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.
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.
Total Cross Sections in Non-Relativistic Scattering Theory.- Classical-Quantum Correspondence in Non-Linear Systems.- Real Axis Asymptotics and Estimates of Hamiltonian Resolvent Kernels.- Statistical Inference in Quantum Mechanics.- Schrodinger Operators with Oscillating Potentials.- Time Delay and Resonance in Simple Scattering.- A Study of the Helmholtz Operator.- Invariant Manifolds and Bifurcations in the Taylor Problem.- On the Dimension of a Finite Difference Approximation to Divergence-Free Vectors.- Intramolecular Dynamics in the Quasiperiodic and Stochastic Regimes.- Problems in the Semiclassical Quantization of Integrable and Nonintegrable Classical Dynamical Systems.- Nonselfadjoint Operators in Diffraction and Scattering.- One-Dimensional Crystals in an External Field.- Dynamics of Forced Coupled Oscillators: Classical Phenomenology of Infrared Multiphoton Absorption.- Irreversibility and Stochasticity of Chemical Processes.- On the Representations of the Local Current Algebra and the Group of Diffeomorphisms (I).- Regular and Chaotic Regimes in Quantum Mechanics.- Barrier Penetration and Exponential Decay in the Stark Effect.- A Nonlinear Schrodinger Equation Yielding the Shape of Molecules by Spontaneous Symmetry Breaking.- Classical Limit of the Number of Quantum States.- Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report.- On a Generalized Weyl-Von Neumann Converse Theorem.- Scattering Theory in Many-Body Quantum Systems. Analyticity of the Scattered Matrix.- Existence, Uniqueness, Stability and Calculation of the Stationary Regimes in Some Nonlinear Systems.- Weighted Trigonometric Approximations in L2(Rn).- The Phenomenon of Magnetic Pairing and Exactly Soluble Models of Magnetic Resonances.- Dyons Solutions to Yang-Mills Equations.- Asymptotic Resonance Properties of the Finite-Dimensional Friedrichs Model.- Self-Consistent Field Methods for Metastable Electronic States: A Promising Extension of the Complex Coordinate Technique.- Timestep Control for the Numerical Solutions of Initial-Boundary-Value Problems.- Quantum Mechanical Angular Distributions and Group Representations on Banach Spaces.- Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients.- Orthogonal Laurent Polynomials and Gaussian Quadrature.- Measurement Theory with Instruments Treated Partially Quantum Mechanically.- Transition Maps and Locality.- The Feynman-Kac Formula for Boson Wiener Processes.- On Nonunitary Equivalence Between Unitary Group of Dynamics and Contraction Semigroups of Markov Processes.
This book covers the entire span of quantum mechanics whose developments have taken place during the early part of the twentieth century up till the present day. We start with the Rutherford-Bohr model of the atom followed by Schrodinger's wave mechanics with its application to the solution of calculating the energy spectrum of a particle in a box, the harmonic oscillator and finally the hydrogen atom. Heisenberg's matrix mechanics and its duality with Schrodinger's wave mechanics, quantum mechanics in the interaction picture. Dirac's relativistic theory of the electron exhibiting the spin of the electron as a relativistic effect when it interacts with an external electromagnetic field. Feynman's path integral approach to non-relativistic quantum mechanics with is a marvellous intuitive interpretation as a sum over paths and how classical mechanics is obtained from its limit as Planck' constant tends to zero, methods for computing the spectra of the Dirac Hamiltonian in a radial potential, quantum field theory as developed by Feynman, Schwinger, Tomonaga and Dyson for describing the interaction between electrons, positrons, and photons via propagators using both the operator theoretic expansions and Feynman's path integral. We also introduce time independent and time dependent perturbation theory in quantum mechanics with applications to quantum gate design for quantum computers forming a major part of the research conducted by the author's research group, Quantum noise introduced into the Schrodinger and Dirac's equation based on the Hudson-Parthasarathy quantum stochastic calculus in Boson Fock space, scattering theory and wave operators with applications to quantum gate design, some aspects of second quantization like the interpretation of Boson Fock space in terms of harmonic oscillator algebras and the BCS theory of superconductivity, Wigner-Mackey-Frobenius theory of induced representations of a group with applications to Wigner's theory of particle classification, Dirac's equation in a gravitational field and Yang-Mills non-Abelian gauge theories with application to the construction of unified quantum field theories and finally, the more recent theory of super-symmetry which is a Boson-Fermion unification theory. We have discussed the statistics of Boson's, Fermions and Maxwell-Boltzmann based on entropy maximization. The book is written in problem-solution format and it would be of use to physicists and engineers interested respectively in developing unified field theories and in the design of quantum gates. Note: T&F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.
