I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II With his Theory of Relativity, Albert Einstein in 1905 put an end to all mechanical ether interpretations of electromagnetic phenomena, such as the ether model shown above. In it, the magnetic field was imagined as a system of molecular vortices rotat ing around the field lines, with 'ball bearings' between vortices consisting of charge particles. The velocity of rotation is to be proportional to the field strength, and when neighboring vortices rotate with differing velocities, the charge particles get displaced. This model was the basis for the derivation of the Maxwell equations. "1 never satisfy myself unless I can make a mechanical model of a thing ... that is whv 1 cannot get the electromagnetic theory ... " (Lord Kelvin, 1884). Roman U. Sexl Helmuth K. Ur bantke Relativity, Groups, Particles Special Relativity and Relativistic Symmetry in Field and Particle Physics Revised and translated from the German by H. K. Ur bantke Springer-Verlag Wien GmbH Dr. Roman U. Sexl t Dr. Helmuth K. Urhantke Institut fUr Theoretische Physik Universitat Wien, Vienna, Austria This edition succeeds the third, revised Gennan-Ianguage edition, Relativitiit, Gruppen, Teilchen, This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photo-copying machines or similar means, and storage in data banks. © 2001 Springer-Verlag Wien Originally published by Springer-Verlag Wien New York in 2001 Camera-ready copies provided by the author Printed on acid-free and chlorine-free bleached paper SPIN 10756865 With 56 figures and a frontispiece Library of Congress Cataloging-in-Publication Data Sexl, Roman Ulrich. [Relativitat, Gruppen, Teilchen. English] Relativity, groups, particles: special relativity and relativistic symmetry in field and particle physics / Roman U. Sexl, Helmuth K. Urbantke; revised and translated from the German by Helmuth K. Urbantke. - Rev. ed. p. cm. This edition succeeds the third, revised German-language edition, Relativitat, Gruppen, Teilchen, cl992 Springer-Verlag Wien -T.p.verso. Includes bibliographical references and index. ISBN 978-3-211-83443-5 (alk. paper) 1. Relativity (Physics) 2. Field theory (Physics) 3. Representations of groups. 4. Particles (Nuclear physics) I. Urbantke, Helmuth Kurt. II. Title. QCI73.65.S48132000 530.11 - dc21 00-063782 ISBN 978-3-211-83443-5 ISBN 978-3-7091-6234-7 (eBook) DOI 10.1007/978-3-7091-6234-7 Preface and Introduction Like many textbooks, the present one is the outgrowth of lecture courses, mainly given at the University of Vienna, Austria; on the occasion of the English edition, it may be mentioned that our first such lecture course was delivered by my late co author, Roman U. Sexl, during the fall and winter term 1967-68 in the USA-more precisely, at the University of Georgia (Athens). Since then, Particle Physics has seen spectacular revolutions; but its relativistic symmetry has never been shaken. On the other hand, new technological developments have enabled applications like the GPS (Global Positioning System) that, in a sense, brought Relativity to the domain of everyday use. The purpose of the lecture courses, and thus of the book, is to fill a gap that the authors feel exists between the way Relativity is presented in introductory courses on mechanics and/or electrodynamics on the one hand and the way relativistic symmetry is presented in particle physics and field theory courses on the other. The reason for the gap is a natural one: too many other themes have to be addressed in the introductory courses, and too many applications are impatiently waiting for their presentation in the particle and field theory courses. In this text we try to bridge this gap, and guide the reader (him and her, we hope) to more abstract points of view concerning space-time geometry and symmetry wher ever they are useful. At the same time, the reader is introduced to the world of groups and their realizations, particularly Lie groups and Lie algebras. Much of this material could have been omitted given a severe restriction to the groups actually to be dealt with, but a slight broadening was intentional. However, we stress that we certainly do not see the need of entering the realm of the simple Lie algebras of rank greater than one, which would be necessary for the discussion of the inner symmetries of