Foundations of Physics

, Volume 37, Issue 8, pp 1149–1180 | Cite as

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

  • Diego Julio Cirilo-Lombardo

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 \({\lambda=\frac{1}{4}}\) and \({\frac{3}{4}}\) with four possible (non-trivial) fractional representations for the group decomposition of the spin structure. From the theory of semi-groups 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.


quantization supergroups harmonic oscillator geometrical actions Hamiltonian formulation group theory 


03.65.-w 11.30.Pb 42.50.-p 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Bogoliubov Laboratory of Theoretical Physics and Joint Institute for Nuclear ResearchDubnaRussian Federation

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