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| Why Group Theory? | |
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| Finite Groups | |
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| Groups and representations | |
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| Example - Z[subscript 3] | |
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| The regular representation | |
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| Irreducible representations | |
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| Transformation groups | |
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| Application: parity in quantum mechanics | |
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| Example: S[subscript 3] | |
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| Example: addition of integers | |
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| Useful theorems | |
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| Subgroups | |
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| Schur's lemma | |
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| * Orthogonality relations | |
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| Characters | |
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| Eigenstates | |
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| Tensor products | |
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| Example of tensor products | |
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| * Finding the normal modes | |
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| * Symmetries of 2n+1-gons | |
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| Permutation group on n objects | |
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| Conjugacy classes | |
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| Young tableaux | |
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| Example -- our old friend S[subscript 3] | |
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| Another example -- S[subscript 4] | |
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| * Young tableaux and representations of S[subscript n] | |
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| Lie Groups | |
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| Generators | |
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| Lie algebras | |
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| The Jacobi identity | |
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| The adjoint representation | |
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| Simple algebras and groups | |
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| States and operators | |
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| Fun with exponentials | |
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| SU(2) | |
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| J[subscript 3] eigenstates | |
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| Raising and lowering operators | |
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| The standard notation | |
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| Tensor products | |
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| J[subscript 3] values add | |
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| Tensor Operators | |
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| Orbital angular momentum | |
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| Using tensor operators | |
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| The Wigner-Eckart theorem | |
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| Example | |
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| * Making tensor operators | |
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| Products of operators | |
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| Isospin | |
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| Charge independence | |
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| Creation operators | |
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| Number operators | |
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| Isospin generators | |
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| Symmetry of tensor products | |
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| The deuteron | |
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| Superselection rules | |
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| Other particles | |
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| Approximate isospin symmetry | |
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| Perturbation theory | |
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| Roots and Weights | |
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| Weights | |
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| More on the adjoint representation | |
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| Roots | |
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| Raising and lowering | |
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| Lots of SU(2)s | |
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| Watch carefully - this is important! | |
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| SU(3) | |
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| The Gell-Mann matrices | |
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| Weights and roots of SU(3) | |
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| Simple Roots | |
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| Positive weights | |
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| Simple roots | |
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| Constructing the algebra | |
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| Dynkin diagrams | |
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| Example: G[subscript 2] | |
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| The roots of G[subscript 2] | |
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| The Cartan matrix | |
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| Finding all the roots | |
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| The SU(2)s | |
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| Constructing the G[subscript 2] algebra | |
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| Another example: the algebra C[subscript 3] | |
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| Fundamental weights | |
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| The trace of a generator | |
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| More SU(3) | |
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| Fundamental representations of SU(3) | |
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| Constructing the states | |
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| The Weyl group | |
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| Complex conjugation | |
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| Examples of other representations | |
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| Tensor Methods | |
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| Lower and upper indices | |
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| Tensor components and wave functions | |
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| Irreducible representations and symmetry | |
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| Invariant tensors | |
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| Clebsch-Gordan decomposition | |
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| Triality | |
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| Matrix elements and operators | |
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| Normalization | |
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| Tensor operators | |
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| The dimension of (n,m) | |
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| * The weights of (n,m) | |
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| Generalization of Wigner-Eckart | |
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| * Tensors for SU(2) | |
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| * Clebsch-Gordan coefficients from tensors | |
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| * Spin s[subscript 1] + s[subscript 2] - 1 | |
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| * Spin s[subscript 1] + s[subscript 2] - k | |
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| Hypercharge and Strangeness | |
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| The eight-fold way | |
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| The Gell-Mann Okubo formula | |
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| Hadron resonances | |
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| Quarks | |
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| Young Tableaux | |
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| Raising the indices | |
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| Clebsch-Gordan decomposition | |
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| SU(3) [right arrow] SU(2) [times] U(1) | |
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| SU(N) | |
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| Generalized Gell-Mann matrices | |
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| SU(N) tensors | |
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| Dimensions | |
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| Complex representations | |
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| SU(N) [multiply sign in circle] SU(M) [set membership] SU(N +M) | |
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| 3-D Harmonic Oscillator | |
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| Raising and lowering operators | |
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| Angular momentum | |
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| A more complicated example | |
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| SU(6) and the Quark Model | |
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| Including the spin | |
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| SU(N) [multiply sign in circle] SU(M) [set membership] SU(NM) | |
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| The baryon states | |
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| Magnetic moments | |
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| Color | |
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| Colored quarks | |
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| Quantum Chromodynamics | |
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| Heavy quarks | |
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| Flavor SU(4) is useless! | |
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| Constituent Quarks | |
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| The nonrelativistic limit | |
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| Unified Theories and SU(5) | |
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| Grand unification | |
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| Parity violation, helicity and handedness | |
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| Spontaneously broken symmetry | |
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| Physics of spontaneous symmetry breaking | |
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| Is the Higgs real? | |
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| Unification and SU(5) | |
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| Breaking SU(5) | |
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| Proton decay | |
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| The Classical Groups | |
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| The SO(2n) algebras | |
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| The SO(2n + 1) algebras | |
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| The Sp(2n) algebras | |
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| Quaternions | |
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| The Classification Theorem | |
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| II-systems | |
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| Regular subalgebras | |
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| Other Subalgebras | |
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| SO(2n + 1) and Spinors | |
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| Fundamental weight of SO(2n + 1) | |
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| Real and pseudo-real | |
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| Real representations | |
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| Pseudo-real representations | |
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| R is an invariant tensor | |
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| The explicit form for R | |
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| SO(2n + 2) Spinors | |
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| Fundamental weights of SO(2n + 2) | |
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| SU(n) [subset or is implied by] SO(2n) | |
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| Clifford algebras | |
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| [Gamma][subscript m] and R as invariant tensors | |
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| Products of [Gamma][subscript s] | |
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| Self-duality | |
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| Example: SO(10) | |
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| The SU(n) subalgebra | |
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| SO(10) | |
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| SO(10) and SU(4) [times] SU(2) [times] SU(2) | |
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| * Spontaneous breaking of SO(10) | |
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| * Breaking SO(10) [right arrow] SU(5) | |
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| * Breaking SO(10) [right arrow] SU(3) [times] SU(2) [times] U(1) | |
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| * Breaking SO(10) [right arrow] SU(3) [times] U(1) | |
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| * Lepton number as a fourth color | |
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| Automorphisms | |
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| Outer automorphisms | |
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| Fun with SO(8) | |
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| Sp(2n) | |
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| Weights of SU(n) | |
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| Tensors for Sp(2n) | |
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| Odds and Ends | |
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| Exceptional algebras and octonians | |
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| E[subscript 6] unification | |
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| Breaking E[subscript 6] | |
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| SU(3) [times] SU(3) [times] SU(3) unification | |
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| Anomalies | |
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| Epilogue | |
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| Index | |