I: Stochastic Quantum Mechanics.- 1 / Nonrelativistic Stochastic Quantum Mechanics.- 1.1 Stochastic Values and Stochastic Phase Spaces.- 1.2. Sharp Localizability and Systems of Imprimitivity.- 1.3. Stochastic Localizability and Systems of Co variance.- 1.4. Phase-Space Representations of the Galilei Group.- 1.5. Localizability in Nonrelativistic Stochastic Phase Spaces.- 1.6. Stochastic Phase-Space Systems of Covariance.- 1.7. Resolution Generators and Gauge Freedom in Stochastic Phase Spaces.- 1.8. Stochastic Probability Currents and Their Sharp-Point Limits.- 1.9. Propagators in Nonrelativistic Stochastic Phase Spaces.- 1.10. Quantum Stochastic Processes and Phase-Space Path Integrals.- 1.11. Semiclassical Propagators in Stochastic Phase Spaces.- 1.12. The Physical Nature and the Measurement of Proper Wave Functions and of Stochastic Position and Momentum.- Notes.- 2 / Relativistic Stochastic Quantum Mechanics.- 2.1. The Nonexistence of Relativistic Probability Currents at Sharp Space time Points.- 2.2. The Unfeasibility of Sharp Localization in Relativistic Quantum Mechanics.- 2.3. Phase-Space Representations of the Poincare Group.- 2.4. Resolution Generators and Relativistic Phase-Space Representations.- 2.5. Localizability in Relativistic Stochastic Phase Spaces.- 2.6. Systems of Covariance in Relativistic Stochastic Phase Spaces.- 2.7. Relativistic Canonical Commutation Relations and Gauge Freedom.- 2.8. Relativistically Covariant and Conserved Probability and Charge Currents.- 2.9. Propagators in Relativistic Stochastic Phase Spaces.- 2.10. Relativistic Extended Quantum Particles in External Electromagnetic Fields.- Notes.- 3 / Statistical Mechanics on Stochastic Phase Spaces.- 3.1. Phase-Space Representations of Quantum Statistical Mechanics.- 3.2. Informational Equivalence and Completeness of POV Measures.- 3.3. Generalized Stochastic Phase-Space Systems of Covariance.- 3.4. Classical and Quantum Mechanics in L2 (?N).- 3.5. Classical and Quantum Statistical Mechanics in Master Liouville Space ?(?N).- 3.6. Quantum Observables as Density Distributions on Stochastic Phase Spaces.- 3.7. Unified Formulations of Classical and Quantum Boltzmann Equations and Scattering Cross-Sections.- Notes.- II: Quantum Spacetime.- 4 / Reciprocity Theory and the Geometrization of Stochastic Quantum Mechanics.- 4.1. The Operational Foundations of Special and General Relativity.- 4.2. Coordinate Fluctuation Amplitudes for Coherent Flows of Quantum Test Particles.- 4.3. Quantum Lorentz Frames and Minkowski Quantum Spacetime.- 4.4. General Quantum Spacetimes and Stochastic Geometries.- 4.5. Reciprocity Theory and Born's Quantum Metric Operator.- 4.6. Unitary Internal Symmetries and Generalized Representations of Relativistic Canonical Commutation Relations.- Notes.- 5 / Field Theory on Quantum Spacetime.- 5.1. Free Scalar Fields on Minkowski Quantum Spacetime.- 5.2. LSZ Formalism and Causality on Quantum Spacetime.- 5.3. Lagrangian Field Theory on Quantum Spacetime.- 5.4. Dirac Bispinors and Free Fields on Quantum Spacetime.- 5.5. Photon Localizability and the Electromagnetic Field on Quantum Spacetime.- 5.6. Reciprocally Invariant Quantum Electrodynamics for Extended Fermions.- Notes.- Appendix A / Elements of Measure Theory.- Appendix B / Elements of Operator Theory on Hilbert Spaces.- Notes.- References.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities
We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schrödinger, Pauli and Dirac equations, as well as for the Maxwell equations. We explain in detail all basic theoretical concepts. We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the Hamilton-Jacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, four conservation laws (energy, momentum, angular momentum and charge), Lie algebra of angular momentum and spherical functions, scattering theory (limiting amplitude principle and limiting absorption principle), the Lienard-Wiechert formulas, Lorentz group and Lorentz formulas, Pauli theorem and relativistic covariance of the Dirac equation, etc. We give a detailed oveview of the conceptual development of the quantum mechanics, and expose main achievements of the ``old quantum mechanics'' in the form of exercises. One of our basic aim in writing this book, is an open and concrete discussion of the problem of a mathematical description of the following two fundamental quantum phenomena: i) Bohr's quantum transitions and ii) de Broglie's wave-particle duality. Both phenomena cannot be described by autonomous linear dynamical equations, and we give them a new mathematical treatment related with recent progress in the theory of global attractors of nonlinear hyperbolic PDEs.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.