Publications

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2018
Giardino, S.  2018.  ``Non-anti-hermitian Quaternionic Quantum Mechanics''. Adv. Appl. Clifford Algebras. AbstractWebsite

The breakdown of Ehrenfest’s theorem imposes serious limitations on quaternionic quantum mechanics (QQM). In order to determine the conditions in which the theorem is valid, we examined the conservation of the probability density, the expectation value and the classical limit for a non-anti-hermitian formulation of QQM. The results also indicated that the non-anti-hermitian quaternionic theory is related to non-hermitian quantum mechanics, and thus the physical problems described with both of the theories should be related.

2017
Giardino, S.  2017.  ``Four-dimensional conformal field theory using quaternions''. Advances in Applied Clifford Algebras. 27(3):2457–2471. AbstractWebsite

We have built a constrained four-dimensional quaternion-parametrized conformal field theory using quaternion holomorphic functions as the generators of quaternionic conformal transformations. With the two-dimensional complex-parametrized conformal field theory as our model, we study the stress tensor, the conserved charge, the symmetry generators, the quantization conditions and several operator product expansions. Future applications are also addressed.

Giardino, S.  2017.  ``Möbius transformation for left-derivative quaternion holomorphic functions''. Adv. Appl. Clifford Algebras (aceito para publica\c cão). 27(2):1161–1173. AbstractWebsite

Holomorphic quaternion functions only admit affine functions; thus, the Möbius transformation for these functions, which we call quaternionic holomorphic transformation (QHT), only comprises similarity transformations. We determine a general group X which has the group G of QHT as a particular case. Furthermore, we observe that the Möbius group and the Heisenberg group may be obtained by making X more symmetric. We provide matrix representations for the group X and for its algebra x. The Lie algebra is neither simple nor semi-simple, and so it is not classified among the classical Lie algebras. We prove that the group G comprises SU(2,C) rotations, dilations and translations. The only fixed point of the QHT is located at infinity, and the QHT does not admit a cross-ratio. Physical applications are addressed at the conclusion.

Giardino, S.  2017.  ``Quaternionic Ahraronov-Bohm effect''. Adv. Appl. Clifford Algebras. 27(3):2445–2456. AbstractWebsite

A quaternionic analog of the Aharonov–Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics. A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.

2016
Giardino, S.  2016.  ``Quaternionic particle in a relativistic box''. Found. Phys.. 46(4):473-483., Number 4 AbstractWebsite

This study examines quaternion Dirac solutions for an infinite square well. The quaternion result does not recover the complex result within a particular limit. This raises the possibility that quaternionic quantum mechanics may not be understood as a correction to complex quantum mechanics, but it may also be a structure that can be used to study phenomena that cannot be described through the framework of complex quantum mechanics.

Giardino, S, Rivelles V.  2016.  ``Tunnelling of Pulsating Strings in Deformed Minkowski Spacetime''. Eur. Phys. J.. C76:234., Number 5 Abstractwebsite

Using the WKB approximation we analyse the tunnelling of a pulsating string in deformed Minkowski spacetime.

2015
Giardino, S, DeLeo S, Ducati G.  2015.  ``Quaternionic Dirac Scattering''. J. Phys. Math.. 6(1):1000130., Number 1 AbstractWebsite

The scattering of a Dirac particle has been studied for a quaternionic potential step. In the potential region an additional diffusion solution is obtained. The quaternionic solution which generalizes the complex one presents an amplification of the reflection and transmission rates. A detailed analysis of the quaternionic spinorial velocities shed new light on the additional solution. For pure quaternionic potentials, the interesting and surprising result of total transmission is found. This suggests that the presence of pure quaternionic potentials cannot be seen by analyzing the reflection or transmission rates. It has been observed by measuring the mean value of some operator.

2014
Giardino, S.  2014.  ``Axisymmmetric empty space: light propagation, orbits and dark matter''. J. Mod. Phys.. 5(15):1402-1411. AbstractWebsite

This study presents an axisymmetric solution of the Einstein equations for empty space. The geometry is studied by determining its Petrov classification and killing vectors. Light propagation, orbital motion and asymptotic and Newtonian limits are also studied. Additionally, cosmological applications of the geometry are also outlined as an alternative model for the inflationary universe and as a substitute for dark matter and quintessence.

Giardino, S, DeLeo S.  2014.  ``Dirac solutions for quaternionic potentials''. J. Math. Phys.. 55(2):022301-10. AbstractWebsite

The Dirac equation is solved for quaternionic potentials, i V0 + j W0 (V0∈ℝ,W0∈ℂ). The study shows two different solutions. The first one contains particle and anti-particle solutions and leads to the diffusion, tunneling, and Klein energy zones. The standard solution is recovered taking the complex limit of this solution. The second solution, which does not have a complex counterpart, can be seen as a V0-antiparticle or |W0|-particle solution.

