Symmetries in physics. Proceedings of the international symposium, held in honor of Professor Marcos Moshinsky at Cocoyoc, Morelos, México, June, 3-7, 1991 (original) (raw)
Related papers
The harmonic oscillator behind all aberrations
2010
The group-theoretical structure of the harmonic oscillator appears in many guises. Originally developed by Marcos Moshinsky among several others for applications in nuclear physics, we point out here that the harmonic oscillator structure appears in aberrations of geometric optics, particularly in their classification by rank, symplectic spin and weight. And further, the finite harmonic oscillator appears again in the nonlinear transformations of finite Hamiltonian systems, when applied to the parallel processing of signals.
On a hidden symmetry of quantum harmonic oscillators
Journal of Difference Equations and Applications, 2013
We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.
Commensurate harmonic oscillators: Classical symmetries
Journal of Mathematical Physics, 2002
The symmetry properties of a classical N -dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurate oscillator possesses the same number of globally defined constants of motion as an isotropic oscillator. In both cases invariant phase-space functions form the algebra su(N ) with respect to the Poisson bracket. In the isotropic case, the phase-space flows generated by the invariants can be integrated globally to a set of finite transformations isomorphic to the group SU (N ). For a commensurate oscillator, however, the group SU (N ) of symmetry transformations is found to exist only on a reduced phase space, due to unavoidable singularities of the flow in the full phase space. It is therefore crucial to distinguish carefully between local and global definitions of symmetry transformations in phase space. This result solves the longstanding problem of which symmetry to associate with a commensurate harmonic oscillator.
HNPS Proceedings
The concept of bisection of a harmonic oscillator or hydrogen atom, vised in the past in establishing the connection between U(3) and 0(4), is generalized into multisection (trisection, tetrasection, etc). It is then shown that all symmetries of the N-dimensional anisotropic harmonic oscillator with rational ratios of frequencies (RHO), some of which are underlying the structure of superdeformed and hyperdeformed nuclei, can be obtained from the U(N) symmetry of the corresponding isotropic oscillator with an appropriate combination of multisections. Furthermore, it is seen that bisections of the N-dimensional hydrogen atom, which possesses an 0(N+1) symmetry, lead to the U(N) symmetry, so that further multisections of the hydrogen atom lead to the symmetries of the N-dim RHO. The opposite is in general not true, i.e. multisections of U(N) do not lead to 0(N+1) symmetries, the only exception being the occurence of 0(4) after the bisection of U(3).
Geometry, Symmetries, and Classical Physics: A Mosaic
Geometry, Symmetries, and Classical Physics: A Mosaic, 2021
This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume. - - Key features: -> Contains a modern, streamlined presentation of classical topics, which are normally taught separately. -> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity. -> Focuses on the clear presentation of the mathematical notions and calculational technique. - - - Table of Contents: - Chapter 1. Manifolds and Tensors. - Chapter 2. Geometry and Integration on Manifolds. - Chapter 3. Symmetries of Manifolds. - Chapter 4. Newtonian Mechanics. - Chapter 5. Lagrangian Methods and Symmetry. - Chapter 6. Relativistic Mechanics. - Chapter 7. Lie Groups. - Chapter 8. Lie Algebras. - Chapter 9. Representations. - Chapter 10. Rotations and Euclidean Symmetry. - Chapter 11. Boosts and Galilei Symmetry. - Chapter 12. Lorentz Symmetry. - Chapter 13. Poincare Symmetry. - Chapter 14. Conformal Symmetry. - Chapter 15. Lagrangians and Noether's Theorem. - Chapter 16. Spacetime Symmetries of Fields. - Chapter 17. Gauge Symmetry. - Chapter 18. Connection and Geodesics. - Chapter 19. Riemannian Curvature. - Chapter 20. Symmetries of Riemannian Manifolds. - Chapter 21. Einstein's Gravitation. - Chapter 22. Lagrangian Formulation. - Chapter 23. Conservation Laws and Further Symmetries. - - Appendices - A) Notation and Conventions. - Physical Units and Dimensions. - Mathematical Conventions. - Abbreviations. - B) Mathematical Tools. - Tensor Algebra. - Matrix Exponential. - Pauli and Dirac Matrices. - Dirac Delta Distribution. - Poisson and Wave Equation. - Variational Calculus. - Volume Element and Hyperspheres. - Hypersurface Elements. - C) Weyl Rescaling Formulae. - D) Spaces and Symmetry Groups. - - Bibliography. - - Index.
2005
The concept of symmetries in physics is briefly reviewed. In the first part of these lecture notes, some of the basic mathematical tools needed for the understanding of symmetries in nature are presented, namely group theory, Lie groups and Lie algebras, and Noether's theorem. In the second part, some applications of symmetries in physics are discussed, ranging from isospin and flavor symmetry to more recent developments involving the interacting boson model and its extension to supersymmetries in nuclear physics.