par ticle physics. Naturally, mathematical developments tend to occupy a large amount of space here, but we hope that the gradual transition from the explicit component matrix format to the more abstract version of linear algebra will, in the end, work against loss of sight of the basic concepts. Motivation and heuristic considerations are in the foreground, and our presentation will essentially remain at the heuristic level whenever functional analysis would be needed to cope with the infinite-dimensional spaces that occur. Also, the precise definition of manifolds is not given, although we try to give the reader at least a vague impression of group manifolds, covering spaces, fiber bundles, etc., since these objects are there and should be named for ease of addressing. Moreover, all these concepts pervade modern theoretical physics in many other places. For their precise definition, the reader is referred to suitable mathematical textbooks, some of which we quote. However, basic group theory and abstract (multi)linear algebra are summarized in two of the appendices. At this point, we may list things the reader should be acquainted with. On the mathematical side, these include linear algebra (first only in three but later in arbi- VI Preface and Introduction trary dimensions), multivariable calculus, and a rudimentary knowledge of the Dirac delta function; the basic definitions from group theory are useful to be known already as well. On the physics side, they include the basic concepts of theoretical mechanics, electrodynamics, and quantum theory (on a level that assumes multi variable calcu lus); thus, e.g., small parts of the well-known books by Goldstein, Jackson, and Schiff will suffice. Enough experimental background is assumed, so that our only very occa sional mention of experiments suffices to assure the reader that we are indeed talking about physics rather than pure mathematics. Throughout this book, particularly so in its first half, we have interpolated his torical remarks: if short enough, they appear in small print paragraphs interspersed in the main text; if longer, they take the form of whole sections (namely, sects. 1.6 and 2.11, written together with R. Mansouri, now at Sharif University of Technology, Teheran, who also contributed to sect. 10.3). Similarly, mathematical asides of inter est or of relevance in later sections may appear in small print paragraphs. These may be omitted on a first reading of the section they appear in, but must sometimes be (re)turned to on studying later sections. (In other words, there is no strict separation in the book enabling a "track one" and a "track two" reading.) In any case, they are hoped to whet the reader's appetite and to allow looking at some of the features of Relativity from a "higher" point of view. The table of contents gives a general overview of our subject matter, so here we make only a few general remarks on how the development proceeds. Chapter 1 gives a "derivation" of the Lorentz transformation starting from the usual "axioms" (which are not to be understood in the sense of logicians). The role of group structure should already be apparent in this stage, even if that term is introduced only later. The role of the rotation group of Euclidean 3-space is very much in the foreground here, which is perhaps somewhat unusual. Chapter 2 discusses standard elementary consequences of the Lorentz transformation, including Thomas rotation. The sections on superluminal phenomena and non-Einstein synchronized reference frames may appear somewhat outside the canonical textbook content. Chapters 3, 4, and 5 are standard, but the latter includes, in a semi-historical section, the history of 'classical electron theory' and the role played by relativistic covariance in the later developments of that theory. With chapter 6, we enter the group-representation part of the book, and a reader who knows standard relativistic mechanics and electrodynamics might well begin with this chapter, perhaps first reading sections 1.5, 2.9, 2.10 and the introduction to chapter 3. Chapter 6 includes an investigation of the structure of the Lorentz group (its quasidirect product structure in particular, since that is closely related to our initial derivation of the group) as well as the basic definitions and theorems from the theory of representations. All