Giardino, S.  2014.  ``The Static String''. Mod. Phys. Lett.. A29(03):1450018. AbstractWebsite

In this paper, the quantum fluctuation of a rigid and static string is reported to be identical to a free quantum particle. Solutions similar to this static string have already been found in the semiclassical quantization of pulsating strings, and our results show that the semiclassical quantization of pulsating strings is, in some cases, a perturbation of static strings. We also interpret the energy of the static string as a lower bound for the pulsating string and speculate about a description of quantum mechanics in terms of semiclassical string theory.

2013
Giardino, S, Teotônio-Sobrinho P.  2013.  ``A non-associative quaternion scalar field theory''. Mod. Phys. Lett.. A28(35):1350163., Number 35 AbstractWebsite

A nonassociative Groenewold–Moyal (GM) plane is constructed using quaternion-valued function algebras. The symmetrized multiparticle states, the scalar product, the annihilation/creation algebra and the formulation in terms of a Hopf algebra are also developed. Nonassociative quantum algebras in terms of position and momentum operators are given as the simplest examples of a framework whose applications may involve string theory and nonlinear quantum field theory.

Giardino, S.  2013.  ``Angular invariant quantum mechanics in arbitrary dimension''. Rev. Bras. Ens. Fis.. 35:3307. AbstractWebsite

One dimensional quantum mechanics problems, namely the infinite potential well, the harmonic oscillator, the free particle, the Dirac delta potential, the finite well and the finite barrier are generalized for finite arbitrary dimension in a radially symmetric, or angular invariant, manner. This generalization enables the Schrödinger equation solutions to be visualized for Bessel functions and Whittaker functions, and it also enables connections to multi-dimensional physics theories, like string theory.

Giardino, S.  2013.  ``Semi-classical strings in (2+1)-dimensional backgrounds''. ISRN High Energy Phys.. 2013:517858. AbstractWebsite

This study analyzes the geometrical relationship between a classical string and its semiclassical quantum model. From an arbitrary (2+1)-dimensional geometry, a specific ansatz for a classical string is used to generate a semi-classical quantum model. In this framework, examples of quantum oscillations and quantum free particles are presented that uniquely determine a classical string and the space-time geometry where its motion takes place.

2011
Giardino, S, Carrión HL.  2011.  ''Classical strings in AdS4 x CP3 with three angular momenta''. JHEP. 1108:057. AbstractWebsite

In this paper, rotating strings in three directions of AdS4×CP3 geometry are studied; its divergent energy limit, and conserved charges are also determined. An interpretation of these configurations as either giant magnons or spiky strings is discussed.

Giardino, S.  2011.  ``Divergent energy strings in AdS_5 x S5 with three angular momenta''. JHEP. 1112:022. AbstractWebsite

In this paper, novel solutions for strings with three angular momenta in AdS5 × S 5 geometry are presented; the divergent energy limit and the corresponding conserved charges, as well as dispersion relation are also determined. Interpretations of these configurations as either a giant magnon (GM) or a spiky string (SS) are discussed.

Giardino, S, Carrión HL.  2011.  ``Lie 3-algebra and super-affinization of split-octonions''. Mod. Phys. Lett.. A26(35):2663-2675. AbstractWebsite

The purpose of this study is to extend the concept of a generalized Lie 3-algebra, known to the divisional algebra of the octonions

Giardino, S, Rivelles V.  2011.  ``Pulsating Strings in Lunin-Maldacena Backgrounds''. JHEP. 1107:057. AbstractWebsite

We consider pulsating strings in Lunin-Maldacena backgrounds, specifically in deformed Minkowski spacetime and deformed AdS5 × S 5. We find the relation between the energy and the oscillation number of the pulsating string when the deformation is small. Since the oscillation number is an adiabatic invariant it can be used to explore the regime of highly excited string states. We then quantize the string and look for such a sector. For the deformed Minkowski background we find a precise match with the classical results if the oscillation number is quantized as an even number. For the deformed AdS5 × S 5 we find a contribution which depends on the deformation parameter.

1997
Giardino, S, Vicentini G, Isolani PC.  1997.  ``Synthesis and characterization of lanthanide picrate complexes with tripiperidinophosphine oxide''. J. Alloys Comp.. 249(1-2):91-93. AbstractWebsite

Compounds with the general formula Ln(pic)3·2tpppo (Ln: La-Lu, Y; pic=picrate; tpppo=tripipendinophosphine oxide) were synthesized by reaction of the hydrated picrates with tpppo in absolute ethanol. IR absorption spectra were interpreted in terms of the coordination of tpppo through the oxygen and picrate anions that are bidentate, coordinated to the central ions through the phenoxo group and one oxygen of an ortho-nitro group. Conductance measurements in acetonitrile solutions show that the complexes behave as non-electrolytes. X-ray powder patterns show only one isomorphous series. Phosphorus-31 NMR showed a signal shift to lower field with relation to the free ligand, showing evidence of bonding through the phosphoryl group. The parameters obtained from the absorption spectrum of the Nd compound indicate that the metal-ligand bonds present some covalent character. The emission spectra of the europium and samarium adducts were determined. The spectrum of the europium compound was interpreted in terms of D2d distorted towards C2v symmetry for the Eu(O)8 chromophore.