arXiv subjects

Sergio Giardino

Publications and source records attributed to Sergio Giardino.

At least 19 recordsLinked to original sources

arXiv2026

The quantum harmonic oscillator and the real Hilbert space

The harmonic oscillator is considered within generalized frameworks using complex and quaternionic numbers. The classical oscillator is considered in terms of a complex position function, and quantum oscillators are examined in terms of complex wave functions, and in terms of quaternionic wave functions as well. Both of the quantum solutions are obtained within the real Hilbert space formalism. The results reveal the complex and quaternionic descriptions as suitable frameworks for non-stationary processes, including damped oscillations, forced oscillations, and additionally self-interacting processes that cannot be appropriately described otherwise.

quant-ph↗

arXiv2026

Deformed angular momentum algebra within the real Hilbert space

Starting from generalized position operators, we derive complex and quaternionic angular momentum operators along with their commutation algebra as well. These algebras differ from the standard Hermitian ones, especially in terms of commutation relations involving partial and total angular momentum operators. Despite these differences, the effective quantum expectation values obtained from slightly deformed algebras align with those from the conventional Hermitian algebra. This suggests that even though the wave functions and resulting dynamics differ from standard quantum Hermitian behavior, these deformed algebras can still be effectively understood as valid angular momentum algebras.

quant-ph↗

arXiv2026

Quantum self-interaction within an infinitely deep cavity

One examines the infinitely deep quantum cavity, also known as the quantum infinite square well, within the framework of the real Hilbert space. The solutions are considered in terms of complex wave functions, and also in terms of quaternionic wave functions. The complex results reproduce the usual achievements established in the complex Hilbert space, but also extend them to non-stationary solutions, as well as to distorted stationary solutions, different energy spectra, and dislocated observed position. The quaternionic cases further admit the incidence of self-interaction, something that cannot be observed in complex solutions. Therefore, both the complex and quaternionic solutions are more general than previous cases, thus opening the way to further one-dimensional solutions to be researched in the non-relativistic theory.

quant-ph↗

arXiv2025

Self-interacting quantum particles and the Dirac delta potential

The Dirac delta function potential is considered within the real Hilbert space approach for complex wave functions, as well as quaternionic wave functions. As has been previously determined, the real Hilbert space approach enables the possibility of self-interacting physical systems. The self-interaction precludes confining states, and also imposes non-stationary quantum states, both of them representing novel situations that cannot be observed in terms of quantum wave functions. These results remark the differences between quaternionic quantum mechanics ($\mathbbm H$QM) and complex quantum mechanics ($\mathbbm C$QM), and also establish a method of solving the wave equation that may be applied to a variety of different cases.

quant-ph↗

arXiv2025

Complex Lagrangian dynamics

In this article one introduces a formalism of classical mechanics where complex Lagrangian functions are admitted. The results include complex versions of the Lagrangian function, of the Euler-Lagrange equation, of the Hamilton principle, a geometric formulation, and the relation to a previous complex Hamiltonian formalism. The framework is particularly suitable for non-stationary motion, and various pathways can be followed in future investigation.

math-ph↗

arXiv2025

Klein-Gordon equation within the real Hilbert space formalism

Within this article one finds the statement of the Klein-Gordon problem within the real Hilbert space formalism ($\mathbbm R$HS) in terms of complex wave functions, and in terms of quaternionic wave functions as well. The complex formulation comprises hermitian and non-hermitian cases, while the quaternionic solutions additionally set in motion self-interacting particles. The non-hermitian cases comprise non-conservative processes, while the self-interaction physically implies the increase of the effective mass of the particle, an effect that cannot be reproduced using a complex wave function. The obtained autonomous particle solutions, as well as the Klein problem agree to the previously discovered self-interacting non-relativistic particle, and thus reinforce $\mathbbm R$HS as viable and consistent way to explore open problems in quantum mechanics. Also important, the negative energy problem that plagues the usual formalism is eliminated within this approach.

quant-ph↗

arXiv2025

Expectation value dynamics within real Hilbert space quantum mechanics

Dynamic equations concerning physical expectation values have been examined in terms of the real Hilbert space approach to quantum mechanics. The considered cases involve complex wave functions, as well as quaternionic wave functions. The consistency of the formalism has been verified in terms of the continuity equation, the classical limit, and generalizations of the quantum Lorentz force, and the Virial theorem. Besides testing the consistency of the real Hilbert space approach, generalized position and angular momentum operators have been introduced, and inspire exciting directions for further research.

quant-ph↗

arXiv2025

Classical and quantum complex dynamics

A generalization of classical mechanics is obtained from a complex parametrization of the phase space. The formalism supports complex Hamiltonian functions describing non-conservative classical mechanical systems. A quantization scheme that is general enough to incorporate non-stationary physical processes is also achieved.

