Applications of the Theory of Groups in Mechanics and Physics

Fundamental Theories of Physics

An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor: ALWYN VAN DER MERWE, University of Denver; U.S.A.

Editorial Advisory Board: GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D .. JOSEPHSON, University of Cambridge, u.K. CLIVE KILMISTER, University of London, u.K. PEKKA J. LAHTI, University ofTurku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada FRANCO SELLERI, Universita di Bara, Italy TONY SUDBURY, University of York, u.K. HANS-JURGEN TREDER, Zentralinstitut for Astrophysik der Akademie der

Wissenschaften, Germany

Volume 140

Applications of the Theory of Groups in Mechanics and Physics

by

Petre P. Teodorescu Faculty of Mathematics, University of Bucharest, Bucharest, Romania

and

Nicolae-Alexandru P. Nicorovici School of Physics, The University of Sydney, New South Wales, Australia

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6581-0 ISBN 978-1-4020-2047-6 (eBook) DOI 10.1007/978-1-4020-2047-6

Printed on acid-free paper

All Rights Reserved © 2004 Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2004 Softcover reprint of the hardcover 1 st edition 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exc1usive use by the purchaser of the work.

CONTENTS

PREFACE

INTRODUCTION

1. ELEMENTS OF GENERAL THEORY OF GROUPS

1 Basic notions

2

3

1.1 Introduction of the notion of group 1.2 1.3 1.4

Basic definitions and theorems Representations of groups

The S3 group

Topological groups 2.1 Definitions. Generalities. Lie groups

2.2 Lie algebras. Unitary representations

Particular Abelian groups

3.1 The group of real numbers

3.2

3.3

The group of discrete translations

The SO(2) and en groups

2. LIE GROUPS

The SO(3) group 1.1 Rotations 1.2 Parametrization of SO(3) and 0(3) 1.3 Functions defined on 0(3). Infinitesimal generators

2 The SU(2) group 2.1 Parametrization of SU(2) 2.2 Functions defined on SU(2). Infinitesimal generators

3 The SU(3) and GL(n, q groups 3.1 SU(3) Lie algebra

v

IX

xi

4 19 33

40 40 50

52

52 54

58

61

61 61 70 74 76 76 82

89 89

VI APPLICATIONS OF THE THEORY OF GROUPS

3.2 Infinitesimal generators. Parametrization of SU(3) 102 3.3 The GL(n, q and SU(n) groups 107

4 The Lorentz group 111 4.1 Lorentz transformations 111 4.2 Parametrization and infinitesimal generators 118

3. SYMMETRY GROUPS OF DIFFERENTIAL EQUATIONS 123

1 Differential operators 123 1.1 The SO(3) and SO(n) groups 123 1.2 The SU(2) and SU(3) groups 127

2 Invariants and differential equations 133 2.1 Preliminary considerations 133 2.2 Invariant differential operators 138

3 Symmetry groups of certain differential equations 149 3.1 Central functions. Characters 149 3.2 The SO(3), SU(2), and SU(3) groups 151 3.3 Direct products of irreducible representations 159

4 Methods of study of certain differential equations 176 4.1 Ordinary differential equations 176 4.2 The linear equivalence method 177 4.3 Partial differential equations 186

4. APPLICATIONS IN MECHANICS 201

1 Classical models of mechanics 201 1.1 Lagrangian formulation of classical mechanics 201 1.2 Hamiltonian formulation of classical mechanics 206 1.3 Invariance of the Lagrange and Hamilton equations 213 1.4 Noether's theorem and its reciprocal 220

2 Symmetry laws and applications 230

2.1 Lie groups with one parameter and with m parameters 230 2.2 The Symplectic and Euclidean groups 235

3 Space-time symmetries. Conservation laws 244 3.1 Particular groups. Noether's theorem 244 3.2 The reciprocal of Noether's theorem 251 3.3 The Hamilton-Jacobi equation for a free particle 259

4 Applications in the theory of vibrations 263 4.1 General considerations 263 4.2 Transformations of normal coordinates 265

