6,907 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
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
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.
Study to determine suitable high temperature, high altitude, total temperature sensors Final report
High temperature, high altitude total temperature sensor development - thermocouple devic
Phases and Transitions in Phantom Nematic Elastomer Membranes
Motivated by recently discovered unusual properties of bulk nematic
elastomers, we study a phase diagram of liquid-crystalline polymerized phantom
membranes, focusing on in-plane nematic order. We predict that such membranes
should enerically exhibit five phases, distinguished by their conformational
and in-plane orientational properties, namely isotropic-crumpled,
nematic-crumpled, isotropic-flat, nematic-flat and nematic-tubule phases. In
the nematic-tubule phase, the membrane is extended along the direction of {\em
spontaneous} nematic order and is crumpled in the other. The associated
spontaneous symmetries breaking guarantees that the nematic-tubule is
characterized by a conformational-orientational soft (Goldstone) mode and the
concomitant vanishing of the in-plane shear modulus. We show that long-range
orientational order of the nematic-tubule is maintained even in the presence of
harmonic thermal luctuations. However, it is likely that tubule's elastic
properties are ualitatively modified by these fluctuations, that can be studied
using a nonlinear elastic theory for the nematic tubule phase that we derive at
the end of this paper.Comment: 12 pages, 4 eps figures. To appear in PR
Imaginary in all directions: an elegant formulation of special relativity and classical electrodynamics
A suitable parameterization of space-time in terms of one complex and three
quaternionic imaginary units allows Lorentz transformations to be implemented
as multiplication by complex-quaternionic numbers rather than matrices.
Maxwell's equations reduce to a single equation.Comment: 8 page
Quantum models related to fouled Hamiltonians of the harmonic oscillator
We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator
which provide, at the classical level, the same equation of motion as the
conventional Hamiltonian. These Hamiltonians, say and , result
to be explicitly time-dependent and can be expressed as a formal rotation of
two cubic polynomial functions, and , of the canonical variables
(q,p).
We investigate the role of these fouled Hamiltonians at the quantum level.
Adopting a canonical quantization procedure, we construct some quantum models
and analyze the related eigenvalue equations. One of these models is described
by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a
discrete spectrum on the real line. A self-adjoint extension is fixed by
choosing the spectral parameter of the associated eigenvalue
equation equal to zero. The spectral problem is discussed in the context of
three different representations. For , the eigenvalue equation is
exactly solved in all these representations, in which square-integrable
solutions are explicity found. A set of constants of motion corresponding to
these quantum models is also obtained. Furthermore, the algebraic structure
underlying the quantum models is explored. This turns out to be a nonlinear
(quadratic) algebra, which could be applied for the determination of
approximate solutions to the eigenvalue equations.Comment: 24 pages, no figures, accepted for publication on JM
Induced paramagnetic states by localized -loops in grain boundaries
Recent experiments on high-temperature superconductors show paramagnetic
behavior localized at grain boundaries (GB). This paramagnetism can be
attributed to the presence unconventional d-wave induced -junctions. By
modeling the GB as an array of and conventional Josephson junction we
determine the conditions of the occurrence of the paramagnetic behavior.Comment: 4 pages, 4 figures, submitted to Phys. Rev. Let
Representations of U(1,q) and Constructive Quaternion Tensor Products
The representation theory of the group U(1,q) is discussed in detail because
of its possible application in a quaternion version of the Salam-Weinberg
theory.
As a consequence, from purely group theoretical arguments we demonstrate that
the eigenvalues must be right-eigenvalues and that the only consistent scalar
products are the complex ones. We also define an explicit quaternion tensor
product which leads to a set of additional group representations for integer
``spin''.Comment: 28 pages, Latex, Dipartimento di Fisica, Universita di Lecce
INFN-Sezione di Lecc
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