The Minkowski and Conformal Superspaces: The Classical and Quantum Descriptions PDF
Preview The Minkowski and Conformal Superspaces: The Classical and Quantum Descriptions
8972_9789814566339_TP.indd 1 16/2/15 9:51 am May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk World Scientific 8972_9789814566339_TP.indd 2 16/2/15 9:51 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Fioresi, Rita, 1966– The Minkowski and conformal superspaces : the classical and quantum descriptions / by Rita Fioresi (Università di Bologna, Italy), María Antonia Lledó (Universitat de València, Spain). pages cm Includes bibliographical references and index. ISBN 978-9814566339 (hardcover : alk. paper) 1. Supersymmetry. 2. Generalized spaces. 3. Quantum groups. 4. Minkowski geometry. I. Lledó, María Antonia. II. Title. QC174.17.S9F56 2015 539.7'25--dc23 2015000733 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore LaiFun - The Minkowski and Conformal.indd 1 16/2/2015 4:00:36 PM February10,2015 10:42 BC:8972-TheMinkowskiandConformalSuperspaces minkconfws-corrected pagev Preface This monograph originated from the desire to describe, in a mathemati- cal rigorous way, the construction of the Minkowski and conformal super- spacesashomogeneousspacesforthePoincar´eandconformalsupergroups, respectively. Many authors, at different times, have contributed to the development of the modern theory of the Minkowski superspace since the pioneering works in supersymmetry [77,151,158]. Very little is novel in these notes. Our work has grown out from our papers [26,27,66] on this subject and we have drawn heavily, especially in the treatment of the infinitesimal setting, from the comprehensive and beautiful monograph by V. S. Varadarajan[147]. Ideally these notes are aimed at graduate students in mathematics and physics. We have made an effort to keep our treatment as elementary as possible andself contained, providingthe reader with a terse, but intuitive introduction to those advanced topics (like the theory of sheaves or the functor of points approach) which are necessary for our purposes. As for the physics part, we try to build the physical intuition from the beginning, withintroductionstospecialrelativity,quantummechanicsand even quantum field theory. This aims to describe the fundamentals, so one cannotexpect a treatmentof supersymmetricfield theories here. Even when our presentation greatly differs from the way in which supersymme- try is usually introduced by physicists, we try to make contact with their language. WeareverygratefultoProf. V.S.Varadarajanforhisconstanthelpand encouragement while this book was written. We have learnt this subject following his 2000 seminar at UCLA and we are deeply indebted to him. v February10,2015 10:42 BC:8972-TheMinkowskiandConformalSuperspaces minkconfws-corrected pagevi vi The Minkowski and Conformal Superspaces We also wish to thank Prof. A. Waldron, Prof. A. Schwarz and Dr. E. Latini for their enthusiasm for this work; without them this book would not have been written. We finally wish to thank Prof. L. Andrianopoli, Prof. A. Brini, Prof. C. Carmeli, Prof. A. Cattaneo, Prof. N. Ciccoli, Prof. F. Gavarini, Dr. S. D. Kwok, Prof. L. Migliorini and Prof. J. Navarro-Salas for helpful discussions andremarks. We also wantto thank the UCLA Departmentof Mathematics, the Dipartimento di Matematica, Universit`a di Bologna and theDepartamentdeF´ısicaTeo`rica,UniversitatdeVal`enciaforsupportand hospitality during the realization of this work. February10,2015 10:42 BC:8972-TheMinkowskiandConformalSuperspaces minkconfws-corrected pagevii Introduction In the sixties and seventies particle physics underwent a very fast develop- ment with many discoveries both, at the theoretical and the experimental level. The Standard Model, the model for electromagnetic and nuclear interactions, was taking its first steps. There are two different nuclear interactions: the weak interaction and the strong interaction. The weak interaction is responsible for the radioactive decays of nuclei of unstable isotopes of atoms by emission of particles and for the nuclear fusion, in which two nuclei colliding at high speed form a new nucleus. Nuclearfusionoftwohydrogennucleiintooneheliumnucleusistheprocess that fuels the Sun. With the works of Glashow, Weinberg and Salam, the electromagneticandtheweaknuclearinteractionswereunifiedintoasingle theory called the theory of the electroweak interactions. The strong nuclear interaction is the force that keeps together the pro- tons and neutrons (nucleons) in the nucleus in spite of the electric repul- sion of the positively chargedprotons. Not only nucleons, but other heavy particles, generically called hadrons, were being discovered, and it seemed that they could not all be fundamental particles. Instead, one could as- sume the existence of only three fundamental particles, called quarks, that would combine to produce all the different hadrons. The quarks are not observed directly and their interactions are described by a theory that, in its currentform, is called QuantumCromodynamics (QCD). QCD has not been unified with the electroweak theory, at least not in the way in which electromagnetic and weak interactions are unified, but they are both very similar at the mathematical level. They are non abelian Yang-Mills the- ories with gauge symmetry groups SU(2)×U(1) (electroweak) and SU(3) (QCD). The word ‘non abelian’ simply