Non-compact Groups, Coherent States, Relativistic Wave Equations and the Harmonic Oscillator

Abstract

Relativistic geometrical action for a quantum particle in the superspace is analyzed from theoretical group point of view. To this end an alternative technique of quantization outlined by the authors in a previous work and, that is, based in the correct interpretation of the square root Hamiltonian, is used. The obtained spectrum of physical states and the Fock construction consist of Squeezed States (SS) which correspond to the representations with the lowest weights λ = 1 4 and 3 4 with four possible (non-trivial) fractional representations for the group decomposition of the spin structure. From the theory of semigroups the analytical representation of the radical operator in the superspace is constructed, the conserved currents are computed and a new relativistic wave equation is proposed and explicitly solved for the time dependent case. The relation with the Relativistic Schrödinger equation and the Time-dependent Harmonic Oscillator (TDHO) is analyzed and discussed.

Dynamical symmetries, coherent states and nonlinear realizations: The SO(2,4) case

“…i) The Inönu-Wigner contraction [43] in order to pass from SL (2C) to the super-Poincare algebra (corresponding to the original symmetry of the model of refs. [42,44]) then, the even part of the curvature is split into a R 3,1 part R α → SO (3, 1) + R 3,1 . Than we rewrite the superalgebra (59) as…”

“…Note that the "Dirac-like" fermionic part is obviously under the square root because it is a part of the full curvature, fact that was not advertised by the authors in [45] (see also [29]) that doesn't take into account the geometrical origin of the action. An interesting point is to perform the same quantization as in the first part of the research given in [44] in order to obtain and compare the spectrum of physical states with the one obtained in ref. [45].…”