of these are well-illustrated with reference to material in previous chapters. Chapter 7 is preparatory to chapters 8 and 9; in particular, section 7.10 on mul tivalued representations may be helpful to some readers. In chapter 8 on the finite dimensional representations of the Lorentz group, we hope we have made clear the often-confused role played by the use of complex numbers in this context; we explain complex structure, real structure, complexification, realification and the job they do for us here. Preface and Introduction Vll Chapter 9 first discusses the representation theoretic aspect of covariant wave equations; after a general discussion of relativistic symmetry in quantum mechanics, it then introduces the well-known Wigner classification. The mention of helicity as a 'topological quantum number' is perhaps not frequently encountered in other texts. Chapter 10, on conservation laws associated with relativistic space-time symme try, can be read almost independently of the preceding ones. Section 10.3 shows an application of a phenomenologically constructed energy-momentum tensor. We have already commented on two of the appendices (A and B); Appendix C continues an already quite lengthy appendix to section 9.1 on Dirac spinors: both are intended to encourage an essentially basis-free attitude towards the 'gamma' matrices, such as would be required when going to the curved space-time of General Relativity. Appendix D tries to give a modest introduction to relativistic covariance in Quantum Field Theory. There are exercises to most sections; in the later chapters, many of them ask the reader to provide proofs, following given hints, for theorems of a general nature that were quoted and applied in the main text. Essentially, these exercises intend to further the reader's intuition about linear spaces. Thanks are due to many persons who contributed in one way or another to the previous (German) editions: their names are listed there. Added here to that list must be my colleague Helmut Kiihnelt, who tried (essentially in vain) to educate me in I5.IE;X and, in any case, helped me, as also did Ulrich Kiermayr, to overcome many difficulties. Of course, the responsibility for any imperfections in typesetting, as well as for infelicities of language and content, is entirely with me. Every new edition gives opportunity not only to eliminate mistakes in the previous one but also to create new ones. At least, a reasonable balance is hoped for. I will be grateful to anybody bringing mistakes and ambiguous or cryptic formulations to my attention, which in our electronic age should be easy using [email protected]; I plan to make the collection of corrections so obtained available via link on the homepage of my institution, http://www.thp.univie.ac.at/. in due time, so that even readers of this present edition may profit from such activity. Our big hope is that the present edition contribute to an increase of joy in physics by widening more people's scope for "seeing" symmetry in nature! Naturally, this edition is dedicated to the memory of my former co-author, teacher and friend, ROMAN ULRICH SEXL whose untimely and tragic death, now 14 years ago, prevents him from greeting the new millennium. Vienna, August 2000 Helmuth K. Urbantke Contents 1 The Lorentz Transformation 1 1.1 Inertial Systems . . . . . . . . . . . . . . . . . 1 1.2 The Principle of Relativity .......... . 3 1.3 Consequences from the Principle of Relativity 4 Appendix 1: Reciprocity of Velocities ..... 7 Appendix 2: Some Orthogonal Concomitants of Vectors 7 1.4 Invariance of the Speed of Light. Lorentz Transformation 8 1.5 The Line Element . . . . . . . . . . . . 10 1.6 Michelson, Lorentz, Poincare, Einstein 13 2 Physical Interpretation 19 2.1 Geometric Representation of Lorentz Transformations. 19 2.2 Relativity of Simultaneity. Causality 21 2.3 Faster than Light . . . . . . . . . . . . . . . . . . . . . 24 2.4 Lorentz Contraction ................... 28 2.5 Retardation Effects: Invisibility of Length Contraction and Apparent Superluminal Speeds 29 2.6 Proper Time and Time Dilation . . . . . . . . 32 2.7 The Clock or Twin Paradox . . . . . . . . . . 34 2.8 On the Influence of Acceleration upon Clocks 37 2.9 Addition of Velocities. . . 38 2.10 Thomas Precession . . . . 