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

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

HI Imaging of LGS 3 and an Apparently Interacting High-Velocity Cloud

We present a 93' by 93' map of the area near the Local Group dwarf galaxy LGS 3, centered on an HI cloud 30' away from the galaxy. Previous authors associated this cloud with LGS 3 but relied on observations made with a 36' beam. Our high-resolution (3.4'), wide-field Arecibo observations of the region reveal that the HI cloud is distinct from the galaxy and suggest an interaction between the two. We point out faint emission features in the map that may be gas that has been tidally removed from the HI cloud by LGS 3. We also derive the rotation curve of the cloud and find that it is in solid-body rotation out to a radius of 10', beyond which the rotation velocity begins to decline. Assuming a spherical geometry for the cloud, the implied mass is 2.8 x 10^7 (d/Mpc) M_{Sun}, where d is the distance in Mpc. The observed HI mass is 5.5 x 10^6 (d/Mpc)^2 M_{Sun}, implying that the cloud is dark-matter dominated unless its distance is at least 1.9 Mpc. We propose that the cloud is a high-velocity cloud that is undergoing a tidal interaction with LGS 3 and therefore is located roughly 700 kpc away from the Milky Way. The cloud then contains a total mass of ~2.0 x 10^7 M_{Sun}, 82% of which consists of dark matter.Comment: 5 pages, 2 color figures. Accepted for publication in ApJ Letter

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

Is Instrumental Music In Christian Worship Scriptural?

https://digitalcommons.acu.edu/crs_books/1389/thumbnail.jp

Contemplations on Dirac's equation in quaternionic coordinates

A formulation of Dirac's equation using complex-quaternionic coordinates appears to yield an enormous gain in formal elegance, as there is no longer any need to invoke Dirac matrices. This formulation, however, entails several peculiarities, which we investigate and attempt to interpret

A Magellanic Origin for the Warp of the Galaxy

We show that a Magellanic Cloud origin for the warp of the Milky Way can explain most quantitative features of the outer HI layer recently identified by Levine, Blitz & Heiles (2005). We construct a model similar to that of Weinberg (1998) that produces distortions in the dark matter halo, and we calculate the combined effect of these dark-halo distortions and the direct tidal forcing by the Magellanic Clouds on the disk warp in the linear regime. The interaction of the dark matter halo with the disk and resonances between the orbit of the Clouds and the disk account for the large amplitudes observed for the vertical m=0,1,2 harmonics. The observations lead to six constraints on warp forcing mechanisms and our model reasonably approximates all six. The disk is shown to be very dynamic, constantly changing its shape as the Clouds proceed along their orbit. We discuss the challenges to MOND placed by the observations.Comment: 4 pages, 3 figures, submitted to ApJ Letters. Additional graphics, 3d visualizations and movies available at http://www.astro.umass.edu/~weinberg/lm