CONTENTS vii

5. APPLICATIONS IN THE THEORY OF RELATIVITY AND THEORY OF CLASSICAL FIELDS 279 1

2

3

Theory of Special Relativity 1.1 Preliminary considerations 1.2 Applications in the theory of Special Relativity

Theory of electromagnetic field 2.1 Noether's theorem for the electromagnetic field 2.2 Conformal transformations in four dimensions

Theory of gravitational field 3.1 General equations 3.2 Conservation laws in the Riemann space

6. APPLICATIONS IN QUANTUM MECHANICS AND PHYSICS OF ELEMENTARY PARTICLES

1 Non-relativistic quantum mechanics

2

3

1.1 Invariance properties of quantum systems 1.2 The angular momentum. The spin

Internal symmetries of elementary particles 2.1 The isospin and the SU(2) group 2.2 The unitary spin and the SU(3) group

Relativistic quantum mechanics 3.1 Basic equations. Symmetry groups 3.2 Elementary particle interactions

REFERENCES

INDEX

279 279 289 302 302 317 324 324 329

335 335 335 358 371 371 383 396 396 407

423

431

PREFACE

The notion of group is fundamental in our days, not only in mathematics, but also in classical mechanics, electromagnetism, theory of relativity, quantum mechanics, theory of elementary particles, etc. This notion has developed during a century and this development is connected with the names of great mathematicians as E. Galois, A. L. Cauchy, C. F. Gauss, W. R. Hamilton, C. Jordan, S. Lie, E. Cartan, H. Weyl, E. Wigner, and of many others. In mathematics, as in other sciences, the simple and fertile ideas make their way with difficulty and slowly; however, this long history would have been of a minor interest, had the notion of group remained connected only with rather restricted domains of mathematics, those in which it occurred at the beginning. But at present, groups have invaded almost all mathematical disciplines, mechanics, the largest part of physics, of chemistry, etc. We may say, without exaggeration, that this is the most important idea that occurred in mathematics since the invention of infinitesimal calculus; indeed, the notion of group expresses, in a precise and operational form, the vague and universal ideas of regularity and symmetry.

The notion of group led to a profound understanding of the character of the laws which govern natural phenomena, permitting to formulate new laws, correcting certain inadequate formulations and providing unitary and non­contradictory formulations for the investigated phenomena. In this way, unitary methods are obtained for solving problems of different physical nature but hav­ing common mathematical features; the investigation of these problems in a unitary way permits a better understanding of the special phenomena under consideration.

The present volume is a new edition of a volume published in 1985 ("Aplicatii ale teoriei grupurilor in mecanidi ~i fizidi", Editura Tehnidi, Bucharest, Ro­mania). This new edition contains many improvements concerning the presen­tation, as well as new topics using an enlarged and updated bibliography. We outline the manner in which the group theory can be applied to the solution and

ix

x APPLICATIONS OF THE THEORY OF GROUPS

systematization of several problems in the theory of differential equations, in mechanics, and physics. Having in view the great number of published works (as it can be seen in the bibliography), we wish to give the reader a number of preliminary notions and examples which are absolutely necessary for a better understanding of certain works, in different domains, which are based on ap­plications of group theory. Since they are of special importance, being at the same time more accessible, we have considered in detail only the Lie groups, the Lie algebras associated with these groups, and their linear representations.

The work requires as preliminaries only the mathematical knowledge ac­quired by a student in a technical university. It is addressed to a large audience, to all those interested and compelled to use mathematical methods in various fields of research, like: mechanics, physics, engineering, people involved in research or teaching, as well as students.

P. P. TEODORESCU AND N.-A. P. NICOROVICI

INTRODUCTION

To study a natural phenomenon by means of the methods of group theory, we adopt the following methodology: We consider a phenomenon F (mechanical or physical), modelled by a system of differential equations S. If the system S possesses certain symmetry properties, then there exist certain groups of trans­formations of coordinates, which act upon a canonical Hilbert space associated with S (generally, the space of solutions). The representations, the invariants, and covariants of these groups have certain significances directly connected with the properties of S, which can be translated as properties of the phenomenon F (as a rule, conservation laws). In this way, the properties of the phenomenon F can be enounced and analysed by means of the results of group theory.

