Table Of ContentGRADUATE STUDENT SERIES IN PHYSICS
Series Editor:
Professor Douglas F Brewer, MA, DPhil
Emeritus Professor of Experimental Physics, University of Sussex
SYMMETRIES IN
QUANTUM MECHANICS
FROM ANGULAR MOMENTUM TO
SUPERS YMMETRY
MASUD CHAICHIAN
Department of Physics, University of Helsinki
and
Helsinki Institute of Physics
ROLF HAGEDORN
Retired Staff Member of Theory Division, CERN, Geneva
INSTITUTE OF PHYSICS PUBLISHING
Bristol and Philadelphia
CONTENTS
Preface xiii
1 Introduction 1
1.1 Notation 1
1.2 Some basic concepts in quantum mechanics 4
1.3 Some basic objects of group theory 7
1.3.1 (cid:9) Groups: finite, infinite, continuous, Abelian, non-
Abelian; subgroup of a group, cosets 7
1.3.2 (cid:9) Isomorphism, automorphism, homomorphism 8
1.3.3 (cid:9) Lie groups and Lie algebras 9
1.3.4 (cid:9) Representations: faithful, irreducible, reducible,
completely reducible (decomposable), indecomposable,
adjoint, fundamental 10
1.3.5 (cid:9) Relation between Lie algebras and Lie groups, Casimir
operators, rank of a group 11
1.3.6 (cid:9) Schur's lemmas 12
1.3.7 (cid:9) Semidirect sum of Lie algebras and semidirect product
of Lie groups (inhomogeneous Lie algebras and groups) 12
1.3.8 (cid:9) The Haar measure 13
1.4 Remark about the introduction of angular momentum 14
2 Symmetry in Quantum Mechanics 16
2.1 Definition of symmetry 16
2.1.1 (cid:9) General considerations 16
2.1.2 (cid:9) Formal definition of symmetry; ray correspondence 20
2.2 Wigner's theorem: the existence of unitary or anti-unitary
representations 21
2.3 Continuous matrix groups and their generators 28
2.3.1 (cid:9) General considerations 28
2.3.2 (cid:9) Continuous matrix groups; decomposition into pieces 29
2.3.3 (cid:9) The Lie algebra (Lie ring, infinitesimal ring) 30
2.3.4 (cid:9) Canonical coordinates 32
2.3.5 (cid:9) The structure of the group and its infinitesimal ring 36
2.3.6 (cid:9) Summary: continuous matrix groups and their Lie algebra 38
2.3.7 (cid:9) Group representations 39
2.4 The physical significance of symmetries 42
vii
viii CONTENTS
2.4.1 (cid:9) Continuous groups connected to the identity; Noether's
theorem 42
2.4.2 (cid:9) Pieces not connected to the identity; discrete groups 45
2.4.3 (cid:9) Super-selection rules 45
2.4.4 (cid:9) Complete symmetry group, complete sets of commuting
observables, complete sets of states 49
2.4.5 (cid:9) Summary of the chapter 52
3 Rotations in Three Dimensional Space 53
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3.1 General remarks on rotations 53
3.1.1 (cid:9) Interpretation 53
3.1.2 (cid:9) Parameters describing a rotation 54
3.1.3 (cid:9) Representation of a rotation 55
3.2 Sequences of rotations 56
3.2.1 (cid:9) Considering the 'abstract' rotations R 57
3.2.2 (cid:9) Considering the 3 x 3 rotation matrices M(R) 59
3.3 The Lie algebra and the local group 63
3.3.1 (cid:9) The rotation matrix Mp(q) 63
3.3.2 (cid:9) The generators of the rotation group 65
3.3.3 (cid:9) The local group 67
3.3.4 (cid:9) Canonical parameters of the first and the second kind 68
3.4 The unitary representation U(R) induced by the three-
dimensional rotation R 69
4 Angular Momentum Operators and Eigenstates 72
4.1 The operators of angular momentum J1, .12 and .13 72
4.1.1 (cid:9) The physical significance of J 72
4.1.2 (cid:9) The angular momentum component in a direction n 75
4.2 Commutation relations for angular momenta 75
4.3 Direct sum and direct product 80
4.4 Angular momenta of interacting systems 85
4.5 Irreducible representations; Schur's lemma 87
4.6 Eigenstates of angular momentum 91
4.7 Orbital angular momentum 99
4.7.1 (cid:9) Angular momentum operators in polar coordinates 100
4.7.2 (cid:9) Construction of the eigenfunctions 102
4.7.3 (cid:9) Orbital angular momenta have only integer eigenvalues 104
4.7.4 (cid:9) Spherical harmonics 105
4.7.5 (cid:9) The phase convention 108
4.7.6 (cid:9) Parity 108
4.7.7 (cid:9) Particular cases 108
