Non-compact groups, Coherent States, Relativistic Wave equations and the Harmonic Oscillator

TL;DR

This paper explores a novel approach to quantizing a relativistic particle in superspace, revealing unique squeezed state spectra, constructing a radical operator, and proposing a new relativistic wave equation linked to harmonic oscillators.

Contribution

It introduces an alternative quantization technique for relativistic particles in superspace, deriving a new wave equation and analyzing its connection to harmonic oscillators and fractional group representations.

Findings

Spectrum includes squeezed states with fractional spin representations.

Constructed an analytical radical operator in superspace.

Proposed and solved a new relativistic wave equation.

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 which correspond to the representations with the lowest weights s=1/4 and s=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 Schr\"odinger equation and the Time-dependent Harmonic Oscillator is analyzed and discussed.