Group Theory and Its Applications: Volume III, Volume 3Ernest M. Loebl Group Theory and its Applications, Volume III covers the two broad areas of applications of group theory, namely, all atomic and molecular phenomena, as well as all aspects of nuclear structure and elementary particle theory. This volume contains five chapters and begins with an introduction to Wedderburn’s theory to establish the structure of semisimple algebras, algebras of quantum mechanical interest, and group algebras. The succeeding chapter deals with Dynkin’s theory for the embedding of semisimple complex Lie algebras in semisimple complex Lie algebras. These topics are followed by a review of the Frobenius algebra theory, its centrum, its irreducible, invariant subalgebras, and its matric basis. The discussion then shifts to the concepts and application of the Heisenberg-Weyl ring to quantum mechanics. Other chapters explore some well-known results about canonical transformations and their unitary representations; the Bargmann Hilbert spaces; the concept of complex phase space; and the concept of quantization as an eigenvalue problem. The final chapter looks into a theoretical approach to elementary particle interactions based on two-variable expansions of reaction amplitudes. This chapter also demonstrates the use of invariance properties of space-time and momentum space to write down and exploit expansions provided by the representation theory of the Lorentz group for relativistic particles, or the Galilei group for nonrelativistic ones. This book will prove useful to mathematicians, engineers, physicists, and advance students. |
Contents
| 1 | |
The Algebra a5 SU6 as a Physically Significant Example | 95 |
Chapter 3 Frobenius Algebras and the Symmetric Group | 143 |
Chapter 4 The HeisenbergWeyl Ring in Quantum Mechanics | 189 |
Chapter 5 Complex Extensions of Canonical Transformations and Quantum Mechanics | 249 |
Chapter 6 Quantization as an Eigenvalue Problem | 333 |
Chapter 7 Elementary Particle Reactions and the Lorentz and Galilei Groups | 369 |
| 465 | |
| 473 | |
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Common terms and phrases
algebra G analytic functions angular momentum asymptotic atomic Bargmann Hilbert space Barut basis functions classical coefficients commutes complex phase space configuration consider coordinates corresponding cosh defining matrix denote differential equation Dirac double coset Dynkin diagram eigenfunctions eigenvalues embedding energy finite Frobenius algebra given group algebras Group Theory Hamiltonian Hermitian Hilbert space hyperboloid idempotent invariant subalgebra irreducible representation isospin-free Lemma Lie algebra linear canonical transformations Lorentz amplitudes Math matric basis elements matrix matrix elements maximal subalgebras momenta nilpotent obtain operators orbital orthogonal partial-wave particles permutation perturbed phase space Phys physical point group point transformations Poisson bracket poles polynomials problem Proof properties quantization quantum mechanics quantum numbers relations relativistic scalar scattering amplitude Section semisimple algebra sequence adapted simple sinh spin-free square integrable subgroup subspace symmetric group symmetry adapted Theorem tion two-variable expansions unitary representations variables vector space wave function Weyl zero-order