quant-ph↗

arXiv2024

Self-interacting quantum particles

The real Hilbert space formalism developed within the quaternionic quantum mechanics ($\mathbb H$QM) is fully applied to the simple model of the autonomous particle. This framework permits novel insights within the usual description of the complex autonomous particle, particulaly concening the energy of a non-stationary motion. Through the appraisal of the physical role played by a fully quaternionic scalar potential, a original self-interaction within the quaternionic autonomous particle has been determined as well. Scattering processes are considered to illustrate these novel features.

quant-ph↗

arXiv2024

Differential geometry using quaternions

This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and torsion concepts, differential forms, directional derivatives and the structural equations. The analogy between the quaternionic and the real geometries is obtained using a matrix representation of quaternions. The results evidences the quaternionic formalism as a suitable language to differential geometry that can be useful in various directions of future investigation.

math.DG↗

arXiv2023

Generalized imaginary units in quantum mechanics

The generalization of the imaginary unit is examined within the instances of the complex quantum mechanics ($\mathbb C$QM), and of the quaternionic quantum mechanics ($\mathbb H$QM) as well. Whereas the complex theory describes non-stationary quantum processes, the quaternionic theory does not admit such an interpretation, and associates the generalized imaginary unit to a novel time evolution function. Various possibilities are opened as future directions for future research.

quant-ph↗

arXiv2023

Spin and angular momentum in quaternionic quantum mechanics

We present two novel solutions of real Hilbert state quaternionic quantum mechanics ($\mathbb H$QM). Firstly, we observe that the angular momentum operator admits two different classes of physically non-equivalent free particles. As a second result, we study the Larmor precession to observe that it has a quaternionic solution where a novel phenomenological interpretation is possible, as well as a different of spin is possible, and these results may encourage experimental and theoretical investigations of the quaternionic theory.

quant-ph↗

arXiv2022

Quaternionic fermionic field

The second quantization of the quaternionic fermionic field is undertaken using the real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM). The solution responds to an open problem of quaternionic quantum theory, and launches the basis to the development of the quaternionic interaction theory.

hep-th↗

arXiv2022

Attenuated gravitational radiation

The hypothesis of an alternative way of obtaining gravitational waves is the physical motivation of this article. Using the linear field approximation and a symmetry transformation of the affine connection, new field equations and new gauge conditions have been obtained. Solutions to these field equations have been considered in the empty space and in the wave zone, and in both of them the oscillation amplitudes of their solutions are attenuated exponentially. We expect that these solutions can be useful for building more sophisticated gravitational wave models, and also as an impulse for researching further symmetry transformations of general relativity.

physics.gen-ph↗

arXiv2022

Quaternionic scalar field in the real Hilbert space

Using the complex Klein-Gordon field as a model, we quantize the quaternionic scalar field in the real Hilbert space. The lagrangian formulation has accordingly been obtained, as well as the hamiltonian formulation, and the energy and charge operators. Conversely to the complex case, the quaternionic quantization admits two quantization schemes, concerning either two or four components. Therefore, the quaternionic field permits a richer structure of states, if compared to the complex scalar field case. Moreover, the quaternionic theory admits as a further novel feature a non-associative algebraic structure in their complex components, something not observed in the complex case.

quant-ph↗

arXiv2022

Winding number and homotopy for quaternionic curves

Following a recent approach to quaternionic curves, we defined the quaternionic polar angle that enabled us to define global properties of quaternionic curves, namely the winding number and the homotopy concept. The results admit various applications, including further analogies to plane curves, and physical applications.

math-ph↗

arXiv2021

Quaternionic Dirac free particle

We solve the quaternionic Dirac equation ($\mathbbm H$DE) in the real Hilbert space, and we ascertain that their free particle solutions set comprises eight elements in the case of a massive particle, and a four elements solution set in the case of a massless particle, a richer situation when compared to the four elements solutions set of the usual complex Dirac equation ($\mathbbm C$DE). These free particle solutions were unknown in the previous solutions of anti-hermitian quaternionic quantum mechanics, and constitute an essential element in order to build a quaternionic quantum field theory ($\mathbbm H$QFT).

quant-ph↗

arXiv2021

A complementary covariant approach to gravito-electromagnetism

From a previous paper where we proposed a description of general relativity within the gravito-electromagnetic limit, we propose an alternative modified gravitational theory. As in the former version, we analyze the vector and tensor equations of motion, the gravitational continuity equation, the conservation of the energy, the energy-momentum tensor, the field tensor, and the constraints concerning these fields. The Lagrangian formulation is also exhibited as an unified and simple formulation that will be useful for future investigation.

gr-qc↗