8,834 research outputs found
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.Comment: 24 pages, AMS-Te
Solving simple quaternionic differential equations
The renewed interest in investigating quaternionic quantum mechanics, in
particular tunneling effects, and the recent results on quaternionic
differential operators motivate the study of resolution methods for
quaternionic differential equations. In this paper, by using the real matrix
representation of left/right acting quaternionic operators, we prove existence
and uniqueness for quaternionic initial value problems, discuss the reduction
of order for quaternionic homogeneous differential equations and extend to the
non-commutative case the method of variation of parameters. We also show that
the standard Wronskian cannot uniquely be extended to the quaternionic case.
Nevertheless, the absolute value of the complex Wronskian admits a
non-commutative extension for quaternionic functions of one real variable.
Linear dependence and independence of solutions of homogeneous (right) H-linear
differential equations is then related to this new functional. Our discussion
is, for simplicity, presented for quaternionic second order differential
equations. This involves no loss of generality. Definitions and results can be
readily extended to the n-order case.Comment: 9 pages, AMS-Te
Quaternions and Special Relativity
We reformulate Special Relativity by a quaternionic algebra on reals. Using
{\em real linear quaternions}, we show that previous difficulties, concerning
the appropriate transformations on the space-time, may be overcome. This
implies that a complexified quaternionic version of Special Relativity is a
choice and not a necessity.Comment: 17 pages, latex, no figure
Quaternionic potentials in non-relativistic quantum mechanics
We discuss the Schrodinger equation in 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.Comment: 21 pages, 16 figures (ps), AMS-Te
The octonionic eigenvalue problem
By using a real matrix translation, we propose a coupled eigenvalue problem
for octonionic operators. In view of possible applications in quantum
mechanics, we also discuss the hermiticity of such operators. Previous
difficulties in formulating a consistent octonionic Hilbert space are solved by
using the new coupled eigenvalue problem and introducing an appropriate scalar
product for the probability amplitudes.Comment: 21 page
Quaternionic Electroweak Theory
We explicitly develop a quaternionic version of the electroweak theory, based
on the local gauge group . The need of a complex
projection for our Lagrangian and the physical significance of the anomalous
scalar solutions are also discussed.Comment: 12 pages, Revtex, submitted to J. Phys.
Quaternionic eigenvalue problem
We discuss the (right) eigenvalue equation for , and
linear quaternionic operators. The possibility to introduce an
isomorphism between these operators and real/complex matrices allows to
translate the quaternionic problem into an {\em equivalent} real or complex
counterpart. Interesting applications are found in solving differential
equations within quaternionic formulations of quantum mechanics.Comment: 13 pages, AMS-Te
Amplification of coupling for Yukawa potentials
It is well known that Yukawa potentials permit bound states in the
Schrodinger equation only if the ratio of the exchanged mass to bound mass is
below a critical multiple of the coupling constant. However, arguments
suggested by the Darwin term imply a more complex situation. By numerically
studying the Dirac equation with a Yukawa potential we investigate this
amplification effect.Comment: 7 pages, 2 figure
Quaternionic Electroweak Theory and CKM Matrix
We find in our quaternionic version of the electroweak theory an apparently
hopeless problem: In going from complex to quaternions, the calculation of the
real-valued parameters of the CKM matrix drastically changes. We aim to explain
this quaternionic puzzle.Comment: 8, Revtex, Int. J. Theor. Phys. (to be published
Translations between Quaternion and Complex Quantum Mechanics
While in general there is no one-to-one correspondence between complex and
quaternion quantum mechanics (QQM), there exists at least one version of QQM in
which a {\em partial} set of {\em translations} may be made. We define these
translations and use the rules to obtain rapid quaternion counterparts (some of
which are new) of standard quantum mechanical results.Comment: 15 pages, LaTeX, ULTH-93-3
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