Lie Algebras in Particle Physics From Isospin to Unified Theories

ISBN-10: 0738202339
ISBN-13: 9780738202334
Edition: 2nd 1999 (Revised)
Authors: Howard Georgi
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Book details

List price: $75.00
Edition: 2nd
Copyright year: 1999
Publisher: Perseus Books Group
Publication date: 10/22/1999
Binding: Paperback
Pages: 344
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 1.034
Language: English

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

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