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Lecture Notes in Physics Lecture Notes in Physics Volume 823 Founding Editors W. Beiglböck J. Ehlers K. Hepp H. Weidenmüller Editorial Board B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany M. Hjorth-Jensen, Oslo, Norway R. A. L. Jones, Sheffield, UK H. v. Löhneysen, Karlsruhe, Germany M. S. Longair, Cambridge, UK M. L. Mangano, Geneva, Switzerland J.-F. Pinton, Lyon, France J.-M. Raimond, Paris, France A. Rubio, Donostia, San Sebastian, Spain M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany For further volumes: http://www.springer.com/series/5304 The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opments in physics research and teaching—quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany [email protected] Giovanni Costa • Gianluigi Fogli Symmetries and Group Theory in Particle Physics An Introduction to Space-time and Internal Symmetries 123 Prof. Giovanni Costa Prof. Gianluigi Fogli Dipartmento di Fisica Dipartimento die Fisica Università di Padova Università die Bari Via Marzolo 8 Via Amendola 173 35131 Padova 70126 Bari Italy Italy ISSN 0075-8450 e-ISSN 1616-6361 ISBN 978-3-642-15481-2 e-ISBN 978-3-642-15482-9 DOI 10.1007/978-3-642-15482-9 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2011945259 Ó Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To our children Alessandra, Fabrizio and Maria Teresa Preface The aim in writing this book has been to give a survey of the main applica- tions of group and representation theory to particle physics. It provides the essential notions of relativistic invariance, space-time symmetries and inter- nal symmetries employed in the standard University courses of Relativistic Quantum Field Theory and Particle Physics. However, we point out that this is neither a book on these subjects, nor it is a book on group theory. Specifically, its main topics are, on one side, the analysis of the Lorentz and Poincar´egroups and, on the other side, the internal symmetries based mainly on unitary groups, which are the essential tools for the understanding of the interactions among elementary particles and for the construction of the present theories. At the same time, these topics give important and enlighten- ing examples of the essential role of group theory in particle physics. We have attempted to present a pedagogical survey of the matter, which should be useful to graduate students and researchers in particle physics; the only pre- requisite is some knowledge of classical field theory and relativistic quantum mechanics. In the Bibliography, we give a list of relevant texts and mono- graphs, in which the reader can find supplements and detailed discussions on the questions only partially treated in this book. One of the most powerful tools in dealing with invariance properties and symmetries is group theory. Chapter 1 consists in a brief introduction to group and representation theory; after giving the basic definitions and discussing the main general concepts, we concentrate on the properties of Lie groups and Lie algebras. It should be clear that we do not claim that it gives a self-contained account of the subject, but rather it represents a sort of glossary, to which the reader can refer to recall specific statements. Therefore, in general, we limit ourselves to define the main concepts and to state the relevant theorems without presenting their proofs, but illustrating their applications with specific examples. In particular, we describe the root and weight diagrams, which provide a useful insight in the analysis of the classical Lie groups and their representations; moreover, making use of the Dynkin diagrams, we present a classification of the classical semi-simple Lie algebras and Lie groups. VII VIII Preface The book is divided into two parts that, to large extent, are independent from one another. In the first part, we examine the invariance principles re- lated to the symmetries of the physical space-time manifold. Disregarding gravitation, we consider that the geometry of space-time is described by the Minkowski metric and that the inertial frames of reference of special relativ- ity are completely equivalent in the description of the physical phenomena. The co-ordinate transformations from one frame of reference to another form the so-called inhomogeneous Lorentz group or Poincar´egroup, which contains the space-time translations, besides the pure Lorentz transformations and the space rotations. The introductory and didactic nature of the book influenced the level of the treatment of the subject,for which we renounced to rigorousness and completeness, avoiding, whenever possible, unnecessary technicalities. In Chapter 2 we give a short account of the three-dimensional rotation group , not only for its important role in different areas of physics, but also as a specific illustration of group theoretical methods. In Chapter 3 we consider the main properties of the homogeneous Lorenz group and its Lie algebra. First, we examine the restricted Lorentz group, which is nothing else that the non-compact version SO(3, 1) of the rotation group in four dimensions. In particular we consider its finite dimensional irreducible representations: they are non unitary, since the group is non compact, but they are very useful, in particle physics, for the derivation of the relativistic equations. Chapter 4 is devoted to the Poincar´egroup, which is most suitable for a quantum mechan- ical description of particle states. Specifically, the transformation properties of one-particle and two-particle states are examined in detail in Chapter 5. In this connection, a covariant treatment of spin is presented and its physical meaning is discussed in both cases of massive and massless particles. In Chap- ter 6 we consider the transformation properties of the particle states under the discrete operations of parity and time reversal, which are contained in the homogeneous Lorentz group and which have important roles in particle physics. In Chapter 7, the relativistic wave functions are introduced in con- nection with one-particle states and the relative equations are examined for the lower spin cases, both for integer and half-integer values. In particular, we give a group-theoretical derivation of the Dirac equation and of the Maxwell equations. The second part of the book is devoted to the various kinds of internal symmetries, which were introduced during the extraordinary development of particle physics in the second half of last century and which had a fundamental role in the construction of the present theories. A key ingredient was the use of the unitary groups, which is the subject of Chapter 8. In order to illustrate clearly this point, we give a historical overview of the different steps of the process that lead to the discovery of elementary particles and of the properties of fundamental interactions.
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