Theory of Groups and Symmetries: Representations of Groups and Lie Algebras, Applications 9811217408, 9789811217401

Citation preview

11749_9789811217401_tp.indd 1

6/7/20 4:20 PM

Other Related Titles from World Scientific

Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras by Alexey P Isaev and Valery A Rubakov ISBN: 978-981-3236-85-1 Introduction to the Theory of the Early Universe: Hot Big Bang Theory Second Edition by Valery A Rubakov and Dmitry S Gorbunov ISBN: 978-981-3209-87-9 ISBN: 978-981-3209-88-6 (pbk) Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory Second Edition by Valery A Rubakov and Dmitry S Gorbunov ISBN: 978-981-3275-62-1 ISBN: 978-981-3276-69-7 (pbk) Mathematical Foundations of Quantum Field Theory by Albert Schwarz ISBN: 978-981-3278-63-9 ISBN: 978-0-00-098924-6 (pbk)

KahFee - 11749 - Theory of Groups and Symmetries.indd 1

15/6/2020 9:44:33 am

11749_9789811217401_tp.indd 2

6/7/20 4:20 PM

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

Library of Congress Cataloging-in-Publication Data Names: Isaev, Alexey P., 1957– author. | Rubakov, V. A., author. Title: Theory of groups and symmetries : representations of groups and lie algebras, applications / Alexey P. Isaev, N.N. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia, M.V. Lomonosov Moscow State University, Russia, Valery A. Rubakov, Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia, M.V. Lomonosov Moscow State University, Russia. Description: New Jersey : World Scientific Publishing Co. Pte. Ltd., [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020019144 | ISBN 9789811217401 (hardcover) | ISBN 9789811217418 (ebook for institutions) | ISBN 9789811217425 (ebook for individuals) Subjects: LCSH: Group theory. | Symmetry (Physics) | Group algebras. | Lie algebras. Classification: LCC QC20.7.G76 I84 2021 | DDC 512/.2--dc23 LC record available at https://lccn.loc.gov/2020019144 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2021 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. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11749#t=suppl Desk Editor: Ng Kah Fee Typeset by Stallion Press Email: [email protected] Printed in Singapore

KahFee - 11749 - Theory of Groups and Symmetries.indd 2

15/6/2020 9:44:33 am

July 8, 2020

13:21

Theory of Groups and Symmetries - Vol. II

To Elvira and Helena

v

9in x 6in

b3865-fm

page v

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

This page intentionally left blank

b2530_FM.indd 6

01-Sep-16 11:03:06 AM

July 8, 2020

13:21

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-fm

Contents

Preface 1

Dirac Notations 1.1 1.2 1.3

2

xiii 1

Bra and Ket Vectors. Coordinate and Momentum Representations . . . . . . . . . . . . . . . . . . . . Fock Representation. (Anti)holomorphic Representations . . . . . . . . . . . . . . . . . . . . Clifford Algebra and Free Fermion Algebra . . . . . 1.3.1 Clifford algebra . . . . . . . . . . . . . . . . 1.3.2 Fock representation for free fermion algebra . 1.3.3 Grassmann algebra . . . . . . . . . . . . . .

. . . . . . . .

. . . . .

. . . . .

Finite-Dimensional Representations of Lie Algebras su(2) and s(2, C) and Lie Groups SU (2) and SL(2, C) 2.1 2.2

Finite-Dimensional Representations of Lie Algebras su(2) and s(2, C) . . . . . . . . . . . . . . . . . . . . Differential Realization of Lie Algebra s(2, C) and Highest Weight Representations . . . . . . . . . . . . 2.2.1 Realization of Lie algebra s(2, C) in terms of differential operators . . . . . . . . . . . . . . . 2.2.2 Highest weight representations of differential realizations of s(2, C) . . . . . . . . . . . . . . 2.2.3 Coherent states for Lie algebra s(2, C) . . . .

vii

1 6 14 14 19 22

29

. .

29

. .

37

. .

37

. . . .

42 47

page vii

July 8, 2020

13:21

viii

9in x 6in

b3865-fm

Theory of Groups and Symmetries — Volume II

2.3

2.4

2.5

3

Theory of Groups and Symmetries - Vol. II

Finite-Dimensional Representations of Groups SU (2) and SL(2, C) . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Parameterizations of Group SU (2) . . . . . . . 2.3.2 Finite-dimensional representations of groups SU (2), SL(2, C) and SO(3). Tensor representations — Wigner functions . . . . . . 2.3.3 Spherical functions on S 2 = SU (2)/U (1). Laplace operators on SU (2) and SU (2)/U (1) . Tensor Product of Representations of SU (2) and Clebsch–Gordan Series . . . . . . . . . . . . . . . . . 2.4.1 Clebsch–Gordan expansion . . . . . . . . . . . 2.4.2 Highest weight representations in T (j1 ) ⊗ T (j2 ) 2.4.3 Heisenberg spin chain . . . . . . . . . . . . . . 2.4.4 Calculating Clebsch–Gordan coefficients . . . . 2.4.5 Properties of Clebsch–Gordan coefficients and 3-j symbols . . . . . . . . . . . . . . . . . Tensor Operators and 3n-j Symbols . . . . . . . . . . 2.5.1 Tensor operators and Wigner–Eckart theorem 2.5.2 Racah coefficients and 3n-j symbols . . . . . . 2.5.3 6-j symbols and associativity of product of representations . . . . . . . . . . . . . . . . 2.5.4 Calculating 6-j symbols. Schwinger method . .

. . . .

53 53

. .

58

. .

68

. . . . .

. . . . .

76 76 80 90 93

. . . .

. . . .

100 104 104 108

. . 116 . . 128

Representations of Simple Lie Algebras. Weight Theory 3.1

3.2 3.3 3.4

Root Systems of Simple Lie Algebras . . . . . . . . . 3.1.1 Root systems of Lie algebra s(n, C), so(n, C) and sp(2r, C) . . . . . . . . . . . . . . . . . . . 3.1.2 Root systems of exceptional Lie algebras . . . 3.1.3 Weyl group. Dual root systems . . . . . . . . . Representations and Weights . . . . . . . . . . . . . . Weight Lattice . . . . . . . . . . . . . . . . . . . . . . Classification of Finite-Dimensional Irreducible Representations . . . . . . . . . . . . . . . . . . . . . 3.4.1 Highest weight representations . . . . . . . . . 3.4.2 Fundamental weights and representations of Lie algebras s(n, C), so(n, C) and sp(2r, C) . . . . . . . . . . . . . . . . . . . 3.4.3 Quadratic Casimir operator . . . . . . . . . . .

137 . . 137 . . . . .

. . . . .

137 144 150 157 169

. . 175 . . 175

. . 183 . . 203

page viii

July 8, 2020

13:21

Theory of Groups and Symmetries - Vol. II

9in x 6in

Contents

3.5

4

4.3

4.4

4.5

ix

Weyl Character Formula for Representations of Compact Simple Lie Groups . . . . . . . . . . . . . . . 206 3.5.1 Weyl denominator and Weyl character formula . . . . . . . . . . . . . . . . . . . . . . . . 206 3.5.2 Applications. Explicit formulas for characters and dimensions of representations of groups SU (r + 1), SO(n) and U Sp(2r) . . . . . . . . . . 218

Finite-Dimensional Representations of Algebras s(N , C), su(N ) and Groups SL(N , C) and SU (N ) 4.1 4.2

b3865-fm

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . Action of Group Sr in Tensor Product of Defining Representations . . . . . . . . . . . . . . . . . . . . . Representations of Symmetric Group I. Young Symmetrizers . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Young tableaux and Young symmetrizers . . . 4.3.2 Young symmetrizers and idempotents. Irreducible representations of group Sr and their dimensions . . . . . . . . . . . . . . Finite-Dimensional Irreducible Representations of Groups SU and SL . . . . . . . . . . . . . . . . . . 4.4.1 Finite-dimensional irreducible representations of SL(N, C) and SU (N ) in spaces of symmetrized tensors . . . . . . . . . . . . . 4.4.2 Dimensions of irreducible representations of groups SL(N, C) and SU (N ) . . . . . . . . 4.4.3 Co-defining and adjoint representations of groups SL(N, C) and SU (N ) . . . . . . . . 4.4.4 Quarks, SU (3)-symmetry and its breaking . . Representations of Symmetric Group II. Young–Frobenius Theory . . . . . . . . . . . . . . . . 4.5.1 Idempotents and irreducible representations of associative algebras. Peirce decomposition . 4.5.2 Orthogonality and completeness of Young symmetrizers . . . . . . . . . . . . . . . . . . . 4.5.3 Schur–Weyl duality . . . . . . . . . . . . . . .

