Lie Algebras in Particle Physics

Transcription

1 Lie Algebras in Particle Physics Second Edition Howard Georgi S WieW Advanced Book Program A Member of the Perseus Books Group

2 Contents Why Group Theory? 1 1 Finite Groups Groups and representations Example - Z The regular representation Irreducible representations Transformation groups Application: parity in quantum mechanics Example:.S Example: addition of integers, Useful theorems Subgroups Schur's lemma * Orthogonality relations Characters Eigenstates t Tensor products Example of tensor products * Finding the normal modes * Symmetries of 2n+l-gons Permutation group on n objects Conjugacy classes Young tableaux Example our old friend Another example * Young tableaux and representations of S n 38 xili

3 xiv CONTENTS Lie Groups Generators Lie algebras The Jacobi identity The adjoint representation Simple algebras and groups States and operators Fun with exponentials 53 SU(2) J 3 eigenstates Raising and lowering operators The standard notation Tensor products J3 values add 64 Tensor Operators Orbital angular momentum Using tensor operators The Wigner-Eckart theorem Example * Making tensor operators Products of operators 77 Isospin Charge independence Creation operators Number operators Isospin generators Symmetry of tensor products Thedeuteron Superselection rules Other particles Approximate isospin symmetry Perturbation theory 88 Roots and Weights Weights More on the adjoint representation Roots Raising and lowering 93

4 CONTENTS xv 6.5 LotsofSC/(2)s Watch carefully - this is important! 95 7 SU(3) The Gell-Mann matrices Weights and roots of SU{3) Simple Roots Positive weights Simple roots Constructing the algebra Dynkin diagrams Ill 8.5 Example: G The roots of G The Cartan matrix Finding all the roots TheSU(2)s Constructing the G 2 algebra Another example: the algebra C Fundamental weights The trace of a generator MoreSU(3) Fundamental representations of SU(3) Constructing the states The Weyl group Complex conjugation Examples of other representations Tensor Methods lower and upper indices Tensor components and wave functions Irreducible representations and symmetry Invariant tensors Clebsch-Gordan decomposition Triality Matrix elements and operators Normalization Tensor operators lOThe dimension of (n,m) * The weights of (n,m) 146

5 xvi CONTENTS 10. ^Generalization of Wigner-Eckart * Tensors for SU(2) * Clebsch-Gordan coefficients from tensors * Spin S1+S2-I * Spin si +s 2 -k Hypercharge and Strangeness The eight-fold way The Gell-Mann Okubo formula Hadron resonances Quarks Young Tableaux Raising the indices Clebsch-Gordan decomposition SU(3) -+ SU(2) x C/(l) SU{N) Generalized Gell-Mann matrices SU{N) tensors Dimensions Complex representations., SU{N) SU{M) E SU{N + M) D Harmonic Oscillator Raising and lowering operators Angular momentum A more complicated example.a *7(6) and the Quark Model Including the spin SU(N) SU(M) e SU{NM) The baryon states Magnetic moments Color Colored quarks Quantum Chromodynamics Heavy quarks Flavor SU{4) is useless! 219

6 CONTENTS xvll 17 Constituent Quarks The nonrelativistic limit Unified Theories and SU(b) Grand unification Parity violation, helicity and handedness Spontaneously broken symmetry Physics of spontaneous symmetry breaking Is the Higgs real? Unification and SU{5) Breaking SU{5) Proton decay.... ' The Classical Groups The SO(2n) algebras The S0(2n + 1) algebras The Sp(2n) algebras Quaternions The Classification Theorem II-systems Regular subalgebras Other Subalgebras SO{2n + 1) and Spinors Fundamental weight of S0(2n + 1) Real and pseudo-real Real representations \.... ' Pseudo-real representations R is an invariant tensor The explicit form for R O(2n + 2) Spinors Fundamental weights of SO(2n + 2) SU(n)c SO{2n) Clifford algebras T m and R as invariant tensors Products of Ts Self-duality Example: 5O(10) 279

7 xvlli CONTENTS 23.6 The SU{n) subalgebra (10) (10) and SU(4) x SU(2) x SU{2), * Spontaneous breaking of 50(10) * Breaking 50(10) -» SU(b) * Breaking 50(10) -> 5(7(3) x SU{2) x U{1) * Breaking 50(10) -» 5 7(3) x 7(1) * Lepton number as a fourth color Automorphisms Outer automorphisms Fun with 50(8) p(2n) Weights of SU(n) Tensors for 5p(2n) Odds and Ends Exceptional algebras and octonians E 6 unification Breaking E SU{3) x 5*7(3) x 5f7(3) unification Anomalies 309 Epilogue 311 Index 312