Stefano De Leo

Right eigenvalue equation in quaternionic quantum mechanics

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n -dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.

THE QUATERNIONIC DETERMINANT

The determinant for complex matrices cannot be extended to quaternionic matrices. Instead, the Study determinant and the closely related q-determinant are widely used. We show that the Study determinant can be characterized as the unique functional which extends the absolute value of the complex determinant and discuss its spectral and linear algebraic aspects.

Quaternionic eigenvalue problem

We discuss the (right) eigenvalue equation for H, C and R linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows us to translate the quaternionic problem into an equivalent real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.

Quaternions and special relativity

We reformulate Special Relativity by a quaternionic algebra on reals. Using real linear quaternions, we show that previous difficulties, concerning the appropriate transformations on the 3+1 space–time, may be overcome. This implies that a complexified quaternionic version of Special Relativity is a choice and not a necessity.

Quaternionic potentials in non-relativistic quantum mechanics

We discuss the Schr¨ odinger equation in the presence of quaternionic potentials. The study is performed analytically as long as it proves possible, when not, we resort to numerical calculations. The results obtained could be useful to investigate an underlying quaternionic quantum dynamics in particle physics. Experimental tests and proposals to observe quaternionic quantum effects by neutron interferometry are briefly reviewed.

ABOVE BARRIER POTENTIAL DIFFUSION

The stationary phase method is applied to diusion by a potential barrier for an incoming wave packet with energies greater than the height of the barrier. It is observed that a direct application leads to paradoxical results. The correct solution, conrmed by numerical calculations is the creation of multiple peaks as a consequence of multiple reections. Lessons concerning the use of the stationary phase method are drawn.

Quaternionic differential operators

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential equations with constant coefficients. We overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. The resolution of the complex linear Schrodinger equation, in the presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.

Zeros of unilateral quaternionic polynomials

The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigenvectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) n-order polynomials. To understand the strength of this method, it is compared with the Niven algorithm and it is shown where this (full) matrix approach improves previous methods based on the use of the Niven algorithm. For convenience of the readers, some examples of second and third order unilateral quaternionic polynomials are explicitly solved. The leading idea of the practical solution method proposed in this work can be summarized in the following three steps: translating the quaternionic polynomial in the eigenvalue problem for its companion matrix, finding its eigenvectors, and, finally, giving the quaternionic solution of the unilateral polynomial in terms of the components of such eigenvectors. A brief discussion on bilateral quaternionic quadratic equations is also presented.

OCTONIONIC DIRAC EQUATION

In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and

Quaternionic electroweak theory

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