GROUP REPRESENTATION THEORY FOR PHYSICISTS

Transcription

1 GROUP REPRESENTATION THEORY FOR PHYSICISTS JIN-QUAN CHEN Vfe World Scientific wl Singapore New Jersey London Hong Kong

2 Contents Foreword Preface Glossary v vii xix Introduction 1 Chapter 1 Elements of Group Theory 1.1. Definition of group Permutation group S n Definition of S n Permutation expressed in terms of cycles and transpositions Subgroup Isomorphism and homomorphism Conjugate classes Cosets, Lagrange theorem Invariant subgroup Factor group* Direct product and semi-direct product 17 Chapter 2 Group Representation Theory 2.1. Linear vector space Definition of linear vector space Covariant and contravariant Metrie tensor Linear Operator and its representation Complete set of commuting Operators Eigenspace of self-adjoint Operators Complete set of commuting Operators (CSCO) Group representations Unitary representation Regulär Rep and group algebra Definition of regulär representation 30 IX

3 x Group Representation Theory for Physicists Group space Group algebra Space of functions on the group Equivalent representations and characters Reducible and irreducible representations Subduced and induced representations Schur's lemma Appendix: Non-orthonormal basis Two definitions of the representation of an Operator Representation of an adjoint Operator Representation of a unitary Operator Representation transformation Eigenvectors of a self-adjoint Operator 43 Chapter 3 Representation Theory for Finite Groups 3.1. Class space and class Operators Class Operators Class algebra Space of functions on classes The natural representation of a class algebra The first kind of CSCO of G (CSCO-I) Reduction of the natural representation of the class algebra The CSCO-I of G The CSCO of a direct product group G± x G The case of non-self-adjoint class Operators The groups S$ and CQ V Projection Operator ph Decomposition of the regulär rep into inequivalent reps of G Label for irreps Decomposition of an arbitrary rep space Reduction of representations of C3«, #2 and The group C 3v The group S The group S3 in the configuration a 2 ß The group S 3 in the configuration aßi State permutation group Reduction of the regulär rep of Intrinsic group Definition of the intrinsic group Regulär representation of intrinsic group Action of intrinsic group elements on functions on the group Properties of the intrinsic group Some remarks Intrinsic State (regulär rep case) Intrinsic permutation group and state permutation group 78

4 Contents xi 3.8. CSCO-II and CSCO-III of G Füll reduction of the regulär representation Eigenvectors of the CSCO-III of G The representations >(")* (G) and 0<">m(G) The Standard phase choice for Pm The irreducibility of >(") (G) The EFM for G D G{s) irreducible matrices Generalized irreducible matrices Reduction of the regulär representation in configuration space Example: the group Generalized projection Operator Properties of P^l A recursive method for P^ Eigenfunction method for characters The applications of simple characters Reduction of non-regular reps (EFM for irreducible basis) Canonical subgroup chains with r v = Canonical subgroup chain with r v > Non-canonical subgroup chain Projection Operator method Kronecker product of representations Clebsch-Gordan series Symmetrized and antisymmetrized Squares The CG coefficients Definition and properties of the CG coefficients The EFM for CG coefficients Isoscalar factors Irreducible tensor of a group G Definition of irreducible tensor The Wigner-Eckart theorem Symmetries of the CG coefficients and ISF Applications of group theory in quantum mechanics When G is the symmetry group of the Hamiltonian Splitting of the energy level due to a perturbation Dynamical symmetry General case Selection rule Summary 114 Chapter 4 Representation Theory of the Permutation Group 4.1. Partition, Young diagrams and eigenvalues of CSCO-I Characters of permutation group Character of conjugate representations Branching law, Young-Yamanouchi basis and Young tableaux The Young-Yamanouchi basis 120

5 xii Group Representation Theory for Physicists 4.4. Yamanouchi matrix elements The CSCO-II of permutation groups The EFM for Yamanouchi basis (I) The CSCO-III of the permutation group CSCO-III The labeling for Yamanouchi basis of S n and S n Phase Convention and the principle term The matrix elements of conjugate irreps The symmetrizer and anti-symmetrizer Quasi-standard basis of the permutation group State permutation group (for the case with repeated state labeis) The quasi-standard basis of the permutation group Projection Operator and quasi-standard basis The labelling of the quasi-standard basis The EFM for Standard basis (II) The inner product and the CG series of permutation groups Calculation of the CG coefficients of the permutation group Properties of the CG coefficients of permutation groups Tables of the CG coefficients for Outer-product of the permutation group and the Littlewood rule The calculations of the IDC of S n The properties of IDC Tables of the IDC for S 3 S !+., => S ni S, irreducible basis The S ni+n, Z> S ni S n, subduced basis Transformation between the Standard and non-standard basis of S n The calculation of the SDC Tables of the SDC for S 3 S S n D S ni S n2 isoscalar factor* The S n ^ S _i ISF Phase Convention The properties of the S n Z> S^j ISF Special case Tables of the S 1 «Z) 5 _i ISF TheS^+n, ^ S ni S n, ISF* Appendix: Derivation of Yamanouchi matrix elements by the EFM 202 Chapter 5 Lie Groups 5.1. Tensor Vector (Tensor of rank one) Tensors with rank higher than one Metrie tensor Metrie space Definition of Lie group and some examples Lie algebra 211

