Lie Algebras and ApplicationsSpringer, 22.02.2007 - 196 Seiten In the second part of the 20th century, algebraic methods have emerged as a powerful tool to study theories of physical phenomena, especially those of quantal systems. The framework of Lie algebras, initially introduced by - phus Lie in the last part of the 19th century, has been considerably expanded to include graded Lie algebras, in?nite-dimensional Lie algebras, and other algebraic constructions. Algebras that were originally introduced to describe certainpropertiesofaphysicalsystem,inparticularbehaviorunderrotations and translations, have now taken center stage in the construction of physical theories. This book contains a set of notes from lectures given at Yale Univ- sity and other universities and laboratories in the last 20 years. The notes are intended to provide an introduction to Lie algebras at the level of a one-semester graduate course in physics. Lie algebras have been particularly useful in spectroscopy, where they were introduced by Eugene Wigner and Giulio Racah. Racah’s lectures were given at Princeton University in 1951 (Group Theory and Spectroscopy) and they provided the impetus for the initial applications in atomic and nuclear physics. In the intervening years, many other applications have been made. This book contains a brief account of some of these applications to the ?elds of molecular, atomic, nuclear, and particle physics. The application of Lie algebraic methods in Physics is so wide that often students are overwhelmed by the sheer amount of material to absorb. |
Inhalt
Basic Concepts | 1 |
Semisimple Lie Algebras | 15 |
3 | 26 |
Irreducible Bases Representations | 39 |
Casimir Operators and Their Eigenvalues | 63 |
Tensor Operators | 75 |
Boson Realizations | 91 |
Fermion Realizations | 131 |
Differential Realizations | 147 |
Matrix Realizations 155 | 154 |
Spectrum Generating Algebras and Dynamic Symmetries | 163 |
Degeneracy Algebras | 173 |
189 | |
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Abelian algebra g algebra so(3 Algebras and Applications Algorithm annihilation operators antisymmetric basis Berlin Heidelberg 2006 bilinear products boson operators branching called canonical chain Cartan Casimir operator characterized commutation relations complex Consider constructed coordinates Coulomb problem coupling coefficients creation and annihilation degeneracy algebra denoted Differential realization discussed in Chap dynamic symmetries dynamical algebra eigenvalues Example fermion fermion operators formula Francesco Iachello fundamental representations Gel'fand GL(n Hamiltonian harmonic oscillator integer introduce irreducible representations isomorphic Ji J2 lattice of algebras Lie algebra u(n Lie group linear matrix elements matrix realization metric tensor missing label non-canonical chains notation Notes Phys number field obtained particles physics quadratic Casimir operator quantum mechanics quantum numbers Racah form real Lie algebra recoupling representations of so(3 root diagram rotation semisimple Lie algebras so(n sp(n su(n subalgebra chains tensor operator tensor product transform vector Wigner Young tableau