Particles in the early universe : high-energy limit of the standard model from the contraction of its gauge group PDF
Preview Particles in the early universe : high-energy limit of the standard model from the contraction of its gauge group
PARTICLES IN THE EARLY UNIVERSE High-Energy Limit of the Standard Model from the Contraction of Its Gauge Group 11537_9789811209727_TP.indd 1 29/11/19 9:17 AM b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM PARTICLES IN THE EARLY UNIVERSE High-Energy Limit of the Standard Model from the Contraction of Its Gauge Group Nikolai A Gromov Russian Academy of Sciences, Russia World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 11537_9789811209727_TP.indd 2 29/11/19 9:17 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 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. PARTICLES IN THE EARLY UNIVERSE High-Energy Limit of the Standard Model from the Contraction of Its Gauge Group Copyright © 2020 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. ISBN 978-981-120-972-7 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11537#t=suppl Typeset by Stallion Press Email: [email protected] Printed in Singapore YongQi - 11537 - Particles in the Early Universe.indd 1 26-11-19 2:48:00 PM November29,2019 15:13 ParticlesintheEarlyUniverse–9inx6in b3760-fm pagev Preface Group-TheoreticalMethodsareanessentialpartofmoderntheoreticaland mathematical physics. The most advanced theory of fundamental inter- actions, namely Standard Model, is a gauge theory with gauge group SU(3)×SU(2)×U(1).Alltypesofclassicalgroupsofinfiniteseries:orthog- onal, unitary and symplectic as well as inhomogeneous groups, which are semidirectproductsoftheirsubgroups,areusedindifferentareasofphysics. Euclidean,Lobachevsky,Galilean,Lorentz,Poincar´e,(anti)deSittergroups arethebasesforspaceandspace-timesymmetries.Supergroupsandsuper- symmetric models in the field theory predict the existence of new super- symmetric partners of knownelementary particles. Quantumdeformations of Lie groupsandLie algebrasleadto non-commutativespace-timemodels (or kinematics). Contractions of Lie groups is the method for receiving new Lie groups from the initial ones. In the standard E. Wigner and E. Ino¨nu¨ approach, [In¨onu¨ and Wigner (1953)] continuous parameter (cid:2) is introduced in such a way that in the limit (cid:2) → 0 group operation is changed but Lie group structure and its dimension are conserved. In general, a contracted group is a semidirect product of its subgroups. In particular, a contraction of semisimple groups gives non-semisimple ones. Therefore, the contraction method is atoolforstudying non-semisimplegroupsstartingfromthe well known semisimple (or simple) Lie groups. The method of contractions (limit transitions) was extended later to other types of groups and algebras.Gradedcontractions [de Montigny and Patera (1991); Moody and Patera (1991)] additionally conserve grading of Lie algebra. Lie bialgebra contractions [Ballesteros, Gromov, Herranz, del Olmo and Santander (1995)] conserve both Lie algebra structure and cocommutator. Contractions of Hopf algebras (or quantum groups) are v November29,2019 15:13 ParticlesintheEarlyUniverse–9inx6in b3760-fm pagevi vi Particlesin the Early Universe introduced in such a way [Celeghini, Giachetti, Sorace and Tarlini (1990, 1992)] that in the limit (cid:2) → 0 new expressions for coproduct, counit and antipode appear which satisfies Hopf algebra axioms. All these give rise to the following generalization of the notion of group contraction on contrac- tion of algebraic structures [Gromov (2004)]. Definition 0.1. Contraction of algebraic structure (M,∗) is the map φ(cid:2) dependent on parameter (cid:2) φ(cid:2) :(M,∗)→(N,∗(cid:2)), (0.1) where(N,∗(cid:2))isanalgebraicstructureofthesametype,whichisisomorphic in (M,∗) when (cid:2)(cid:4)=0 and non-isomorphic when (cid:2)=0. There is another approach [Gromov (2012)] to the description of non- semisimpleLiegroups(algebras)basedontheirconsiderationoverPimenov algebraPn(ι) with nilpotent commutative generators.In this approachthe motion groups of constant curvature spaces (or Cayley–Klein groups) are realized as matrix groups of special form over Pn(ι) and can be obtained from the simple classical orthogonal group by substitution of its matrix elements for Pimenov algebraelements.It turns outthat such substitution coincides with the introduction of Wigner–Ino¨nu¨ contraction parameter (cid:2). So our approach demonstrates that the existence of the corresponding structures over algebra Pn(ι) is the mathematical base of the contraction method. It should be noted that both approaches supplement each other and in the final analysis give the same results. Nilpotent generators are more suitableinthemathematicalconsiderationofcontractionswhereasthecon- traction parameter continuously tending to zero corresponds more to the physical intuition where a physical system continuously changes its state and smoothly goes into its limit state. In chapter 5, devoted to the appli- cations of the contraction method to the modern theory of the elementary particles, the traditional approach is used. It is well known in geometry (see, for example, review [Yaglom, Rosen- feldandYasinskaya(1964)])thatthereare3ndifferentgeometriesofdimen- sion n, which admit the motion group of maximal order. R.I. Pimenov suggested [Pimenov (1959, 1965)] a unified axiomatic description of all 3n geometriesofconstantcurvature(orCayley–Kleingeometries)anddemon- strated