6 Contents xiii Generators of Lie groups Finite transformations Correspondence between Lie groups and Lie algebras Linear transformation groups Infinitesimal Operators for linear transformation groups Metrie tensor in n-dimensional space and infinitesimal Operators Unitary groups Infinitesimal Operators of SU n The group U(n, m) Orthogonal group O n The real orthogonal group 0(n, m) Sympletic group Infinitesimal Operators in group parameter space Isomorphism and anti-isomorphism of Lie groups and Lie algebras Invariant Integration Representation of compact Lie groups Fundamental representation Adjoint representation Metrie tensor in the r-dimensional vector space The invariants and Casimir Operators of Lie groups Intrinsic Lie groups Definition and Interpretation of the intrinsic Lie group Infinitesimal Operators of intrinsic groups in group parameter space The CSCO approach to the rep theory of Lie group Irreducible tensors of Lie groups and intrinsic Lie groups The Cartan-Weyl basis Theorems on roots Root diagram The Dynkin diagram and simple root representation The Cartan matrix Theorems on weights The Dynkin representation Algorithms for Computing the roots and weights The fundamental weight System Fundamental weight System representation and Cartesian representation Comparison of the different representations The characters and CG series of Lie algebras The characters of Lie groups The CG series of Lie groups 279 Chapter 6 The Rotation Group 6.1. The differential Operators of J x,y,z and J x,y,z in group parameter space Irreps of the SO2 group The CSCO-I and characters of S The CSCO-III and irreducible matrix element of S

7 xiv Group Representation Theory for Physicists 6.5. The CSCO-II and irreducible bases of S The intrinsic State of S The projection State of S Irreducible tensors of S0 3 and SO The irreducible tensor of the adjoint rep of S0 3 and SO Irreducible tensor of S0 3 and S0 3 in general cases 295 Chapter 7 The Unitary Groups 7.1. Unitary groups in coordinate space and State space Relations between the CSCO-I and generators of unitary groups and permutation groups The Gel'fand invariants The relation between the CSCO-I of permutation groups and unitary groups Relations between the generators of unitary groups and permutation groups The CSCO-II and CSCO-III of U n and SU n The Gel'fand basis and Gel'fand matrix elements The Gel'fand basis of unitary groups and quasi-standard basis of permutation groups The CSCO-II of unitary groups and CSCO of the broken chains of permutation groups The labeling and finding of the Gel'fand basis The contragredient representation The CG coefficients of SU n group The CG coefficients of U n and the IDC of the permutation group The procedure for evaluating the SU n CG coefficients Phase Convention The CG coefficients of SU n and the S f D S fl S f, irreducible basis The SU mn 2> SU m x SU n irreducible basis The CG coefficients of S f and the SU mn D SU m x SU n irreducible basis The irreps ([Pi]t[pä]) of the groups SU m and SU n contained in the irrep [i>] of SU mn Representation transformation between the SU mn D SU m X SU n irreducible basis and the SU mn Gel'fand basis The 5t/ in,, D SU ni x SU ni X SU n, irreducible bases and the Racah coefficients of permutation groups* SU nin^n3ni D SU ni X SU n2 X SU n, x SU ni irreducible basis and the 9u coefficients of the permutation group* SU m+n D SU m SU n irreducible basis The IDC of permutation groups and SU m+n D SU m (81 SU n irreducible bases The content of irreps ([ui], [v 2 \) of SU m <8> SU n in the irrep of SU m+n 338

8 Contents xv The representation transformation between the SU m+n D SU m SU n irreducible basis and the Gel'fand basis of SU m + n The isoscalar factors and the fractional parentage coefficients Isoscalar factors The orbital fractional parentage coefficients (CFP) The spin-isospin CFP The total CFP Eigenfunction method for evaluating the CFP Sf Z> Sf 1 (g) Sf 2 Sf, irreducible basis and SU n Racah coefficients* Sf D 5/j ig) Sf 7 <g> Sf 3 Sft irreducible basis and the 9i/ coefficients of SU n * The 9z/ coefficients of SU n Evaluation of the Racah coefficients and 9u coefficients of SU n SU mn D SU m x SU n CFP SU mn 3 SU m x SU n CFP and S fl+h O S fl S fi ISF The evaluation of the SU mn D SU m X SU n many-particle CFP The symmetries of the SU mn D SU m x SU n ISF More examples C/ 4 (2i+i) => {SU 2l+1 D S0 3 ) x (SU 4 D SU 2 x SU 2 ) ISF and total CFP The SU m+n D SU m SU n CFP* The Sf D Sf-i outer-product ISF (The SU f D SU f -x U x ISF) The S f ^ S fll S fai outer-product ISF {SUf D SU fl2 SU 34 ISF) The SU m+n D SU n ISF and S f D 5 /l3 5 /j4 outer-product ISF The evaluation of S U m+n D Si/,«5 Zi n ISF Symmetries of the SU m+n D SU m SU n ISF The SU n singlet factor Second quantized expression for the CFP One-particle CFP Two-particle CFP The CFP in the interacting boson model 378 Chapter 8 The Point Groups 8.1. Basic Operations of point groups Basic Operations and their faithful reps Some commonly used point groups The CSCO-I and CSCO-II of point groups The conventional labeling for point group irreps (Mullikan notation) The CSCO-II for the commonly used point groups Irreducible matrix elements and irreducible basis of point groups G Irreducible matrices Irreducible basis The Splitting of atomic levels and the O3 D G D G(s) basis The CG coefficients of point groups The CG series of point groups The CG coefficients of point groups 403