means that the gauge symmetry vii February10,2015 10:42 BC:8972-TheMinkowskiandConformalSuperspaces minkconfws-corrected pageviii viii The Minkowski and Conformal Superspaces groups are non abelian. Electroweaktheory and QCD form what is known as the Standard Model (SM) of particle interactions. It is worth to point out that quarks also undergo electroweak interactions. Thisleavesoutthefourthinteraction,gravity,describedbyacompletely different theory: Einstein’s General Relativity (GR). This is a theory in which the geometry of the spacetime becomes the dynamical variable, and itsmathematicaltreatmentgreatlydiffersfromtheSM.Nevertheless,since itincorporatesthespecialtheoryofrelativity(onesaysata‘locallevel’)one can think on the Poincar´e algebra (the Lorentz algebra plus translations) as an algebra of infinitesimal symmetries. In this state of things, physicists were looking for new constructions using bigger symmetry groups to help with unresolved problems. Nowa- days we know six different types of quarks: we say that quarks come in six different ‘flavors’, up (u), down (d), strange (s), charm (c), bottom (b) andtop(t), togetherwiththeir antiparticles,u¯,d¯,s¯,c¯,¯b,t¯. The firstquarks discovered were the up and down quarks. They have similar masses and there is an approximate SU(2) symmetry of the Standard Model in which (u,d) conform a doublet of SU(2), while (u¯,d¯) are in the dual or contra- gradient representation. This symmetry is called isospin: it is in fact an oldacquaintance,sinceitismanifestedinthesimilarmassesandbehaviour of the neutron (ddu) and the proton (uud), which also form a doublet of SU(2). Wigner was the first to propose, back in 1936, a theory that mixed the SU(2) isospin symmetry of nucleons together with the non relativistic SU(2)spin,aclassicremnantofthespinofthePoincar´egroup. Wignerpro- posedanSU(4)symmetrygroupthatwouldcontainbothSU(2)’s. Perhaps wishfully, or ahead of his time, Wigner called the SU(4) representations or multiplets of particles, supermultiplets. The isospin group SU(2) was enhanced to SU(3) with the prediction of the strange quark, s. Then, (u,d,s) would be a triplet of SU(3). Hadrons then could be organized in other representations of SU(3) which are ob- tained by tensoring the fundamental representation of SU(3) and its dual several times. This successful classification of the hadrons was known as theeightfold way,inreferencetothe8-dimensionaladjointrepresentationof SU(3). Itwasproposedindependently byGell-MannandNe’emanin1961. Soon after, Wigner’s supermultiplets were upgraded to representations of an SU(6) symmetry group which would contain the SU(2) spin group and the SU(3) flavor group [76,130]. The quarksu,d,s arethe lightquarks, being the c, b and t increasingly February10,2015 10:42 BC:8972-TheMinkowskiandConformalSuperspaces minkconfws-corrected pageix Introduction ix heavier and behaving too differently as to suppose that there is an even approximate symmetry among them. Although this theory explained some facts, it presented problems of interpretation regarding the spin-statistics connection. Nevertheless, this lineofresearchcametoanendin1967,withtheno-gotheoremofColeman and Mandula [33], which stated that, under reasonable assumptions, the spacetime symmetries (the Poincar´e group in this case) and the ‘internal’ symmetriessuchasflavororgaugesymmetries,couldonlyappeartogether combined as a direct product. This was a very strong result that also seemedto putatheoreticalbarrierforthe unificationofGeneralRelativity with the Standard Model. Gol’fand and Likhtman [77] were the first to use a new object to try to circumventtheColeman-Mandulatheorem. TheyproposedaLiesuperalge- bra, as opposedto a Lie algebra,as the algebraof infinitesimal symmetries of a physical theory. Lie superalgebras are vector spaces with a bilinear bracket that is symmetric or antisymmetric depending on the elements on which it acts. The Jacobi identity needs only a modification to work also on Lie superalgebras. Pioneering works on superalgebras applied to physical theories are Volkov and Akulov [151], Wess and Zumino [158,159], Ferrara and Zu- mino [49] and Salam and Strathdee [131,132] to mention just a few. This prompted a development of physics in a different direction and ultimatelyledtothediscoveryofsupersymmetrictheoriesandsupergravity. In supergravity, all of the interactions stem out of a geometric principle so they are all treated in the same way. The most obvious restriction that physics imposes on the suitable Lie superalgebras is that the elements or generators that have a symmetric bracket (or anticommutator) among them must be in a spinorial repre- sentation of the Lorentz algebra, so(1,3). These are representations that do not lift to representations of the Lorentz group SO(1,3) but of its spin group,SL(2,C)R,whichisitsdoublecoverand,consequently,hasthesame Lie algebra. A simpler example of this phenomenon is found with the group SO(3). It has the same Lie algebra as SU(2), its double (universal) cover,butthe doubletofSU(2)isnotarepresentationofSO(3). Then, the doublet is a spinorial representation of the orthogonal Lie algebra so(3). The restriction comes from the spin-statistics connection: ordinary (non spinorial) representations are associated to generators with antisymmetric bracketandspinorialrepresentationstogeneratorswithsymmetricbracket
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