229 . . 230 . . 233 . . 242 . . 242

. . 253 . . 264

. . 264 . . 272 . . 279 . . 282 . . 302 . . 303 . . 319 . . 330

page ix

July 8, 2020

13:21

x

9in x 6in

b3865-fm

Theory of Groups and Symmetries — Volume II

4.6

4.7 5

Theory of Groups and Symmetries - Vol. II

Representations of Symmetric Group III. Okounkov–Vershik Approach . . . . . . . . . . . . . 4.6.1 Jucys–Murphy elements and intertwining operators in algebra C[Sn ] . . . . . . . . . . 4.6.2 Idempotents and spectrum of Jucys–Murphy operators . . . . . . . . . . . . . . . . . . . . 4.6.3 Colored Young graph and branching rule for representations . . . . . . . . . . . . . . . 4.6.4 Young graph and inductive construction of idempotents . . . . . . . . . . . . . . . . . 4.6.5 Projection operators and characters of irreducible representations of U (N ). Symmetric polynomials . . . . . . . . . . . . Concluding Remarks. Gelfand–Tsetlin Basis . . . .

. . . 333 . . . 333 . . . 338 . . . 352 . . . 359

. . . 365 . . . 377

Finite-Dimensional Representations of Groups SO, Sp and Lie Algebras so, sp 5.1

5.2

Tensor Representations of Groups O(N, C), SO(N, C) and Their Subgroups O(p, q), SO(p, q) . . . . . . . . . 5.1.1 Pseudo-orthogonal group O(p, q) and Lie algebra so(p, q) . . . . . . . . . . . . . . . . 5.1.2 Tensor representations of groups O(p, q) . . . . . 5.1.3 Extracting irreducible representations of groups O(p, q) and SO(p, q) from representation T ⊗r . . 5.1.4 Irreducible tensor representations of orthogonal groups. Oscillating Young tableaux . . . . . . . Brauer Algebra Brn and Its Representations . . . . . . 5.2.1 Brauer algebra Brn . Jucys–Murphy elements for Brn . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Intertwining elements and idempotents in algebra Brn . Spectrum of Jucys–Murphy operators . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Oscillating Young tableaux and their content vectors . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Oscillating Young graph for algebra Brn . . . . 5.2.5 Primitive idempotents in Brauer algebra and invariant projectors for representations of orthogonal groups . . . . . . . . . . . . . . .

381 . 382 . 382 . 386 . 389 . 400 . 405 . 406

. 412 . 417 . 419

. 422

page x

July 8, 2020

13:21

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-fm

xi

Contents

5.3 5.4

6

. . . . 427 . . . . 435 . . . . 435 . . . . 440

Groups Spin(p, q) and Their Finite-Dimensional Representations 6.1

6.2

6.3

6.4

7

Tensor Representation of Groups Sp(2r, C) and Their Subgroups Sp(2r, R), U Sp(2r), Sp(p, r − p) Spinor Representations of Lie Algebras so(N, C) 5.4.1 Spinor representations of Lie algebras so(2r, C) . . . . . . . . . . 5.4.2 Spinor representations of Lie algebras so(2r + 1, C) . . . . . . . .

Clifford Algebras and Their Representations . . . . . . . 6.1.1 Real Clifford algebras C(p,q) . . . . . . . . . . . . 6.1.2 Matrix representations of complex Clifford algebras CN and their real forms C(p,q) . . . . . 6.1.3 Weyl representations of Clifford algebras CN and C(p,q) . . . . . . . . . . . . . . . . . . . . . . Spinor Groups Pin(p, q) and Spin(p, q) . . . . . . . . . . . 6.2.1 Definitions of spinor groups Pin(p, q) and Spin(p, q) . . . . . . . . . . . . . . . . . . . . . 6.2.2 Representations of algebras spin(p, q) and groups Spin(p, q) . . . . . . . . . . . . . . . . Conjugation Matrices . . . . . . . . . . . . . . . . . . . . 6.3.1 Conjugation matrices B, C, D for representations of algebra C(p,q) , and their properties . . . . . . . . . . . . . . . . . 6.3.2 Conjugation matrices B, C, D and structure of groups Spin(p, q). Group Spin(8) . . . . . . . . . Dirac, Weyl and Majorana Spinors in Spaces Rp,q . . . . 6.4.1 Spinors in spaces Rp,q and their tensor products . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Charge conjugation of spinors in spaces Rp,q . . . 6.4.3 Algebra C(1,N −1) and spinor group Spin(1, N − 1). Spinors in Minkowski space R1,N −1 . . . . . . . . 6.4.4 Fierz identities for multi-dimensional spinors . . .

Solutions to Selected Problems 7.1 7.2 7.3

445 445 445 451 463 468 468 481 486

486 499 510 510 517 523 528 539

Problem 1.3.6 of Section 1.3.3 . . . . . . . . . . . . . . . 539 Problem 1.3.7 of Section 1.3.3 . . . . . . . . . . . . . . . 540 Problem 2.2.13 of Section 2.2.3 . . . . . . . . . . . . . . . 541

page xi

July 8, 2020

13:21

xii

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-fm

Theory of Groups and Symmetries — Volume II

7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28

Problem 2.3.4 of Section 2.3.2 . . . . . . Problem 2.3.6 of Section 2.3.3 . . . . . . Problem 2.3.7 of Section 2.3.3 . . . . . . Problem 2.3.11 of Section 2.3.3 . . . . . . Problem 2.5.4 of Section 2.5.2 . . . . . . Problem 2.5.15 of Section 2.5.4 . . . . . . Problem 3.3.1 of Section 3.3 . . . . . . . Problem 3.4.5 of Section 3.4.2 . . . . . . Problems 4.3.2 and 4.3.3 of Section 4.3.1 Problem 4.3.10 of Section 4.3.2 . . . . . . Problem 4.3.11 of Section 4.3.2 . . . . . . Problem 4.5.7 of Section 4.5.1 . . . . . . Problem 4.6.9 of Section 4.6.5 . . . . . . Problem 4.6.11 of Section 4.6.5 . . . . . . Problem 4.7.1 of Section 4.7 . . . . . . . Problem 5.1.3 of Section 5.1.3 . . . . . . Problem 5.1.5 of Section 5.1.3 . . . . . . Problem 5.3.4 of Section 5.3 . . . . . . . Problem 6.2.2 of Section 6.2.1 . . . . . . Problem 6.2.4 of Section 6.2.1 . . . . . . Problem 6.3.5 of Section 6.3.2 . . . . . . Problem 6.3.6 of Section 6.3.2 . . . . . . Problem 6.3.10 of Section 6.3.2 . . . . . . Problem 6.4.3 of Section 6.4.1 . . . . . . Problem 6.4.13 of Section 6.4.3 . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

543 544 546 547 552 554 555 559 560 563 565 566 568 569 571 573 576 579 581 582 583 585 585 586 587

