Diego Julio Cirilo-Lombardo

Non-Abelian Born–Infeld action, geometry and supersymmetry

In this work, we propose a new non-Abelian generalization of the Born–Infeld Lagrangian. It is based on a geometrical property of the Abelian Born–Infeld Lagrangian in its determinantal form. Our goal is to extend the Abelian second-type Born–Infeld action to the non-Abelian form preserving this geometrical property, which permits us to compute the generalized volume element as a linear combination of the components of metric and the Yang–Mills energy–momentum tensors. Under the BPS-like condition, the action proposed reduces to that of the Yang–Mills theory, independently of the gauge group. New instanton-wormhole solution and static and spherically symmetric solution in curved spacetime for an SU(2) isotopic ansatz are solved and the N= 1 supersymmetric extension of the model is performed.

Geometrical properties of Riemannian superspaces, observables and physical states

Classical and quantum aspects of physical systems that can be described by Riemannian non-degenerate superspaces are analyzed from the topological and geometrical points of view. For the N=1 case the simplest supermetric introduced by Cirilo-Lombardo (Phys. Lett. B 661:186, 2008) have the correct number of degrees of freedom for the fermion fields and the super-momentum fulfills the mass shell condition, in sharp contrast with other cases in the literature where the supermetric is degenerate. This fact leads a deviation of the 4-impulse (e.g. mass constraint) that can be mechanically interpreted as a modification of the Newton law. Quantum aspects of the physical states and the basic states, and the projection relation between them, are completely described due the introduction of a new Majorana–Weyl representation of the generators of the underlying group manifold. A new oscillatory fermionic effect in the B0 part of the vacuum solution involving the chiral and antichiral components of this Majorana bispinor is explicitly shown.

Non-compact Groups, Coherent States, Relativistic Wave Equations and the Harmonic Oscillator II: Physical and Geometrical Considerations

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 Mp(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.

Unified Field Theoretical Models from Generalized Affine Geometries

New model of a non-dualistic Unified Theory is analyzed. This model is based in a manifold equipped with an underlying hypercomplex structure and zero non-metricity, that makes it geometricaly and physically consistent. Wormhole solution from this new model is presented and is explicitly compared with our previous one coming from the Einstein-Non Abelian Born-Infeld theory (in Class. Quantum Gravity 22:4987–5004, 2005). We find that the torsion plays in this unified theory similar role that Yang Mills type strength field coming from the non-Abelian Born-Infeld energy momentum tensor. The meaning of the Yang-Mills ansatz based in the alignment of the isospin with the frame geometry of the spacetime is discussed.

Information metric from Riemannian superspaces

Abstract The Fisher information metric is introduced in order to find the meaning of the probability distribution in classical and quantum systems described by Riemannian non-degenerated superspaces. In particular, the physical role played by the coefficients a and a ⁎ of the fermionic part of an emergent metric solution, obtained previously (Cirilo-Lombardo, 2012 [1] ) is explored. We find that the metric solution of the superspace establishes a connexion between the Fisher metric and its quantum counterpart, corroborating early conjectures by Caianiello et al. This quantum mechanical extension of the Fisher metric is described by the C P 1 structure of the Fubini–Study metric, with coordinates a and a ⁎ .

Algebraic Structures, Physics and Geometry from a Unified Field Theoretical Framework

Starting from a Unified Field Theory (UFT) proposed previously by the author, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in this UFT general context that the underlying basis of the single geometrical structure P(G, M) (the principal fiber bundle over the real spacetime manifold M with structural group G) reflecting the symmetries of the different fields carry naturally a biquaternionic structure instead of a complex one. This fact allows us to analyze algebraically and to interpret physically in a straighforward way the Majorana and Dirac representations and the relation of such structures with the spacetime signature and non-hermitian (CP) dynamic operators. Also, from the underlying structure of the tangent space, the existence of hidden (super) symmetries and the possibility of supersymmetric extensions of these UFT models are given showing that Rothstein’s theorem is incomplete for that description. The importance of the Clifford algebras in the description of all symmetries, mainly the interaction of gravity with the other fields, is briefly discussed.

Quantum Gravity: physics from supergeometries

We show that the metric (line element) is the first geometrical object to be associated to a discrete (quantum) structure of the spacetime without necessity of black hole-entropy-area arguments, in sharp contrast with other attempts in the literature. To this end, an emergent metric solution obtained previously in [ Phys. Lett. B661 (2008) 186–191] from a particular non-degenerate Riemannian superspace is introduced. This emergent metric is described by a physical coherent state belonging to the metaplectic group Mp(n) with a Poissonian distribution at lower n (number basis) restoring the classical thermal continuum behavior at large n(n → ∞), or leading to non-classical radiation states, as is conjectured in a quite general basis by means of the Bekenstein–Mukhanov effect. Group-dependent conditions that control the behavior of the macroscopic regime spectrum (thermal or not), as the relationship with the problem of area/entropy of the black hole are presented and discussed.

ON THE LORENTZ GROUP SO(3, 1), GEOMETRICAL SUPERSYMMETRIC ACTION FOR PARTICLES AND SQUARE ROOT OPERATORS II. SQUEEZED STATES AND RELATIVISTIC WAVE EQUATIONS

The geometrical relativistic superparticle action is analyzed from the theoretical group point of view. To this end an alternative technique of quantization, outlined by the authors in a previous work and based on the correct interpretation of the square-root Hamiltonian, is used. We show that the obtained spectrum of physical states and the Fock construction in this previous work consist of squeezed states with the even and odd representations with the lowest weights λ = 1/4 and λ= 3/4 corresponding to four possible (nontrivial) fractional representations for the group decomposition of the spin structure. The conserved currents are computed, and a new relativistic wave equation is proposed and explicitly solved for the time-dependent case. The relation between the relativistic Schrodinger equation and the time-dependent harmonic oscillator is analyzed and discussed.

On the Lorentz group SO(3, 1), geometrical supersymmetric action for particles, and square root operators

In this work the problem of the square-root quantum operators is analyzed from the theoretical group point of view. To this end, we considered the relativistic geometrical action of a particle in the superspace in order to quantize it and to obtain the spectrum of physical states with the Hamiltonian remaining in the natural square-root form. The generators of group SO(3, 1) are introduced and the quantization of this model is performed completely. The obtained spectrum of physical states and the Fock construction for the physical states from the Hamiltonian operator in square-root form was proposed, explicitly constructed, and compared with the spectrum and Fock construction obtained from the Hamiltonian in the standard form (i.e., quadratic in momenta). We show that the only states that the square-root Hamiltonian can operate with correspond to the representations with the lowest weights λ = 1/4 and 3/4 with four possible (nontrivial) fractional representations for the group decomposition of the spin structure.

Higher spin particles in the discrete string model approach

A new method is proposed to describe higher spin composite systems. Our goal is to introduce a discrete string model representing bound states in an N-relativistic particle system. The spectrum of the physical states is obtained for an arbitrary number N of particles in an arbitrary spacetime dimension. Particular illustrative examples of the new method for N = 2, 3, 4 and 5 are given, and the conditions on the existence of physical states with a precise spin content in any particular number of spacetime dimensions are clearly established.