that all these geometries can be locally simulated in some region of n-dimension sphericalspace with named coordinates,which can be real, imaginary and nilpotent ones. According to Erlanger program by F. Klein November29,2019 15:13 ParticlesintheEarlyUniverse–9inx6in b3760-fm pagevii Preface vii the main content in geometry is its motion group whereas the properties of transforming objects are secondary. The motion group of n-dimensional sphericalspaceisisomorphictotheorthogonalgroupSO(n+1).Thegroups obtained from SO(n+1) by contractions and analytical continuations are isomorphic to the motion groups of Cayley–Klein spaces. This correspon- dence provides the geometrical interpretation of Cayley–Klein contraction scheme.By analogythis interpretationis transferredto the contractionsof other algebraic structures. TheaimofthisbookistodevelopcontractionmethodforCayley–Klein orthogonaland unitary groups (algebras) and apply it to the investigation of physical structures. The contraction method apart from being of inter- est to group theory itself is of interest to theoretical physics too. If there is a group-theoretical description of a physical system, then the contrac- tion of its symmetry group corresponds to some limit case of the system underconsideration.Sothereformulationofthesystemdescriptioninterms of this method and the subsequent physical interpretations of contraction parameters give an opportunity to study different limit behaviours of the physical system. Examples of such an approach are given in chapter 2 for the space–time models, in chapter 3 for the Jordan–Schwinger representa- tions of groups which are closely connected to the properties of stationary quantumsystemswhoseHamiltoniansarequadraticincreationandannihi- lationoperators.Inchapter5themodifiedStandardModelwithcontracted gauge group is considered. The contraction parameter tending to zero is associated with the inverse average energy (temperature) of the Universe which makes it possible to re-establish the evolution of particles and their interactions in the early Universe. Itislikelythatthedevelopedformalismisessentialinconstructing“gen- eral theory of physical systems” where it will necessarily turn the group- invariant study of a single physical theory in Klein’s understanding (i.e. characterizedby symmetrygroup)toa simultaneousstudy ofa setoflimit theories. Then some physical and geometricalproperties will be the invari- ant properties of all sets of theories and they should be considered in the first place. Other properties will be relevant only for particular represen- tatives and will change under limit transition from one theory to another [Zaitsev (1974)]. In chapter 1, definitions of Cayley–Klein orthogonal and unitary groupsaregiven;their generators,commutatorsandCasimiroperatorsare obtained by transformations of the corresponding quantities of classical groups. November29,2019 15:13 ParticlesintheEarlyUniverse–9inx6in b3760-fm pageviii viii Particles inthe Early Universe In chapter 4, contractions of irreducible representations of unitary and orthogonal algebras in Gel’fand–Tsetlin basis [Gromov (1991, 1992)], which are especially convenient for applications in quantum physics, are considered. Possible contractions, which give different representations of contracted algebras, are found and general contractions leading to repre- sentations with nonzero eigenvalues of Casimir operators are studied in detail. For multi-parametric contractions, when a contracted algebra is a semidirect sum of a nilpotent radical and a semisimple subalgebra, our method ensures the construction of irreducible representations for such algebras starting from well known irreducible representations of classical algebras.Whenalgebrascontractedondifferentparametersareisomorphic, the irreducible representations in different bases (discrete and continuous) are obtained. The list of references does not pretend to be complete and reflects the author’s interests. N. A. Gromov November29,2019 15:13 ParticlesintheEarlyUniverse–9inx6in b3760-fm pageix Contents Preface v 1. The Cayley–Kleingroups and algebras 1 1.1 Dual numbers and the Pimenov algebra . . . . . . . . . . 1 1.1.1 Dual numbers . . . . . . . . . . . . . . . . . . . . 1 1.1.2 The Pimenov algebra . . . . . . . . . . . . . . . . 3 1.2 The Cayley–Klein orthogonalgroups and algebras . . . . 4 1.2.1 Three fundamental geometries on a line . . . . . . 4 1.2.2 Nine Cayley–Kleingroups . . . . . . . . . . . . . 8 1.2.3 Extension to higher dimensions . . . . . . . . . . 14 1.3 The Cayley–Klein unitary groups and algebras . . . . . . 17 1.3.1 Definitions, generators,commutators . . . . . . . 17 1.3.2 The unitary group SU(2;j1) . . . . . . . . . . . . 20 1.3.3 Representations of the group SU(2;j1) . . . . . . 22 1.3.4 The unitary group SU(3;j). . . . . . . . . . . . . 28 1.3.5 Invariant operators . . . . . . . . . . . . . . . . . 31 1.4 Classification of transitions between the Cayley–Klein spaces and groups . . . . . . . . . . . . . . . . . . . . . . 32 2. Space–time models 35 2.1 Kinematics groups . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Carroll kinematics . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Non-relativistic kinematics. . . . . . . . . . . . . . . . . . 43 3. The Jordan–Schwingerrepresentations of Cayley–Klein groups 47 3.1 The second quantization method and matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ix
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