Selected Bibliography

591

References

593

Index

597

page xii

July 8, 2020

13:21

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-fm

Preface

This book is a sequel to the book [1] by the same authors entitled Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras, so this book can be viewed as the second part. References to sections and formulas from the first book are labeled by Roman numeral I, e.g., Section I-3.2.3, Eq. (I-7.2.37). The second book is mainly devoted to the theory of representations of Lie groups and Lie algebras and also to applications. We begin with the presentation in Chapter 1 of the Dirac notation, which is illustrated by boson and fermion oscillator algebras and also Grassmann algebra. In Chapter 2, we give a detailed account of finite-dimensional representations of groups SL(2, C) and SU (2) and their Lie algebras. In Chapter 3, we consider the general theory of finite-dimensional irreducible representations of simple Lie algebras based on the construction of highest weight representations. We then give the classification of all finite-dimensional irreducible representations of the Lie algebras of the classical series s(n, C), so(n, C) and sp(2r, C). In Chapter 4, finite-dimensional irreducible representations of linear groups SL(N, C) and their compact real forms SU (N ) are constructed on the basis of the Schur–Weyl duality. In this approach, spaces of irreducible representations are viewed as invariant subspaces in the tensor products V ⊗r of spaces V of defining representations. These invariant subspaces are singled out with the help of special elements (idempotents) of group algebra C[Sr ] of the group of permutations (symmetric group) Sr . Here, special role is played by the theory of representations of the algebra C[Sr ] (Schur– Frobenius theory, Okounkov–Vershik approach), based on combinatorics of Young diagrams and Young tableaux.

xiii

page xiii

July 8, 2020

13:21

xiv

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-fm

Theory of Groups and Symmetries — Volume II

We then turn in Chapter 5 to pseudo-orthogonal groups O(p, q) and SO(p, q), including multidimensional Lorentz groups O(1, N − 1) and SO(1, N − 1) and their Lie algebras. We also consider symplectic groups Sp(p, q), albeit in less detail. The presentation here is based on the appropriate modification of the techniques exposed in Chapter 4. Finally, in Chapter 6 we study the covering groups Spin(p, q) for pseudoorthogonal groups SO↑ (p, q). The groups Spin(p, q) are called spinor groups and are currently in active use in quantum field theory. The construction of the spinor groups Spin(p, q) requires the introduction of Clifford algebras in spaces Rp,q and the discussion of the representations of these algebras. In this book, as done in the first one, we try to prove or at least give hints of proofs for the majority of facts. We emphasize, however, that, similar to the first book, the level of rigor in places is not to the standard of a mathematically oriented reader; our main goal is rather to make the material comprehensible. Text written in small font may be omitted at the first reading. When necessary, we give references to literature, where one can find more detailed analysis of one or another point. This book started from lectures given at the Departments of Theoretical and Nuclear Physics of Dubna International University and at the Departments of Quantum Theory and High Energy Physics, Quantum Statistics and Field Theory, Particle Physics and Cosmology of Physics Faculty of M.V. Lomonosov Moscow State University. We are grateful to S.E. Derkachev, S.A. Fedoruk, E.A. Ivanov, S.O. Krivonos, S.K. Lando, S.A. Mironov, A.I. Molev, O.V. Ogievetsky, G.I. Olshansky, A.V. Silantiev, Y.M. Shnir, V.P. Spiridonov, A.O. Sutulin, V.O. Tarasov, S.V. Troitsky, N.A. Tyurin, M.A. Vasiliev and A.A. Vladimirov, as well as many other colleagues from the Theory Division of the Institute for Nuclear Research of the Russian Academy of Sciences, Moscow and N.N. Bogoliubov Laboratory for Theoretical Physics of the Joint Institute for Nuclear Research, Dubna, for numerous useful discussions of the material presented in this book.

page xiv

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Chapter 1

Dirac Notations

In this chapter, we introduce convenient notations which we use throughout the book for describing basis vectors and matrix elements of operators in the spaces of representations of Lie groups and Lie algebras. These notations were introduced by P. Dirac [30] and they are standard in quantum mechanics. We have used these notations in the first book [1]. Here, we give the general definitions and rules. We illustrate the convenience of these notations by considering examples of coherent states for quantum oscillator algebra and free fermion algebra. 1.1.

Bra and Ket Vectors. Coordinate and Momentum Representations

Consider complex vector space V (finite dimensional or infinite dimensional). We denote vectors in V as |Ψ, where Ψ stands for a set of symbols, indices, numbers, etc., which characterize a given vector.a Consider also a complex vector space V ∗ dual to V. The space V ∗ consists of linear functionals in V. We denote vectors from V ∗ as Φ|. By definition, every Φ| ∈ V ∗ determines a function: V → C, which we denote by Φ|Ψ ∈ C,

∀|Ψ ∈ V.

(1.1.1)

The expression Φ|Ψ is called convolution of vectors |Ψ and Φ|. The convolution (1.1.1) may be understood also from the dual viewpoint, namely, vectors |Ψ ∈ V define linear functionals in V ∗ . a In what follows, we disregard peculiarities occurring in the case of infinite-dimensional spaces, which are related, in particular, to the convergence of relevant integrals and sums; see, however, Definition 1.1.1 and discussion around it.

1

page 1

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

2

Let V have a basis |α, where the index α that labels the basis vectors, is, in general, a set of indices (i.e., α is a multi-index). Indices entering the multi-index α can take both discrete and continuous values. The basis of V ∗ consisting of vectors β| is called dual to basis |α ∈ V if β is a multi-index of the same type as α and the orthonormality condition holds: β|α = δα,β .

(1.1.2)

Here δα,β is a product of Kronecker symbols and delta-functions which involve indices entering α and β, depending on whether these indices take discrete of continuous values. In accordance with (1.1.2), vectors |Ψ ∈ V and Φ| ∈ V ∗ are expanded in basis vectors:   |Ψ = |αα|Ψ, Φ| = Φ|αα|, (1.1.3) α



α

where α denotes a multiple integral and multiple sum over continuous and discrete indices entering the multi-index α. For any linear operator T acting in V, we define an operator T  which acts in V ∗ (and, conversely, given an operator acting in V ∗ , we define an operator in V) as follows: Φ|(T · |Ψ) = (Φ| · T  )|Ψ,

∀|Ψ ∈ V, ∀Φ| ∈ V ∗ ,

(1.1.4)

where it is convenient to think of the operator T as acting in V on the left, and T  acting in V ∗ on the right. In what follows, we use the same notation for T and T  to simplify formulas, and write Φ|T |Ψ instead of (1.1.4). The relations (1.1.3) ensure that the unit operator I acting in spaces V and V ∗ is expanded in terms of the basis vectors in a simple way:  I= |αα|, (1.1.5) α

while the convolution Φ|Ψ takes the form  Φ|Ψ = Φ|αα|Ψ.

