlinearSigmaModel........ 61
5.3 BRSTSymmetryinSU(3)LinearSigmaModel...................... 64
5.4 BRSTExtensionofFaddeevModel.................................... 75
6 Hamiltonian Quantization and BRST Symmetry of
SkyrmionModels ............................................................ 81
6.1 HamiltonianQuantizationofSU(2)Skyrmion ........................ 82
6.2 BRSTSymmetryofSU(2)Skyrmion.................................. 90
6.3 HamiltonianQuantizationofSU(3)Skyrmion ........................ 94
6.4 FlavorSymmetryBreakingEffectonSU(3)Skyrmion............... 103
7 HamiltonianStructureofOtherModels .................................. 111
7.1 BosonizationofQCDatHighDensity................................. 111
7.2 GaugeSymmetryEnhancementofEnlargedCP(N)Model .......... 122
ix