We show how the above methodology can be applied to the Lagrangian and Hamiltonian differential systems, to the Hamilton-Jacobi equation and the La­grange equations for vibrations (in classical mechanics), to the Maxwell equa­tions (in electromagnetism), to the Schmdinger equation (in non-relativistic quantum mechanics), and to the Klein-Gordon and the Dirac equations (in rela­tivistic quantum mechanics). Several symmetry properties ofthese differential systems are analysed by means of certain local analytical groups, namely: the Symplectic group, the Euclidean group and the non-Euclidean group, the or­thogonal group, the unitary group, and the special unitary group. The physical expression of the invariance under group actions of physical entities (for in­stance, linear momentum, angular momentum, electromagnetic or gravitational field, etc.) is, usually, a conservation law. In particular, the connection between the invariance of the Lagrangian of a physical system under a transformation group and the corresponding conservation laws is expressed by Noether's the­orem.

As an example, a mechanical system evolves in the Euclidean space which, in classical mechanics, is assumed to be isotropic and homogeneous, that is invariant with respect to space translations and space rotations, respectively. We may also add the invariance with respect to time translations. In other

Xl

xii APPUCATIONS OF THE THEORY OF GROUPS

words, we assume that there is no preferred point in space such that we may set the origin of the coordinates anywhere, there is no preferred direction, therefore the orientation of axes is arbitrary, and we may choose any moment as the origin of time. These invariance properties dictate the analytic form of the Lagrangian, or the Hamiltonian, that are the basic ingredients for any axiomatic theory. In turn, the symmetry properties of the Lagrangian lead to conservation laws; in our case, the conservation of momentum, angular momentum, and energy. Hence, the most common conservation laws in classical mechanics are actually consequences of the geometrical properties of space-time.

Another example comes out from the use of group theory to classify the elementary particles. Thus, the spin quantum number associated to elementary particles has found a mathematical formalization in the theory of the SU(2) group. By the aid of the same group, Werner Heisenberg (Nobel Prize 1932) in­troduced the isospin quantum number for elementary particles. In 1961, Murray Gell-Mann (Nobel Prize 1969) and Yuval Ne'eman suggested, independently of each other, a way of classifying the hadrons (elementary particles that take part in strong interactions) known at that time; this classification is based on the SU(3) group. The most important confirmation of the profoundness of this idea was obtained in 1962 by the experimental discovery of the hadrons ~ and ~, theoretically predicted by this model. During the same year, the doublet S was also discovered experimentally. Finally, in 1963, the particle n- was found, thus completing a multiplet of 10 hadrons (decuplet) predicted theoretically by the SU(3) classification. This was the proof that the SU(3) symmetry is deeply rooted in the world of hadrons, although a symmetry model does not constitute a complete theory. Nevertheless, by the aid of the SU(3) classification several regularities were discovered concerning the properties of hadrons, like in the case of the Mendeleev table for chemical elements.

In an attempt to model the properties of hadrons, George Zweig and Mur­ray Gell-Mann introduced in 1964 the quark model, based also on the SU(3) symmetry group (now called SU(3) flavour symmetry group). In the original version of this model it is supposed that the hadrons are composed of sev­eral combinations of three fundamental particles and three antiparticles, called quarks and antiquarks, respectively. The most revolutionary aspect of the quark model consists in the fact that these fundamental particles possess fractionary electric charges, unlike the elementary particles whose electric charges are mul­tiples of the electron charge. However, the quark model was used with great success to predict static properties ofhadrons. To avoid some difficulties related to the ground states of hadrons, the quark model was changed by adding the colour quantum number (quantum chromodynamics theory, or QeD, based on the colour symmetry group SU(3)c). Later, in 1974, the experimental discovery of the JhjJ particle required a new quark (and a new quantum number charm), in 1977 the discovery of the Y (Upsilon) particle necessitated another quark

INTRODUCTION xiii

(bottom) with a new quantum number attribute, and, in 1995, the sixth quark (top) has been discovered at Fermilab. Accordingly, higher symmetry groups have been introduced: SU(3)xSU(3), SU(4), SU(6) etc. Even if the free quarks have never been observed, the actual existence of quarks was indirectly demonstrated in 1981 by the experimental discovery of the gluons (particles that ensure the binding between the quarks which form a hadron).