(1.1.6)

α

Making use of the representation (1.1.5) of the unit operator, we write the action of linear operators T on the basis vectors |β ∈ V and on functions Ψ(α) ≡ α|Ψ as follows:   |αα|T |β, β|T |Ψ = β|T |αα|Ψ. (1.1.7) T |β = α

α

page 2

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

Dirac Notations

b3865-ch01

3

This shows that the matrix of the operator T in the basis {|β} has matrix elements β|T |α. In accordance with (1.1.3) and (1.1.7), function Ψ(α) can be understood as the set of coordinates of the vector |Ψ ∈ V, where the index α “labels” these coordinates. Let us establish one-to-one correspondence between vectors in spaces V and V ∗ by requiring that the convolution obeys (Φ|Ψ)∗ = Ψ|Φ,

∀|Ψ ∈ V, ∀Φ| ∈ V ∗ ,

(1.1.8)

where ∗ in the equality denotes complex conjugation. Then the coordinates of vectors Φ| and |Φ are complex conjugate to each other: (α|Φ)∗ = Φ|α, while the convolution (1.1.6) takes the form   (α|Φ)∗ α|Ψ = Φ(α)∗ Ψ(α). Φ|Ψ = α

(1.1.9)

(1.1.10)

α

The relations (1.1.3) and (1.1.9) ensure that the vector Φ| ∈ V ∗ is Hermitian conjugate to the vector |Φ ∈ V, i.e., (|Φ)† = Φ|, while the convolution Φ|Ψ ≡ (|Φ)† |Ψ

(1.1.11)

is interpreted as Hermitian scalar product in the complex space V (see Definition I-1.2.9). In order that the right-hand side of (1.1.10) makes sense, the functions Φ(α) = α|Φ must be square integrable and summable as functions of α. In the space V equipped with Hermitian scalar product (1.1.10), one naturally defines a norm of a vector, ||Ψ|| = Ψ|Ψ. Definition 1.1.1. If a complex vector space V with Hermitian scalar  product (1.1.10) and norm ||Ψ|| = Ψ|Ψ is complete (i.e., any Cauchy sequenceb in V converges, with respect to norm ||Ψ||, to an element of V), then the space V is called Hilbert space. In general, square integrability and summability in (1.1.10) are understood in the sense of distributions, rather than conventional functions, and the space V does not have to be Hilbert space. of vectors {Ψk }∞ k=1 in space V equipped with norm is called Cauchy sequence (or fundamental sequence) if it obeys Cauchy property: for any arbitrarily small  there exists a number N such that ||Ψi − Ψk || <  for all i, k > N .

b Sequence

page 3

July 8, 2020

10:20

4

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

Finally, making use of (1.1.4) and (1.1.11), we define operator H † , which is Hermitian conjugate to H, by the following equality: Ψ|H † |Φ = (Φ|H|Ψ)∗ ⇒ Φ| · H † = (H · |Φ)† . Operator H obeying H † = H ⇔ α|H|β = (β|H|α)∗ is called Hermitian operator. States in quantum mechanics are described by vectors from Hilbert space V and its Hermitian conjugate dual space V ∗ , while observables like coordinate, momentum, energy, angular momentum, etc., are described by Hermitian operators acting in V. When introducing notations Φ| and |Ψ, Dirac [30] called them bra- and ket-vectors, which are two parts of the word “bracket”. It is often convenient in quantum mechanics to choose the basis in V in a particular way, namely, let us single out a complete set of commuting observables Hi = Hi† (i = 1, 2, . . .), [Hi , Hj ] = 0. Denote the eigenvalues of the operators Hi by αi , and their eigenvectors by |α = |α1 , α2 , . . .: Hi |α = αi |α.

(1.1.12)

Problem 1.1.1. Show that the eigenvalues αi of Hermitian operators Hi are real, and that α |α ∝ δα ,α . The set of eigenvectors |α is precisely the particular basis used in quantum mechanics. It depends, of course, on the choice of the complete set of commuting operators Hi . One can choose the normalization of the vectors |α in such a way that they obey orthonormality and completeness conditions, Eqs. (1.1.2) and (1.1.5), respectively. Example. Coordinate and momentum representations States of a quantum mechanical particle in Rd are vectors |Ψ in infinitedimensional vector space V equipped with Hermitian scalar product (1.1.8). Coordinates of the particle are observables which are described by Hermitian operators x ˆk (k = 1, . . . , d), while the components of particle momenta are described by Hermitian operators pˆk which obey commutation relations ˆj ] = 0 = [ˆ pk , pˆj ], [ˆ xk , x

[ˆ xk , pˆj ] = iIδkj ,

(1.1.13)

page 4

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Dirac Notations

5

where I is unit operator in V. The algebra with 2d generators {ˆ xk , pˆk } and defining relations (1.1.13) is called d-dimensional Heisenberg algebra. p: Eigenvectors of operators x ˆk and pˆk in V are denoted by |x and | xˆk |x = xk |x,

pˆk | p = pk | p,

(1.1.14)

where xk , pk ∈ R are the eigenvalues. Particle states given by vectors |x and | p are states with definite coordinates and definite momenta, respectively. Let |y  ∈ V be an eigenvector of all operators x ˆk with eigenvalues yk . Making use of the commutation relations (1.1.13), we obtain ˆk )|y  = (xk + yk )e−ixk pˆk |y. x ˆk · e−ixk pˆk |y  = (xk e−ixk pˆk + e−ixk pˆk · x Hence, vector e−ixk pˆk |y  is the eigenvector of operators x ˆk with eigenvalues (xk + yk ): |x + y = e−ixk pˆk |y .

(1.1.15)

We differentiate the two sides of Eq. (1.1.15) over xk and then set y = 0 to obtain i∂k |x = i∂k e−ixk pˆk |0 = pˆk |x ⇒ x|ˆ pk = −i∂k x|,

(1.1.16)

where ∂k = ∂/∂xk , and the second relation is obtained from the first one by Hermitian conjugation. Every state |Ψ defines a complex-valued function x|Ψ ≡ Ψ(x), which is coordinate representation of vector |Ψ in basis |x. It is called the wave function of the particle. This correspondence is one-to-one. The action of operators x ˆk and pˆk on state |Ψ is written in terms of the wave function x|Ψ as follows (see (1.1.7)):  xk |x x |Ψ = xk x|Ψ, x|ˆ xk |Ψ = dd x x|ˆ  (1.1.17) pk |x x |Ψ = −i∂k x|Ψ, x|ˆ pk |Ψ = dd x x|ˆ where we use the completeness property (1.1.5):  dd x|xx| = I

(1.1.18)

and recall (1.1.14) and (1.1.16). From the viewpoint of the representation theory, Eqs. (1.1.17) define the (differential) coordinate representation ρ of

page 5

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

6

the Heisenberg algebra (1.1.13): ρ(ˆ xk ) = xk ,

ρ(ˆ pk ) = −i∂k ,

(1.1.19)

which acts in the space of wave functions Ψ(x) = x|Ψ. Problem 1.1.2. Construct (differential) momentum representation of the Heisenberg algebra (1.1.13), which acts on momentum space wave functions Ψ( p) =  p|Ψ. The state | p of the particle with well-defined momentum p = (p1 , . . . , pd ) is described by the wave function x| p, which, according to (1.1.14) and (1.1.17), obeys the equations p = pk x| p. −i∂k x| The solution to these equations is x| p =

1 exp(ipk xk ), (2π)d/2

(1.1.20)

where the normalization factor is determined by the orthonormality condition (1.1.2):  p  = dd x p  |x x| p  = δ d ( p − p  ). (1.1.21)  p  | Here we again use the completeness property (1.1.18). Using (1.1.20), we obtain the wave function in momentum representation   1 p |x x|Ψ = dd x exp(−ipk xk )x|Ψ. (1.1.22)  p |Ψ = dd x (2π)d/2 This relates the momentum and coordinate representations of the state vectors. The relation (1.1.22) is the well-known Fourier transformation from coordinate representation to momentum representation. Another example of employing the Dirac notations is given in the following section. 1.2.

Fock Representation. (Anti)holomorphic Representations

Besides coordinate and momentum representations, of particular importance in quantum mechanics are Fock representation and Bargmann–Fock representations (the latter dubbed holomorphic or antiholomorphic). In this section we present the standard material on these representations (see also

page 6

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Dirac Notations

7

books [15–18]). To define these representations, we introduce, instead of x ˆk and pˆk , new dynamical variables 1 ak = √ (iˆ pk + x ˆk ), 2

1 pk + x a†k = √ (−iˆ ˆk ), 2

(1.2.1)

which, according to (1.1.13), commute as follows: [ak , am ] = 0,

[a†k , a†m ] = 0,

[ak , a†m ] = Iδkm .