40 2.11 On Clock Synchronization 43 3 Lorentz Group, Poincare Group, and Minkowski Geometry 49 3.1 Lorentz Group and Poincare Group. . . . . . 50 3.2 Minkowski Space. Four-Vectors .......... 52 3.3 Passive and Active Transformations. Reversals . . 57 3.4 Contravariant and Covariant Components. Fields 59 4 Relativistic Mechanics 63 4.1 Kinematics ..................... . 63 Appendix: Geometry of Relativistic Velocity Space 66 4.2 Collision Laws. Relativistic Mass Increase 67 4.3 Photons: Doppler Effect and Compton Effect . 70 4.4 Conversion of Mass into Energy. Mass Defect . 75 4.5 Relativistic Phase Space . . . 78 Appendix: Invariance of R.,(q) ........ . 83 x Contents 5 Relativistic Electrodynamics 85 5.1 Forces .......... 85 5.2 Covariant Maxwell Equations 86 5.3 Lorentz Force . . . . . . . . . 91 5.4 Tensor Algebra ....... 92 5.5 Invariant Tensors, Metric Tensor. 95 5.6 Tensor Fields and Tensor Analysis. 102 5.7 The Full System of Maxwell Equations. Charge Conservation. 105 5.8 Discussion of the Transformation Properties . . . . . 108 5.9 Conservation Laws. Stress-Energy-Momentum Tensor 115 5.10 Charged Particles. . . . . . . . . . . . . . . . . . . . 122 6 The Lorentz Group and Some of Its Representations 134 6.1 The Lorentz Group as a Lie Group . . . . . 134 6.2 The Lorentz Group as a Quasidirect Product. 139 6.3 Some Subgroups of the Lorentz Group ... 143 Appendix 1: Active Lorentz Transformations . 145 .ct . Appendix 2: Simplicity of the Lorentz Group 146 6.4 Some Representations of the Lorentz Group 148 6.5 Direct Sums and Irreducible Representations 153 6.6 Schur's Lemma ............... 159 7 Representation Theory of the Rotation Group 169 7.1 The Rotation Group SO(3,R) . . . . . . . 170 7.2 Infinitesimal Transformations ... . . . . 173 7.3 Lie Algebra and Representations of SO(3) 176 7.4 Lie Algebras of Lie Groups. . . . . . . . . 179 7.5 Unitary Irreducible Representations of SO(3) . 183 7.6 SU(2), Spinors, and Representation of Finite Rotations 195 7.7 Representations on Function Spaces 206 7.8 Description of Particles with Spin . . . . . 212 7.9 The Full Orthogonal Group 0(3) ..... 218 7.10 On Multivalued and Ray Representations. 224 8 Representation Theory of the Lorentz Group 229 .ct 8.1 Lie Algebra and Representations of 229 8.2 The Spinor Representation . . . . . . . . . . . 236 8.3 Spinor Algebra ................ 242 Appendix: Determination of the Lower Clebsch-Gordan Terms 246 8.4 The Relation between Spinors and Tensors . .. ....... 247 Appendix 1: Spinors and Lightlike 4-Vectors ..... .. 252 Appendix 2: Intrinsic Classification of Lorentz Transformations. 253 8.5 Representations of the Full Lorentz Group . . . . . . . . . 255 Contents xi 9 Representation Theory of the Poincare Group 261 9.1 Fields and Field Equations. Dirac Equation .. 261 Appendix: Dirac Spinors and Clifford-Dirac Algebra. 265 9.2 Relativistic Covariance in Quantum Mechanics . . . . 271 9.3 Lie Algebra and Invariants of the Poincare Group .. 278 904 Irreducible Unitary Representations of the Poincare Group 285 Pt 9.5 Representation Theory of and Local Field Equations 299 9.6 Irreducible Semiunitary Ray Representations of P 313 10 Conservation Laws in Relativistic Field Theory 317 10.1 Action Principle and Noether's Theorem .... 318 10.2 Application to Poincare-Covariant Field Theory 323 10.3 Relativistic Hydrodynamics .......... . 331 Appendices 336 A Basic Concepts from Group Theory 336 A.1 Definition of Groups ........ . 336 A.2 Subgroups and Factor Groups ... . 336 A.3 Homomorphisms, Extensions, Products 337 Ao4 Transformation Groups . . . . . . . . . 339 B Abstract Multilinear Algebra 340 B.1 Semilinear Maps .... . 340 B.2 Dual Space ........ . 341 B.3 Complex-Conjugate Space . 341 Bo4 Transposition, Complex, and Hermitian Conjugation 342 B.5 Bi- and Sesquilinear Forms .. 342 B.6 Real and Complex Structures 343 B.7 Direct Sums . . . 344 B.B Tensor Products ....... . 344 B.9 Complexification ...... . 345 B.1O The Tensor Algebra over a Vector Space 346 B.ll Symmetric and Exterior Algebra ..... 347 B.12 Inner Product. Creation and Annihilation Operators 349 B.13 Duality in Exterior Algebra ........ . 350 B.14 Q-Geometries and Quantities of Type (9, u) ..... 353 C Majorana Spinors, Charge Conjugation, and Time Reversal in Dirac Theory 357 C.1 Dirac Algebra Reconsidered .............. . 357 C.2 Majorana Spinors, Charge Conjugation, Time Reversal 359