9 xvi Group Representation Theory for Physicists 8.6. Molecular orbital theory Single electron SALC Double point group The rep group Symmetry adapted basis for double point groups 421 Chapter 9 Applications of Group Theory to Many-Body Systems 9.1. Pure configuration shell model One-body Operator Two-body Operator Antisymmetric wave functions for a A+B System Transformation between symmetry bases and physical bases in the quark model The CFP for a mixed configuration The dynamic symmetry modeis of nuclei The quasispin model The proton-neutron quasispin model The Elliott model The interacting boson model The SO& and SPQ fermion dynamic symmetry model The generators of SO s and SP The SO s and SP 6 Hamiltonian The FDSM wave functions The molecular shell model The Hamiltonian as a function of infinitesimal Operators of the unitary group Spin-free approximation 458 Chapter 10 The Space Groups The Euclidean group Definition of the Euclidean group Properties of the Euclidean group Operators The lattice group The space group The point group P and the crystal System The Bravais lattice Operators of the space group The properties of group Operators Example: Group D\i The reciprocal lattice vectors Irreps of the lattice group The Brillouin zone The electron state in a periodic potential Representation space of the space group The little group G(k) 475

10 Contents xvii The representation groups Gk and G'k The rep group G k The rep group G' k Special cases of the rep group G' k The irreducible basis and matrices of G' k The group table of G k The CSCO-II and CSCO-III of G k The irreps of G k and the projective irreps of Go(k) The irreducble basis ofg(fc) Examples: the point W of 0 7 h Seeking the CSCO and the characters of the point W of the space group Ol Seeking the CSCO-I from the existing character table Constructing irreps of the rep group G k. The point W of 0 7 h Irreducible basis and representations of the space group The k star The induced rep A simple algorithm for füll rep matrices The G D G(k (T ) O G^) D T irreducible basis The irreducible basis and matrices of C\ v General star: p= (pi,p2,ps) The star T : p = (0,0,0) The star S :p= (pi,0,0) The star X:p = (1/2,0,0) The Clebsch-Gordan coefficients of space groups* The CG series The calculation of the CG coefficients Relative phase of the CG coefficients The füll CG coefficients of space groups Summary of the eigenfunction method for space group CG coefficients Examples: obtaining space group Clebsch-Gordan coefficients* The CG coefficients of 0 for *X(1) *X{2) *X{v") The CG coefficients of O r h for *X(1) *W(l) *A(i/") The double space groups Appendices 513 Appendix Table AI. Dimensions of irreps of the permutation group Sj (/ < 6) and the unitary groups SU n (n < 6) 515

11 xviii Group Representation Theory for Physicists Table A2. Phase factors e\{y\viv} for the permutation group IDC and SU n CG coefficients 515 References 517 Index 531 Contents of Some Important Tables Table The new and old labelling schemes for irreps of permutation groups Table 3.9. The Standard basis V'm [Pm ) of 53 and 3 and the Standard matrix elements 89 Table The phase factor Am, Young tableau Ym and the corresponding eigenvalues A = 27=3(2/ - 5)A/ 122 Table Yamanouchi matrix elements of adjacent transpositions for 52 5s Table 4.8. Normalization factors R^k[ui) 146 Table CG series of S Table Tables of the CG coefficients of the permutation groups S3 S$ 161 Table Outer-product reduction rule 169 Table The ([i^] [i/ 2 ]) \v\ IDC of the permutation groups 175 Table The SDC for S 3 - S Table Tables of S n ^ 5 n _i ISF for n = 3-5 (i.e., tables of the SU mn D SU m X SU n single particle CFP for arbitrary m and n) 196 Table 8.3. The CSCO-I and CSCO-II of point groups and their eigenfunctions: 1. D2,D 2 h', 2. C 2 ; 3. D 2 <i; 4. Dz,D 3 h\ 5. C 3v,D 3 d; 6. C 4 ; 7. D 4( j; 8. Dt,D ih ; 9.Ds,Dsh] 10. C &v,dsd] 11. C ev ; 12. D 6d ; 13. DQ,DQH\ 14. Coc.Ax,«,; 15. T,T h ; 16.T d ; 17. 0,O h 390 Table Group table for the point group O and double point group 0 T, and the correspondence of notations for the group elements 513 Table The Operations of the point group elements on the primitive translation tj 514