arXiv subjects
Publications and source records attributed to Sergio Giardino.
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes benefit of the quaternionic structure and the results are much simpler than the present formulations of quaternionic curves.
We solve Klein-Gordon equation (KGE) in the framework of the real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM). The presented solution is the simplest ever obtained for quaternionic quantum theories, and the closest to the complex solution. The scattering of a quaternionic charged scalar particle from an electric field is also obtained. A remarkable feature of quaternionic scalar particles is the existence of massive light cone particles.
In this article we obtained the harmonic oscillator solution for quaternionic quantum mechanics ($\mathbbm{H}$QM) in the real Hilbert space, both in the analytic method and in the algebraic method. The quaternionic solutions have many additional possibilities if compared to complex quantum mechanics ($\mathbbm{C}$QM), and thus there are many possible applications to these results in future research.
We study the elastic scattering of quantum particles based on a real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM) and derive expression for the wave function, the phase shifts, as well as the optical theorem for the case of a hard sphere scattering potential. The strong agreement between these new quaternionic results and the corresponding results in complex quantum mechanics reinforce the validity of the $\mathbbm H$QM generalization of ordinary complex quantum mechanics ($\mathbbm C$QM).
We develop a quaternionic electrodynamics and show that it naturally supports the existence of magnetic monopoles. We obtained the field equations, the continuity equation, the electrodynamic force law, the Poynting vector, the energy conservation and the stress-energy tensor. The formalism also enabled us to generalize the Dirac monopole and the charge quantization rule.
The one-dimensional infinite square well is the simplest solution of quantum mechanics, and consequently one of the most important. In this article, we provide this solution using the real Hilbert space approach to quaternic quantum mechanics ($\mathbbm{H}$QM). We further provide the one-dimensional finite as well and a method to generate quaternic solutions from non-degenerate complex solutions.
This paper describes general relativity at the gravito-electromagnetic precision level as a constrained field theory. In this novel formulation, the gravity field comprises two auxiliary fields, a static matter field and a moving matter field. Equations of motion, continuity equation, energy conservation, field tensor, energy-momentum tensor, constraints and Lagrangian formulation are presented as a simple and unified formulation that can be useful for future research.
After a review on recent results on quaternic quantum mechanics ($\mathbb{H}$QM), we present further consistency tests that reinforce its compatibility with the usual complex quantum mechanics ($\mathbb{C}$QM). The novel results comprises the Virial theorem, the quantum quaternic Lorentz force, the existence of a quaternic magnetic monopole and the redefinition of the expectation value.
Solutions of quaternionic quantum mechanics (QQM) are difficult to grasp, even in simple physical situations. In this article, we provide simple and understandable free particle quaternionic solutions, that can be easily compared to complex quantum mechanics (CQM). As an application, we study the scattering of quaternionic particles through a scalar step potential. We also provide a general solution method for the quaternionic Schrödinger equation, which can be applied to more sophisticated and physically interesting models.
A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After a brief discussion on unitary operators in this formalism, we conclude that this quantum theory is indeed consistent, and can be a valuable tool in the search for new physics.
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.
If $Ψ$ is a quaternionic wave function, then $iΨ\neq Ψi$. Thus, there are two versions of the quaternionic Schrödinger equation (QSE). In this article, we present the second possibility for solving the QSE, following on from a previous article. After developing the general methodology, we present the quaternionic free particle solution and the scattering of the quaternionic particle through a scalar barrier.
A quaternionic analog of the Aharonov-Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics (QQM). A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.
We build a four-dimensional quaternion-parametrized conformal field theory (QCFT) using quaternion holomorphic functions as the generators of quaternionic conformal transformations. Taking the two-dimensional complex-parametrized conformal field theory (CCFT) as our model, we study the stress tensor, the conserved charge, the symmetry generators, the quantization conditions and several operator product expansions (OPE's). Future applications also are addressed.
Using the WKB approximation we analyse the tunnelling of a pulsating string in deformed Minkowski spacetime.
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 $\mathsf{X}$ which has the group $\mathsf{G}$ of QHT as a particular case. Furthermore, we observe that the Möbius group and the Heisenberg group may be obtained by making $\mathsf{X}$ more symmetric. We provide matrix representations for the group $\mathsf{X}$ and for its algebra $\mathfrak{x}$. The Lie algebra is neither simple nor semi-simple, and so it is not classified among the classical Lie algebras. They prove that the group $\mathsf{G}$ comprises $\mathsf{SU}(2,\mathbb{C})$ 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.
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 sheed 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 be observed by measuring the mean value of some operator.
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.