In the domain of weak interactions, the introduction of group theory was entailed by the discovery of the parity violation in the {3 decay (Tsung Dao Lee and Chen Ning Yang, Nobel Prize 1956), which corresponds to the non­invariance of this process under the discrete group of spatial inversions. In 1967, Steven Weinberg and, independently, some months later, Abdus Salam, introduced a unified theory of electromagnetic and weak interactions, a theory based on an elegant mathematical formalism: the U(l)xSU(2) symmetry (No­bel Prize 1979, together with Sheldon Lee Glashow). The outstanding success of this theory was the experimental discovery of the particles w± (1982) and ZO (1983), predicted theoretically.

All these examples show the importance of the theory of groups as a mathe­matical method in modelling of natural phenomena. Galois invented the group theory and applied it to algebraic equations. It is the merit of Lie who developed a theory based on the assumption that the symmetries are the primary features of natural phenomena, and they dictate the form of dynarnicallaws. Lie applied his theory to dynamical systems modelled by systems of ordinary differential equations, and created the symmetry analysis of ordinary differential equations. There are many extensions of Lie's theory due to different authors, but the most important is the group analysis of partial differential equations, developed by Ovsiannikov. From all these contributions and successive extensions of the original theory, we have today a very powerful mathematical tool which allows the investigation of phenomena pertaining to different domains of physics in a unitary way.

Finally, some comments about notations. The set of all n x n complex matrices M (n) form a Lie algebra with respect to the internal composition law

[a,bJ=ab-ba, a,bEM(n).

The corresponding Lie group is the general linear group in n dimensions, de­noted by GL(n, q, and defined as the set of linear transformations acting in the n-dimensional linear complex space en. To each element in GL(n, q, we may associate a matrix g such that we may define the group GL(n, q in terms of its matrix representation

GL(n, q = {g I g E M(n), det(g) -I- O},

the internal composition law being the product of matrices. Many applications of the group theory in physics involve subgroups of GL( n, q, that are:

XIV APPLICATIONS OF THE THEORY OF GROUPS

• the special (unimodular) linear group

SL(n,q = {g I g E GL(n,q, det(g) = I},

• the unitary group

U(n) = {g I g E GL(n,q, gtg = e},

where the symbol t denotes the Hermitian conjugation, and e is the identity matrix,

• the special unitary group

SU(n) = {g I g E SL(n,q, gtg = e} = U(n) nSL(n,q,

• the linear real group

GL(n,JR) = {g I g E GL(n,q, Img = O},

where I m means the imaginary part of the argument,

• the special linear real group

SL(n, JR) = {g I g E GL(n, JR), det (g) = I},

• the orthogonal group

O(n) = {g I g E GL(n,JR), gTg = e} = U(n) n GL(n,JR),

where the superscript T denotes a transposed matrix,

• the special orthogonal group

SO(n) = {g I g E SL(n,JR), gTg = e} = O(n) n SL(n,JR).

Other groups, like the Symmetric group Sn" the Symplectic group Sp(2n, q, the Euclidean group E(n), etc., will be defined in the corresponding chapters.

Note that there are many different notations for some groups (for instance O+(n) instead ofSO(n), but in this book we used the notations which seems to match better the scheme from above. We also have to point out that, generally we used Einstein's summation convention for dummy indices. However, at some places we explicitly mentioned the ranges for the dummy indices to avoid any confusion.

In the examples of applications of group theory we used the fully integrated environment for technical computing MathematiccfY developed at Wolfram Re­search. Some of the symbolic calculations have been carried out by means of the Mathematica® packages MathLie™ by G. Baumann, and YaLie by J. M. Diaz.