(1.2.2)

Further we denote the identity operator I by 1, and an element c · I, where c is a constant, is written simply as c. Operators (1.2.1) are particularly convenient in studying d-dimensional quantum oscillator whose Hamiltonian reads d d    1  2 1 † 2 ˆ ak ak + H= ˆk = . (1.2.3) pˆk + x 2 2 k=1

k=1

For this reason, the algebra with generators ak , a†m and defining relations (1.2.2) is called d-dimensional quantum oscillator algebra. The space F of states of the d-dimensional quantum oscillator, where the algebra with generators {ak , a†m } acts, is called Fock space. This space is constructed in the following way. One introduces the normalized vacuum vector |0 ∈ F such that ak |0 = 0,

∀k,

0|0 = 1.

(1.2.4)

Accordingly, the conjugate vector 0| = (|0)† obeys 0|a†k = 0. Then one constructs normalized vectors 1 (a† )n1 · (a†2 )n2 · · · (a†d )nd |0, (1.2.5) |n1 , n2 , . . . , nd  = √ n1 ! · · · nd ! 1 which make basis in the Fock space F , i.e., any vector in F can be written as |Ψ =

∞ 

ψn1 ,...,nd |n1 , n2 , . . . , nd  ∈ F,

ψn1 ,...,nd ∈ C.

n1 ,...,nd =0

The Fock space coincides with the space of all states of a quantum mechanical particle in Rd : any state of that particle can be written in the above form. According to (1.2.2), (1.2.5) we have √ ak |n1 , . . . , nk , . . . , nd  = nk |n1 , . . . , nk − 1, . . . , nd , (1.2.6) √ a†k |n1 , . . . , nk , . . . , nd  = nk + 1|n1 , . . . , nk + 1, . . . , nd ,

page 7

July 8, 2020

10:20

8

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

i.e., operator ak decreases nk by one, while a†k increases nk by one. For this reason, the operators ak and a†k are called annihilation and creation operators, respectively. Basis vectors n1 , n2 , . . . , nd | in dual space F ∗ are obtained from vectors (1.2.5) by Hermitian conjugation, and any vector Φ| ∈ F ∗ can be ∞ written as follows: Φ| = n1 ,...,nd =0 φ∗n1 ,...,nd n1 , . . . , nd | = (|Φ)† , where φn1 ,...,nd ∈ C. Problem 1.2.1. Prove that vectors (1.2.5) are orthonormalized: k1 , . . . , kd | ˆ is Hermitian n1 , . . . , nd  = δk1 n1 · · · δkd nd . Prove that the Hamiltonian H  ∗ with respect to the scalar product Φ|Ψ = n1 ,...,nd (φn1 ,...,nd ψn1 ,...,nd ). Check that the vectors (1.2.5) are eigenvectors of the Hamiltonian (1.2.3): 

d  1 ˆ 1 , n2 , . . . , nd  = |n1 , n2 , . . . , nd . nk + (1.2.7) H|n 2 k=1

Problem 1.2.2. Find the wave function x|0 of the oscillator ground state (vacuum) in coordinate representation. Show that the eigenstates |n1 , n2 , . . . , nd  of the Hamiltonian  coordinate have the following form in the d nk 1/2 x|0, where H (x )/(2 n !) representation: x|n1 , . . . , nd  = nk k k k=1 Hn (x) = (−1)n ex

2

dn −x2 dxn e

are Hermite polynomials.

The completeness relation (1.1.5) for vectors |n1 , . . . , nd  is written in the following way: I=

∞ 

|n1 , . . . , nd n1 , . . . , nd |,

(1.2.8)

n1 ,...,nd =0

where I is unit operator in the Fock space. Note that one can invert the relations (1.2.1) and obtain in this way the Fock representation of the Heisenberg algebra: 1 x ˆk = √ (ak + a†k ), 2

1 pˆk = √ (ak − a†k ). i 2

Let us turn to holomorphic and antiholomorphic representations. In analogy to the eigenstates of coordinate and momentum (1.1.14), to obtain the antiholomorphic representation, we introduce the complete set of eigenvectors of the annihilation operators ak : ak |z = zk |z,

(1.2.9)

page 8

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

Dirac Notations

b3865-ch01

9

where |z := |z1 , z2 , . . . , zd , and zk are complex eigenvalues. Vectors |z are called coherent states, and they can be written as (hereafter summation over repeated indices k is assumed) |z = exp(zk a†k )|0,

(1.2.10)

where |0 is the vacuum Fock state. Equation (1.2.9) follows from (1.2.10) in view of (1.2.2). Making use of the representation (1.2.10), we obtain a†k |z =

∂ |z. ∂zk

(1.2.11)

Problem 1.2.3. Find the wave functions x|z of coherent states in coordinate representation. We have similar formulas for conjugate vectors z ∗ | = (|z)† ∂ z ∗ |. ∂zk∗

(1.2.12)

z ∗ |w = 0| exp(zk∗ ak )|w = 0| exp(zk∗ wk )|w = exp(zk∗ wk ),

(1.2.13)

z ∗ | = 0| exp(zk∗ ak ),

z ∗ |a†k = zk∗ z ∗ |,

z ∗ |ak =

The convolution of two coherent states equals

where z, w ∈ C, and we employ Eq. (1.2.9) and the relation 0|w = 1, ∀w ∈ C. It is now clear that for z = w, coherent states |w and |z are not orthogonal. Nevertheless, they can be used for constructing new representations in quantum mechanics. The key point here is the following proposition. Proposition 1.2.1. The completeness condition (1.2.8) is written for coherent states as follows (cf. (1.1.18))  (1.2.14) dμ(z, z ∗ ) exp(−zk∗ zk )|zz ∗ | = I, where the measure dμ(z, z ∗ ) on C d is defined by c dμ(z, z ∗ ) =

d d 1  2 1  d z = dxk dyk , k πd πd k=1

zk = xk + iyk .

(1.2.15)

k=1

mathematical literature (see, e.g., [17]), the often used notation (for measure on C) is d2 z = 2i dz ∧ dz ∗ , which is equivalent to our convention.

c In

page 9

July 8, 2020

10:20

10

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

Proof. We prove (1.2.14) in one dimensional case d = 1; generalizing to arbitrary finite number of dimensions is trivial. We insert the definitions of coherent states (1.2.10), (1.2.12) into the left-hand side of (1.2.14) and write the integral in polar coordinates z = ρeiφ :  ∞  (z ∗ a† )m 1 (za)n 2 2 |00| d z exp(−|z| ) π n! m! m,n=0   2π ∞  2 1 1 ∞ |nm| dρe−ρ ρn+m+1 dφei(n−m)φ √ = π 0 n!m! 0 m,n=0  ∞ ∞ ∞   2 2n+1 −ρ2 = dρρ e |nn| = I. (1.2.16) |nn| = n! 0 n=0 n=0 This is the desired result. Now, like in the case of coordinate or momentum representation, we consider a function z ∗ |Ψ ≡ Ψ(z ∗ ) that corresponds to a state vector |Ψ. This is the wave function in the antiholomorphic representation. In antiholomorphic representation, operators ak and a†k act on a state |Ψ as follows (see (1.2.12)): z ∗ |ak |Ψ =

∂ z ∗ |Ψ, ∂zk∗

z ∗ |a†k |Ψ = zk∗ z ∗ |Ψ,

(1.2.17)

while the scalar product Φ|Ψ, in view of the completeness relation (1.2.14), reads   ∗ ∗ ∗ Φ|Ψ = dμ(z, z ∗ )e−zk zk Φ|zz ∗ |Ψ = dμ(z, z ∗ )e−zk zk (Φ(z ∗ )) Ψ(z ∗ ). (1.2.18) Thus, the formulas (1.2.17) define antiholomorphic representation ρ¯ of the oscillator algebra ak and a†k (cf. (1.1.19)): ρ¯(ak ) =

