On Relativistic Angular Momentum Theory. Quantum Numbers in the Little Groups of the Poincare Group. Poincare and Lorentz-invariant expansions of relativistic amplitudes

This is three papers of P. Winternitz and co-authors which link separation of variables in differential equations with its symmetry group. Here ii abstract of these papers: On Relativistic Angular Momentum Theory P. Winternitz, Ya. A. Smorodinskii, and M. Uhlir Soviet Journal of Nuclear Physics: Volume 1, Number 1, July 1965 (J. Nucl. Phys. (U.S.S.R.) 1, 163-172 January, 1965) The components of the relativistic angular momentum are given explicitly in four coordinate systems in the Lobachevskii space of relativistic velocities. Complete sets of commuting operators are found which define these systems. We consider the classical quantities corresponding to the invariants of subgroups of the Lorentz group, and the electromagnetic fields in which they are integrals of the motion. Quantum Numbers in the Little Groups of the Poincare Group Winternitz, I. Lukac, and Ya. A. Smorodinskii Soviet Journal of Nuclear Physics: Volume 7, Number 1, July 1968 (J_ Yad. Fiz. 7, 192-201 January, 1968) The integrals of motion and the problem of the introduction of quantum numbers in the groups o(3), o(2,1), and E2 are considered. It is shown that to any system of coordinates admitting a separation of variables in the Laplace equation there corresponds an integral of motion which is a homogeneous Hermitian polynomial quadratic in the generators of the corresponding group. Any operator of the type considered is equivalent to one of the polynomials. Poincare and Lorentz-invariant expansions of relativistic amplitudes P. Winternitz, Ya. A. Smorodinskii, and M. B. Sheftel Soviet Journal of Nuclear Physics: Volume 7, Number 6, December 1968 (Yad. Fiz. 7, 1325—1338 (June, 1968) A discussion is presented of the double expansions of relativistic amplitudes in terms of the irreducible representations of the homogeneous Lorentz group, suggested recently for arbitrary values of the kinematic variables s and t. The relation of these expansions to the relativistic phase shift analysis in terms of representations of various little groups of the Poincare group is studied.