4.7.8 (cid:9) Further formulae 111
4.8 Spin4 eigenstates and operators 111
4.9 Double-valued representations; the covering group SU(2) 114
4.10 Construction of the general j, m-state from spin4 states 116
(cid:9)
CONTENTS ix
5 Addition of Angular Momenta 121
5.1 (cid:9) The general problem 121
5.2 (cid:9) Complete sets of mutually commuting (angular momentum)
observables 122
5.3 (cid:9) Combining two angular momenta; Clebsch—Gordan (Wigner)
coefficients 128
5.3.1 (cid:9) Notation 128
5.3.2 (cid:9) Definition and some properties of the Clebsch—Gordan
coefficients 131
5.3.3 (cid:9) Orthogonality of the Clebsch—Gordan coefficients 133
5.3.4 (cid:9) Sketch of the calculation of the Clebsch—Gordan
coefficients; phase convention and reality 134
5.3.5 (cid:9) Calculation of (ji m j2j — mu ff) 138
5.3.6 (cid:9) Obvious symmetry relations for CGCs 140
5.3.7 (cid:9) Wigner's 3j-symbol and Racah's V (.ii /2/31m I m2m3)-
symbol 147
5.3.8 (cid:9) Racah's formula for the CGCs 149
5.3.9 (cid:9) Regge's symmetry of CGCs 154
5.3.10 Collection of formulae for the CGCs; a table of special
values 156
5.4 Combining three angular momenta; recoupling coefficients 159
5.4.1 (cid:9) General remarks; statement of the problem 159
5.4.2 (cid:9) The 6j-symbol and the Racah coefficients 162
5.4.3 (cid:9) Collection of formulae for recoupling coefficients 164
5.5 Combining more than three angular momenta 168
5.6 Numerical tables and important references on addition of angular
momenta 168
6 Representations of the Rotation Group 170
6.1 Active and passive interpretation; definition of D; the
invariant subspaces H 170
6.2 The explicit form of D.(a, /3, y) 173
6.2.1 (cid:9) The spin-i case 173
6.2.2 (cid:9) The general case 175
6.3 General properties of D(i) 177
6.3.1 (cid:9) Relation to the Clebsch—Gordan coefficients 177
6.3.2 (cid:9) Significance of the relation to the CGCs 179
6.3.3 (cid:9) Relation to the eigenfunctions of angular momentum 183
6.3.4 (cid:9) Orthogonality relations and integrals over D-matrices 188
6.3.5 (cid:9) A projection formula 190
6.3.6 (cid:9) Completeness relation for the D-matrices 191
6.3.7 (cid:9) Symmetry properties of the D-matrices 194
7 The Jordan Schwinger Construction and Representations 196
—
7.1 (cid:9) Bosonic operators 196
x (cid:9) CONTENTS
7.2 Realization of su (2) Lie algebra and the rotation matrix in terms
of bosonic operators (cid:9) 199
7.3 A short note about the new field of quantum groups (cid:9) 204
8 Irreducible Tensors and Tensor Operators (cid:9) 207
8.1 Introduction (cid:9) 207
8.2 Definition and properties (cid:9) 209
8.3 Tensor product; irreducible combination of irreducible tensors;
scalar product (cid:9) 211
8.4 Invariants and covariant equations (cid:9) 213
8.5 Spinor and vector spherical harmonics (cid:9) 215
8.6 Angular momenta as spherical tensor operators (cid:9) 219
8.7 The Wigner—Eckart theorem (cid:9) 220
8.8 Examples of applications of the Wigner—Eckart theorem (cid:9) 222
8.8.1 The trace of T (kq) (cid:9) 222
8.8.2 Tensors of rank 0 (scalars, invariants) (cid:9) 223
8.8.3 The angular momentum operators (cid:9) 223
8.9 Projection theorem for irreducible tensor operators of rank 1 (cid:9) 223
9 Peculiarities of Two-Dimensional Rotations: Anyons, Fractional Spin
and Statistics (cid:9) 227
9.1 Introduction (cid:9) 227
9.2 Properties of rotations in two-dimensional space and fractional
statistics (cid:9) 228
9.3 Particle—flux system: example of anyon (cid:9) 232
9.4 Possible role of anyons in physics (cid:9) 237
10 A Brief Glance at Relativistic Problems (cid:9) 239
10.1 Introduction (cid:9) 239
10.2 The generators of the inhomogeneous Lorentz group (Poincaré
group) (cid:9) 240
10.2.1 Translations; four-momentum (cid:9) 241
10.2.2 The homogeneous Lorentz group; angular momentum (cid:9) 242
10.3 The angular momentum operators (cid:9) 248
10.3.1 Commutation relations of the J" with each other (cid:9) 248