∂ , ∂zk∗

ρ¯(a†k ) = zk∗ ,

(1.2.19)

which acts in the Hilbert space of antiholomorphic functions Ψ(z ∗ ) = z ∗ |Ψ with the scalar product (1.2.18). In literature, this representation is also known as Bargmann–Fock representation. By performing the replacement zk∗ ↔ zk , (∀k) in (1.2.12), (1.2.14), (1.2.17) and (1.2.18), we obtain yet another formulation of Bargmann–Fock representation denoted by ρ. It can be called holomorphic representation.

page 10

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Dirac Notations

11

It acts in the Hilbert space of holomorphic functions z|Ψ = Ψ(z) ≡ Ψ(z1 , . . . , zd ) with the scalar product   ∗ ∗ Φ|Ψ = dμ(z, z ∗ )e−zk zk Φ|z ∗ z|Ψ = dμ(z, z ∗ )e−zk zk (Φ(z))∗ Ψ(z), (1.2.20) and the representation of the oscillator algebra (1.2.2) is given byd ρ(ak ) =

∂ , ∂zk

ρ(a†k ) = zk .

(1.2.21)

The completeness condition (1.2.14) reads in the representation ρ as  dμ(z, z ∗ ) exp(−zk∗ zk )|z ∗ z| = I, (1.2.22) while basis vectors (1.2.5) are 1 z|n1 , . . . , nd  = √ z n1 z n2 · · · zdnd , n1 ! · · · nd ! 1 2

(1.2.23)

where we use z|a†k = zk z|; cf. (1.2.12). Proposition 1.2.2. The scalar product (1.2.18), (1.2.20) can be written as follows: (1.2.24) Φ|Ψ = Φ∗ (∂1 , . . . , ∂d ) · Ψ(z)|z1 =···=zd =0 ,  where ∂k = ∂/∂zk . Here, if Φ(z) = Cn1 ,...,d z1n1 . . . zdnd then Φ∗ (∂1 , . . . ,  ∗ n1 nd Cn1 ,...,nd ∂1 . . . ∂d . ∂d ) = Proof. It suffices to prove (1.2.24) in the case when the vector Φ(z) = z|Φ is an arbitrary basis vector (1.2.23). In that case, we integrate by parts and obtain  ∗ n1 , . . . , nd |Ψ = dμ(z, z ∗ )n1 , . . . , nd |z ∗ z|Ψe−zk zk  =

∗ z ∗n1 · · · zd∗nd dμ(z, z ∗ ) √1 z|Ψe−zk zk n1 ! · · · nd !

in the literature one considers the representation ρ (ak ) = zk , ρ (a†k ) = (cf. (1.2.21)) which acts in the space of holomorphic functions Ψ(z) and also is called holomorphic representation. We will not discuss this possibility here.

d Sometimes

− ∂z∂ k

page 11

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

12

 =



dμ(z, z ∗ )(−1)



nk

∂ n1 · · · ∂dnd −zk∗ zk e z|Ψ √1 n1 ! · · · nd !

∗ ∂ n1 · · · ∂dnd z|Ψ dμ(z, z ∗ )e−zk zk √1 n1 ! · · · nd !   ∂1n1 · · · ∂dnd = √ Ψ(z) . n !···n !

=

1

d

(1.2.25)

z1 =···=zd =0

Since an arbitrary holomorphic functions Φ(z) can be written as an expansion in basis monomials (1.2.23), then (1.2.25) gives (1.2.24). Remark 1. The representation (1.2.24) of the scalar product (1.2.20) is particularly convenient from technical viewpoint. We extensively use this representation below, in Sections 2.2.2, 2.4.4 and 2.5.4 (see, e.g., (2.2.28) and (2.4.86)). Making use of (1.2.2), one can write any operator A in Fock space F in normal (Wick) form:  Aμ,ν (a†1 )n1 · (a†2 )n2 · · · (a†d )nd (a1 )m1 · (a2 )m2 · · · (ad )md , (1.2.26) A= μ,ν

where μ and ν are multi-indices, μ = (m1 , . . . , md ) and ν = (n1 , . . . , nd ), and Aμ,ν ∈ C. In this form, all annihilation operators in every term in (1.2.26) are on the right of the creation operators. Antiholomorphic representation is convenient for determining the Wick symbols of operators A in F . To this end, we write the action of the operator (1.2.26) on a vector |Ψ in antiholomorphic representation:  ∗ (1.2.27) w∗ |A|Ψ = dμ(z, z ∗ )e−zk zk w∗ |A|zz ∗ |Ψ, where w∗ |A|z is the kernel of the operator A. The latter is computed straightforwardly by making use of (1.2.9) and (1.2.12): ∗

A(w∗ , z) =



w∗ |A|z = A(w∗ , z)ewk zk , Aμ,ν (w1∗ )n1 · (w2∗ )n2 · · · (wd∗ )nd (z1 )m1 · (z2 )m2 · · · (zd )md .

μν

(1.2.28) The function A(w∗ , z) is dubbed Wick (normal) symbol of the operator A. The only difference of the Wick symbol from the normal form is that the former involves complex numbers wi∗ and zj instead of operators a†i and aj .

page 12

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

Dirac Notations

b3865-ch01

13

Wick symbols are convenient for writing a product of two operators A1 and A2 in normal form (1.2.26). Proposition 1.2.3. Let A1 (w∗ , z) and A2 (w∗ , z) be the Wick symbols of operators A1 and A2 , then the Wick symbol of the operator A1 · A2 is given by  ∗ ∗ ∗ (A1 ·A2 )(w , v) = dμ(z, z ∗ )e−(zk −wk )(zk −vk ) A1 (w∗ , z)A2 (z ∗ , v), (1.2.29) where the measure dμ(z, z ∗ ) is defined in (1.2.15). Proof. In accordance with (1.2.28), we have ∗

(A1 · A2 )(w∗ , v) = w∗ |A1 · A2 |ve−wk vk  ∗ ∗ = dμ(z, z ∗ )w∗ |A1 |zz ∗ |A2 |ve−zk zk −wk vk  ∗ ∗ ∗ ∗ = dμ(z, z ∗ )A1 (w∗ , z)A2 (z ∗ , v)ewk zk +zk vk e−zk zk −wk vk , which coincides with (1.2.29). Thus, Eq. (1.2.29) gives an explicit formula for the Wick symbol of A1 · A2 , which straightforwardly translates to the normal form of A1 · A2 . Remark 2. d-dimensional quantum oscillator algebra (1.2.2) is used for constructing the realizations of the generators of Lie algebras of classic series s (d, C), so(d, C) and sp(d, C). As an example, the generators Eij ∈ s (d, C) can be represented as  δij  † Eij = a†i aj − ak ak , Ekk = 0. (1.2.30) d k

k

Indeed, it is straightforward to check that commutation relations of the operators (1.2.30) coincide with the defining relations for algebra s (d, C) (see Section I-3.3.1 of the first book) [Eij , Ekm ] = δjk Eim − δim Ekj .

