Dynamical groups and generalized symmetries in quantum theory : with applications in atomic and particle physics

Herbert S. Green

Every law of physics is invariant under some group of transformations and is therefore the expression of some type of symmetry. Symmetries are classified as geometrical, dynamical or statistical. At the most fundamental level, statistical symmetries are expressed in the field theories of the elementary particles. This paper traces some of the developments from the discovery of Bose statistics, one of the two fundamental symmetries of physics. A series of generalizations of Bose statistics is described. A supersymmetric generalization accommodates fermions as well as bosons, and further generalizations, including parastatistics, modular statistics and graded statistics, accommodate particles with properties such as 'colour'. A factorization of elements of ggl (nb' n f) can be used to define truncated boson operators. A general construction is given for q-deformed boson operators, and explicit constructions of the same type are given for various 'deformed' algebras; these include a rather simple Q-deformed variety as well as the well known q-deformed variety. A summary is given of some of the applications and potential applications.

Evelyn Aronson, I.A. Malkin, Vladimir I. Man’ko

Florian A. Scheck

Robert J. Kolenkow

Group theory, originating from algebraic structures in mathematics, has long been a powerful tool in many areas of physics, chemistry and other applied sciences, but it has seldom been covered in a manner accessible to undergraduates. This book from renowned educator Robert Kolenkow introduces group theory and its applications starting with simple ideas of symmetry, through quantum numbers, and working up to particle physics. It features clear explanations, accompanying problems and exercises, and numerous worked examples from experimental research in the physical sciences. Beginning with key concepts and necessary theorems, topics are introduced systematically including: molecular vibrations and lattice symmetries; matrix mechanics; wave mechanics; rotation and quantum angular momentum; atomic structure; and finally particle physics. This comprehensive primer on group theory is ideal for advanced undergraduate topics courses, reading groups, or self-study, and it will help prepare graduate students for higher-level courses.

