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

Abstract

The physical meaning of the particularly simple non-degenerate supermetric, introduced in the previous part by the authors, is elucidated and the possible connection with processes of topological origin in high energy physics is analyzed and discussed. New possible mechanism of the localization of the fields in a particular sector of the supermanifold is proposed and the similarity and differences with a 5-dimensional warped model are shown. The relation with gauge theories of supergravity based in the OSP (1/4) group is explicitly given and the possible original action is presented. We also show that in this non-degenerate super-model the physic states, in contrast with the basic states, are observables and can be interpreted as tomographic projections or generalized representations of operators belonging to the metaplectic group M p (2) .The advantage of geometrical formulations based on non-degenerate super-manifolds over degenerate ones is pointed out and the description and the analysis of some interesting aspects of the simplest Riemannian superspaces are presented from the point of view of the possible vacuum solutions .

Riemannian Superspaces, Exact Solutions and the Geometrical Meaning of the Field Localization

“…chiral plus anti-chiral parts). The system of equations for A(t) and (t) that we are looking for was given in [15], and is the following…”

“…The superfield solution for the fields (see the "square states" of Refs. [15][16][17][18][19]) that we are looking for, have the following form…”

“…Motivated by the above, we complete our previous work [15] studying and analyzing from the point of view of the possible vacuum solutions, the simplest non trivial supermetric given by Volkov and Pashnev in [14] that was the "starting point" toy model of [15] ds 2 = ω μ ω μ + aω α ω α − a * ωαωα (1) This particular non-degenerate supermetric contains the complex parameters a and a * that make it different of other more standard supermetrics. Then, our main task is to find the meaning and the role played by these complex parameters from the geometrical and physical point of view.…”

“…To this end, we compare the solution of Ref. [15] that was computed in the N = 1 four dimensional superspace proposed in [14,[16][17][18][19], compactified to one dimension and restricted to the pure time-dependent case with:…”

“…2 we give a brief review, based in a previous work of the author [15], about the N = 1 non-degenerate four dimensional superspace proposed by Volkov and Pashnev and its solution. Section 3 is devoted to analyze the relation of the supermetric under consideration with the superspace (1 | 2) given explicitly under which conditions one is reduced to the other one from the point of view of the obtained vacuum solutions.…”