(1.2.31)

Likewise, the generators Lij ∈ so(d, C) and MAB ∈ sp(2d, C) have the following representations: Lij = a†i aj − a†j ai ,

MAB =

1 (zA zB + zB zA ), 2

(1.2.32)

where A, B, . . . = 1, 2, . . . , 2d and (z1 , . . . , z2d ) = (a†1 , . . . , a†d , a1 , . . . , ad ). Commutation relations of the operators (1.2.32) coincide with the defining

page 13

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

14

relations of algebras so(d, C) and sp(2d, C) (see Section I-3.3.1 of the first book) [Lij , Lk ] = δjk Li + δj Lki + δik Lj + δi Ljk , [MAB , MCD ] = −1 || ||JAC

where

−1 JBC MAD

+

−1 JAC MBD

+

−1 JAD MCB

(1.2.33) +

−1 JBD MCA ,

(1.2.34)

is inverse of the matrix of symplectic metric  0 Id −1 J= . = −J −1 , [zA , zB ] = JAB −Id 0

The representations (1.2.30) and (1.2.32) are called Jordan–Schwinger representations of Lie algebras s (d, C), so(d, C) and sp(d, C). The generators (1.2.30) and (1.2.32) can be written in holomorphic representation (1.2.21) and antiholomorphic representation (1.2.19). As an example, the holomorphic representation of the generators (1.2.30) of the Lie algebra s (d, C) is ∂ ∂ δij  − zk . (1.2.35) Eij = zi ∂zj d ∂zk k

We use this representation below in Section 2.2.2, where we discuss the representations with highest weight for the differential realization of the algebra s (2, C). 1.3.

Clifford Algebra and Free Fermion Algebra

In this section, we consider another important case where Dirac notations are particularly convenient. This example is used below in Sections 5.4 and 6.1 for constructing spinor representations of Lie algebra so(N, C) and its real forms, as well as for describing spinor Lie groups. Somewhat different presentation of the material of this section can be found in books [15–17]. 1.3.1.

Clifford algebra

Definition 1.3.1. Associative algebra C N over the field of complex numbers C with unit element I and generators Γm (m = 1, 2, . . . , N ) obeying the defining relations Γm Γn + Γn Γm = 2δmn I is called complex Clifford algebra.e e More

general definition of Clifford algebras is given in Section 6.1.1.

(1.3.1)

page 14

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Dirac Notations

15

The defining relations (1.3.1) stay intact under any permutation of generators Γm ∈ C N , so the numbering of generators in algebra C N can be chosen arbitrarily. Complex algebra C N has several inequivalent real forms C (p,q) (where p, q ∈ Z≥0 and p + q = N ), i.e., real algebras with the set of generators Γm (m = 1, . . . , N ) obeying defining relations Γm Γn + Γn Γm = 2ηmn I,

ηmn  = diag(1, . . . , 1, −1, . . . , −1),       p

(1.3.2)

q

where ηmn is metric of (pseudo-)Euclidean space R . Clearly, complexifications of real algebras C (p,q) are isomorphic to C N . We discuss in detail and make use of real forms C (p,q) of Clifford algebra below in Chapter 6. p,q

Proposition 1.3.1. Clifford algebra C N with finite number N of its generators is finite dimensional, and its complex dimension is dim(C N ) = 2N .

(1.3.3)

Proof. Elements of algebra C N are linear combinations of monomials Γm1 · Γm2 · · · Γmk . Each of these monomials can be decomposed into symmetric and antisymmetric parts in pairs of indices from the set {m1 , m2 , . . . , mk }. In accordance with (1.3.1), (partially) symmetrized combinations of products of generators Γm are reduced to polynomials of the order (k − 2) and less. Therefore, the basis of algebra C N consists of unit element I and all totally antisymmetric products of generators Γm : 1  (−1)p(σ) Γσ(m1 ) · Γσ(m2 ) · · · Γσ(mk ) (k = 1, . . . , N ), Γ[m1 ...mk ] = k! σ∈Sk

(1.3.4) where summation runs over all permutations σ ∈ Sk of indices {m1 , . . . , mk }, and p(σ) denotes the parity of permutation σ. Note that the set (1.3.4) with k = 1 contains all generators Γm , while due to antisymmetry of Γ[m1 ...mk ] in indices {m1 , . . . , mk } we have Γ[m1 ...mk ] = 0 for k > N . Thus, the number of basis elements Γ[m1 ...mk ] is finite, so algebra C N is finite dimensional. Making use of defining relations (1.3.1) one reduces the set of basis elements to the elements of the form ΓA = {I, Γm1 , Γm1 Γm2 , Γm1 Γm2 Γm3 , . . . , Γ1 Γ2 · · · ΓN } ≡ {I, Γm1 , Γm1 m2 , Γm1 m2 m3 , . . . , Γ12...N }, Γm1 m2 ...mk ≡ Γm1 Γm2 · · · Γmk ,

(1.3.5)

page 15

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

16

where the product of generators Γmi in each monomial Γm1 m2 ...mk is ordered: 1 ≤ m1 < m2 < m3 < · · · ≤ N. Let us find the number of independent elements (1.3.5) and in this way calculate dimension of algebra C N . The number of kth-order monomials Γm1 ...mk in (1.3.5) is given by k CN =

N! , k!(N − k)!

so the dimension of Clifford algebra C N equals dim(C N ) =

N  k=0

N! = (1 + 1)N = 2N , k!(N − k)!

(1.3.6)

which establishes (1.3.3). Let us consider the last, longest basis element in (1.3.5) which can be written either as an ordered product of all generators Γm ∈ C N or as their antisymmetric product: 1 εi ...i Γi · · · ΓiN , (1.3.7) N! 1 N 1 where we use totally antisymmetric tensor (see Section I-1.2.2 of the first book): Γ1...N = Γ1 · Γ2 · · · ΓN =

ε...ik ...i ... = −ε...i ...ik ... ,

ε12...N = +1.

(1.3.8)

In accordance with (1.3.1), element (1.3.7) obeys the identity (1.3.9) (Γ1...N )2 = (−1)N (N −1)/2 (ΓN · · · Γ2 · Γ1 )(Γ1 · Γ2 · · · ΓN ) = (−1)ν I, N  N where ν = 2 is integer part of 2 , i.e., ν is nonnegative integer, and N = 2ν or N = 2ν + 1. It is convenient to define normalized element ΓN +1 = e−i 2 ν Γ1...N = e−i 2 ν Γ1 · Γ2 · · · ΓN ∈ C N , π

π

(1.3.10)

such that Γ2N +1 = I.

(1.3.11)

Let us now consider two cases. 1. The number of generators of C N is even, N = 2ν. In this case we have, due to (1.3.1), the relation ΓN +1 · Γk = −Γk · ΓN +1

(∀k = 1, . . . , N ).

page 16

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

Dirac Notations

b3865-ch01

17

Therefore, using (1.3.11), we find that the elements Γ1 , . . . , ΓN together with ΓN +1 obey relations (1.3.1) for odd number of generators (N + 1) = 2ν + 1 of algebra C 2ν+1 . We see that algebra C 2ν can be considered as quotient of algebra C 2ν+1 with respect to the relation Γ2ν+1 = e−iπν/2 Γ1 · · · Γ2ν (or Γ2ν+1 = −e−iπν/2 Γ1 · · · Γ2ν ; recall that element ΓN +1 ∈ C 2ν is defined in (1.3.10) modulo sign). On the other hand, this means that there exist irreversible homomorphisms ρ± : C 2ν+1 → C 2ν , defined by Γi → Γi (i = 1, . . . , 2ν),

±Γ2ν+1 → e−iπν/2 Γ1 · · · Γ2ν ,

where {Γ1 , . . . , Γ2ν , Γ2ν+1 } generate algebra C 2ν+1 . Under these homomorphisms, two elements e−iπν/2 Γ1 · · · Γ2ν ∈ C 2ν+1 and Γ2ν+1 ∈ C 2ν+1 (or −Γ2ν+1 ∈ C 2ν+1 ) are mapped to one element e−iπν/2 Γ1 · · · Γ2ν ∈ C 2ν . 2. The number of generators of algebra C N is odd, N = 2ν + 1. Then we make use of (1.3.1) and find for the element (1.3.10) that ΓN +1 · Γk = Γk · ΓN +1