Kurt Bernardo Wolf

Group theory of the Morse potential.- Four Euclidean conformal group approach to the multiphoton processes in the H-atom.- Landau theory, symmetry breaking and the chain criterion.- The configuration dN in cubic symmetry: A symmetry adapted weak field approach.- Group theory of the effective potential.- Bloch theorem for crystals with structural distortions.- Global symmetries of spin systems defined on abelian manifolds.- Phases and conjugacy classes.- The probability density in non relativistic quantum mechanics and the (extended) Galilei group.- Quaternionic quantum mechanics and Adler's chromostatics.- Coherent angular momentum states for the isotropic oscillator.- How to measure the canonical momentum p and operators p + f(q).- A uniqueness result for the segal quantization of a classical system with symmetries.- On the singular points of the matrix elements of the resolvent operator.- Limitations on quantum measuring processes.- Classical and quantum physics on phase space: The state spaces.- Quantum dynamical semigroups and complete positivity. An application to isotropic spin relaxation.- On geometric quantization of the rigid body.- A unified theory of simple dynamical systems.- Geometry of the time-dependent variational principle in quantum mechanics.- Variational characterization of Hamiltonian vectorfields as lie algebra of canonical diffeomorphisms.- The lie theory of extended groups in classical mechanics - Is it of relevance to quantum mechanics?.- On the structure of phase space.- Quantization as mapping and as deformation.- Metric and group structures from quantum physics.- Spontaneous symmetry breaking and restricted superposition principle.- Many-body quantum mechanics as symplectic dynamics.- A nonlinear superposition principle for Riccati equations of the conformal type.- Backlund transformations, conservation laws and lie symmetries for the generalized nonlinear Schrodinger equation.- Killing tensors and variable separation for Hamilton-Jacobi equations.- Lie pseudogroups and gauge theory.- Conformal invariance and symmetry breaking for the one-dimensional wave equation.- The flavour sequence and superselection rules.- The Zitterbewegung of the electron and its compact phase space.- New experimental evidence for a new interpretation of flavor SU(3).- Baryon magnetic moments in broken SU(6).- The Racah-Wigner algebra and the masses of exotic hadrons.- Simple construction of a relativistic wave equation with arbitrary mass and spin and belinfante's 1/S-conjecture.- Poincare group, symplectic group, and relativistic extended hadrons consisting of spin 1/2 quarks.- An operator S matrix theory of the strongly interacting particles.- Generalized Galilei invariant partial wave expansions of the scattering amplitude for collisions between two particles with arbitrary spin.- Ternary algebras as the basis of a dynamical theory of subconstituents.- Solitons as sections in non trivial bundles.- Effects of heavy colored higgs scalars in grand unified theories.- Categories of nonlinear group realizations: A possible explanation for the multiple states of charge.- Classical solutions of non-linear ?-models and their quantum fluctuations.- Higgs representations for SO(n) gauges theories.- Multivortex configurations in the Abelian Higgs model.- Quark confinement in field theories with discrete gauge symmetry Z(3).- Proliferation of effective fields by spontaneous symmetry breakdown.- Explicit solutions for the scalar coupled Yang-Mills system in Minkowski space.- Dynamical supersymmetry SU(6/4) in nuclei.- Group theory of the IBA and its applications.- Bases for irreps of U(6) and three body problems.- Intrinsic sphape of nuclei.- Group theory and nuclear high spin phenomena.- SU3 symmetry in resonating group calculations. Toward a microscopic theory of 12C + 12C molecular resonances.- Collective motion, composite particle structure, and symplectic groups in nuclei.- Group theory and confrontation of nuclear collective models.- Matrix elements of generators of U(6) in a U(6)?U(3)?U(2) basis.- Hartree-Fock one-body dynamics and U(n) co-adjoint orbits.- Group-theoretical aspects of the many-body problem in nuclear theory.- The continuing quest for exact solutions of Einstein's field eouations - - new goals for the eighties.- All non-twisting N's with cosmological constant.- Can quantum effects prevent spacetime collapse?.- Complex plane representation of the Geroch group and a proof of a geroch conjecture.- The Geroch group and soliton solutions of the stationary axially symmetric Einstein's equations.- Geometry of projective asymptotic twistor space.- A truly relativistic approach of the concept of time.- Bases in Sp(2n) representation space.- Semiunitary projective representations of non-connected lie groups: A criterion for the choice of the unitary subgroup.- SU(mn) ? SU(m) x SU(n) isoscalar factors and S(f1+f2) ? S(f1) x S(f2) isoscalar factors.- Structure of enveloping algebras of low-rank groups.- Computer programs for the reduction of symmetrised nth powers of space group irreducible representations using gard's subgroup method.- Some remarks on the uniqueness and reality of Clebsch-Gordan coefficients for corepresentations.- On some special relations involving 3-jm symbols.- Two problems related to the Stone-von Neumann-Mackey uniqueness theorem.- Non-complex representations and their relation to antiunitary symmetry.- Representation theory of compact groups.- Representation theory of compact groups / Shift operators for discrete unitary representations of O(p,q).- Representation theory of compact groups / Maximal Abelian subalgebras of sp(2n,?) and their applications in physics.- Representation theory of compact groups / Polynomial space group tensors.- A permutation for SUn consistent with duality.- Integral transform representation of SL (2,R).- The theory of G? supermanifolds.- Casimir operations and the relations of the simplest supersymmetric superalgebra.- On realizations of graded lie algebras and on properties of d-polynomials.- On gauge formulations of gravitation theories.- Lagrangian formalism for superfields with internal symmetry.- Causal propagation in second order Lorentz-invariant wave equations.- The exceptional group E8 for grand unification.- Completely integrable N-body problems in three-dimension and their relativistic generalization.- Ward-Takashi identities as a necessary and sufficient condition for gauce invariance.- Supersymmetric generalization of Riemannian symmetric pairs.- Composite gauge fields and Riemannian geometry.- Gauge-theory ghosts and unitarity - Progress in the geometrical treatment.- Color in internal supersymmetry unification.- Canonical (possibly Lagrangian) realizations of the Poincare' group with increasing mass-spin trajectories.- Similarity analysis of wave propagation in axially nonlinear nonhomogeneous systems.