10.3.2 Commutation relations of the P" with pi'(cid:9) 249
10.4 A complete set of commuting observables (cid:9) 249
10.4.1 The spin four-vector tut' and the spin tensor S" (cid:9) 250
10.4.2 Commutation relations for wi` and S" (cid:9) 252
10.4.3 Construction of a complete set of commuting
observables; helicity (cid:9) 255
10.4.4 Zero-mass particles (cid:9) 260
10.5 The use of helicity states in elementary particle physics (cid:9) 263
10.5.1 Construction of one-particle helicity states of arbitrary p 263
10.5.2 Two-particle helicity states (cid:9) 265
CONTENTS (cid:9) xi
10.5.3 Eigenstates of the total angular momentum (cid:9) 265
10.5.4 The S-matrix; cross-sections (cid:9) 267
10.5.5 Evaluation of cross-section formulae (cid:9) 269
10.5.6 Discrete symmetry relations: parity, time reversal,
identical particles (cid:9) 270
11 Supersymmetry in Quantum Mechanics and Particle Physics (cid:9) 275
11.1 What is supersymmetry? (cid:9) 275
11.2 SUSY quantum mechanics (cid:9) 279
11.3 Factorization and the hierarchy of Hamiltonians (cid:9) 281
11.4 Broken supersymmetry (cid:9) 285
Appendix A(cid:9) 287
Appendix B (cid:9) 291
Bibliography (cid:9) 293
Index (cid:9) 297
PREFACE
Symmetries are of primary importance in physics, particularly in quantum
theory. Among the first symmetries which were remarked upon historically and
conceptually were those of space and time. While Galilei invariance was later
to be generalized into Lorentz invariance, the invariance under spatial rotations
(though also a subgroup of the Lorentz group) has survived as such and has
become an important subject of quantum theory. It is well known, and will also
be discussed extensively in this book, that invariance generates conservation
laws; in the case of rotational invariance the conserved quantity is the angular
momentum. Therefore there is a dense interlacing between
(cid:127) the description of rotations, geometrically and group theoretically
(cid:127) their representations by unitary transformations in the Hilbert space of
quantum mechanical states, and
(cid:127) the quantum theory of angular momentum.
Somehow these are only three different facets of one and the same thing.
According to circumstances one or the other aspect will naturally be pushed to
the foreground.
The main raison d'être, however, of this book is that we have tried to exhibit
the wholeness of this 'one and the same thing', namely symmetry as the basic
concept underlying all the (sometimes tedious) formalisms. This implies, for
instance, that the Wigner theorem'—stating that to every symmetry group there
corresponds a representation (unitary or anti-unitary) in the quantum mechanical
Hilbert space—is, in our opinion, of such fundamental importance that it should
not simply be dealt with by saying 'Wigner has proved that ... '. Therefore we
give an explicit proof of it. There, as on many other occasions, our aim has
been to stress the underlying ideas and motivations: the 'why is' and the 'how
is' ; in short, to make things plausible rather than overburden the reader with a
formal and condensed proof. Lovers of rigour and compactness may be irritated
by our often pedestrian length, as well as by some repetitions in which earlier
arguments come up again and are discussed anew in another context.
Thus, in spite of its mathematical appearance, this is a didactic text written
by physicists for physicists. We took our time in writing it and we hope our
readers will take their time in reading it.
This, then, has been our philosophy in writing the book. (cid:9) It is
necessarily biased and incomplete (for instance we did not include the graphical
representation of formulae). Fortunately there are a sufficient number of other
books on this subject with different aims:
Description:MASUD CHAICHIAN. Department of Physics, University of Helsinki and. Helsinki Institute of Physics. ROLF HAGEDORN. Retired Staff Member of