(∀k = 1, . . . , N = 2ν + 1),

(1.3.12)

so ΓN +1 belongs to the center of algebra C N . It follows from (1.3.11) that the central element ΓN +1 has only two eigenvalues ±1. Consider now, instead of the standard set of generators Γ1 , . . . , Γ2ν+1 ∈ C 2ν+1 , a new set {Γ1 , . . . , Γ2ν , ΓN +1 }, in which the element Γ2ν+1 is substituted by the central element (1.3.10). The new set is indeed a set of generators: any element of the complex algebra C 2ν+1 can be written in terms of the generators from the new set, once one uses (1.3.10) and writes Γ2ν+1 ≡ ΓN → e−i 2 ν ΓN +1 · Γ1 · · · ΓN −1 . π

(1.3.13)

Note that the elements {Γ1 , . . . , ΓN −1=2ν } form subalgebra C 2ν ⊂ C 2ν+1 , so algebra C 2ν+1 with new generators {Γ1 , . . . , Γ2ν , ΓN +1 } can be viewed as central extension of C 2ν . Then one recalls that the central element (1.3.10) obeys the identity Γ2N +1 = I and writes any element a ∈ C 2ν+1 as a = a1 + ΓN +1 · a2 , where a1 , a2 ∈ C 2ν . Therefore, the algebra C 2ν+1 can be decomposed as follows: C 2ν+1 = I · C 2ν + ΓN +1 · C 2ν =

(I + ΓN +1 ) (I − ΓN +1 ) · C 2ν + · C 2ν , 2 2 (1.3.14)

where 12 (I +ΓN +1 ) and 12 (I −ΓN +1 ) are central projectors in algebra C 2ν+1 , which project the whole algebra C 2ν+1 to eigenspaces of the element ΓN +1

page 17

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

18

with ΓN +1 = +I and ΓN +1 = −I, respectively. The formula (1.3.14) shows that there is isomorphism of complex Clifford algebras C 2ν+1 = C 2ν ⊕ C 2ν , whose explicit form is

Γi →

Γ2ν+1=N →

ΓN +1 →

(1.3.15)



Γi

0

0

Γi

,

i = 1, . . . , 2ν, 

e−i 2 ν Γ1 · · · Γ2ν

0

0 

−e−i 2 ν Γ1 · · · Γ2ν

π

I

0

0

−I

π

,

(1.3.16)

.

Here Γi , Γ2ν+1 , ΓN +1 ∈ C 2ν+1 , while Γi and Γi generate the first and second algebras C 2ν in (1.3.15); explicit form of the generator Γ2ν+1 follows from (1.3.13) (or from the definition (1.3.10)). Thus, we see that the study of the “odd” Clifford algebra C 2ν+1 is reduced to the study of “even” algebra C 2ν . Let us consider Clifford algebra C N with even N = 2ν and divide the set of all generators Γm ∈ C N into two groups containing generators with even and odd m. Let us also introduce the set of operators {zα , z¯α } (α = 1, . . . , ν = N/2): zα =

1 (Γ2α−1 + iΓ2α ), 2

z¯α =

1 (Γ2α−1 − iΓ2α ), 2

(1.3.17)

so that Γ2α−1 = zα + z¯α ,

Γ2α = i(¯ zα − zα ).

(1.3.18)

Due to the property (1.3.1), these operators obey [zα , zβ ]+ = 0,

[¯ zα , z¯β ]+ = 0,

[zα , z¯β ]+ = δαβ I

(∀α, β),

(1.3.19)

where bracket [·, ·]+ denotes anticommutator [a, b]+ ≡ a · b + b · a. Note that the operators zα and z¯α are nilpotent, zα2 = 0, z¯α2 = 0 (an operator is called nilpotent if it squares to zero). Definition 1.3.2. Complex algebra Aν with unity I and generators {zα , z¯α } (α = 1, . . . , ν) which obey the relations (1.3.19) is called algebra of ν free fermions or fermion oscillator algebra.

page 18

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

Dirac Notations

b3865-ch01

19

The change {Γm } → {zα , z¯β } is merely the change of the generator basis in complex algebra C 2ν . In accordance with (1.3.18), this change is invertible, = Aν . so it defines isomorphism C 2ν √ Let us set (Γ1 , . . . , Γ2ν ) = 2(z1 , . . . , zν , z¯ν , . . . , z¯1 ). Then Eqs. (1.3.1) and (1.3.19) are written as follows (cf. (1.3.2)): Γm Γn + Γn Γm = 2ηmn I,

ηnm = δn,N −m+1 ,

(1.3.20)

where ||ηmn || is antidiagonal matrix, whose elements are equal to zero except for elements in the antidiagonal, which are equal to 1. This form of Clifford algebra is used below in Section 5.4. Let ρ be an exact matrix representation of algebra C 2ν , in which all basis elements (1.3.5) are linear independent. The minimal complex matrix space with 22ν independent basis elements has dimension 22ν — this space is Mat2ν (C), whose complex dimension is 2ν × 2ν = 22ν . Therefore, the matrices in the representation ρ must be at least 2ν × 2ν , i.e., the space of the representation ρ must have dimension not less than 2ν . In the following section, we construct exact representation of algebra C 2ν whose space has dimension 2ν . This points toward the isomorphism of complex Clifford algebra C 2ν and matrix algebra Mat2ν (C). We establish this isomorphism in Chapter 6, Section 6.1.2. 1.3.2.

Fock representation for free fermion algebra

Due to isomorphism C 2ν = Aν , the problem of constructing exact irreducible representation ρ of Clifford algebra C 2ν is the same as the problem of constructing exact irreducible representation of free fermion algebra (1.3.19). The latter problem is solved by introducing the Fock space F for free fermion algebra (1.3.19). Let us describe the Fock space F making use of the Dirac notations. We define the vacuum (lowest) vector |0 ∈ F as a vector which is annihilated by all operators zα : zα |0 = 0 (∀α = 1, . . . , ν).

(1.3.21)

Operators z¯α act on |0 as creation operators, i.e., they generate, by acting on |0, all vectors of the Fock space F . Since the operator z¯α is nilpotent, it gives zero when acting twice. Therefore, the entire Fock space F , which, by construction, is the space of representation of algebra (1.3.19), consists of linear combinations of nonzerof basis vectors (k = 0, 1, . . . , ν) |α1 , . . . , αk  = z¯α1 · · · z¯αk |0 (1 ≤ α1 < α2 < · · · < αk ≤ ν).

(1.3.22)

f If any of the vectors (1.3.22) vanished, we would act on it by an appropriate combination of annihilation operators zα1 · · · zαk and get |0 = 0.

page 19

July 8, 2020

10:20

Theory of Groups and Symmetries - Vol. II

9in x 6in

b3865-ch01

Theory of Groups and Symmetries — Volume II

20

Thus, any vector |Ψ ∈ F can be written as |Ψ =

ν 



ψα1 ,...,αk |α1 , . . . , αk ,

ψα1 ,...,αk ∈ C.

(1.3.23)

k=0 α1 β2 > β1 ≥ 1).

(1.3.27)

These are generated from dual vacuum vector 0| ∈ F ∗ defined by the relations 0|¯ zβ = 0,

∀β = 1, 2, . . . , ν,

(1.3.28)

and normalized in such a way that 0|0 = 1. Using this normalization and Eqs. (1.3.25), one calculates the convolution of basis vectors in F and F ∗ : β , . . . , β2 , β1 |α1 , α2 , . . . , αk  = δk, (δβ1 α1 · · · δβ α ).

(1.3.29)

We can now define (see Section 1.1) Hermitian conjugation † of vectors in F and F ∗ as mappings F → F ∗ and F ∗ → F : |α1 , α2 , . . . , αk † = αk , . . . , α2 , α1 |, αk , . . . , α2 , α1 |† = |α1 , α2 , . . . , αk , ν   ψα∗ 1 ,...,αk αk , . . . , α2 , α1 |, |Ψ = Ψ|† . Ψ| = |Ψ† = k=0 α1