82 research outputs found
The symmetry structure of the heavenly equation
We show that excitations of physical interest of the heavenly equation are
generated by symmetry operators which yields two reduced equations with
different characteristics. One equation is of the Liouville type and gives rise
to gravitational instantons, including those found by Eguchi-Hanson and
Gibbons-Hawking. The second equation appears for the first time in the theory
of heavenly spaces and provides meron-like configurations endowed with a
fractional topological charge. A link is also established between the heavenly
equation and the socalled Schr{\"o}der equation, which plays a crucial role in
the bootstrap model and in the renormalization theory.Comment: LaTex, 13 page
Pseudohermitian Hamiltonians, time-reversal invariance and Kramers degeneracy
A necessary and sufficient condition in order that a (diagonalizable)
pseudohermitian operator admits an antilinear symmetry T such that T^{2}=-1 is
proven. This result can be used as a quick test on the T-invariance properties
of pseudohermitian Hamiltonians, and such test is indeed applied, as an
example, to the Mashhoon-Papini Hamiltonian.Comment: 6 page
On the pseudo-Hermitian nondiagonalizable Hamiltonians
We consider a class of (possibly nondiagonalizable) pseudo-Hermitian
operators with discrete spectrum, showing that in no case (unless they are
diagonalizable and have a real spectrum) they are Hermitian with respect to a
semidefinite inner product, and that the pseudo-Hermiticity property is
equivalent to the existence of an antilinear involutory symmetry. Moreover, we
show that a typical degeneracy of the real eigenvalues (which reduces to the
well known Kramers degeneracy in the Hermitian case) occurs whenever a
fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an
antilinear symmetry like the time-reversal operator . Some consequences and
applications are briefly discussed.Comment: 22 page
Alternative Descriptions in Quaternionic Quantum Mechanics
We characterize the quasianti-Hermitian quaternionic operators in QQM by
means of their spectra; moreover, we state a necessary and sufficient condition
for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian
with respect to a uniquely defined positive scalar product in a infinite
dimensional (right) quaternionic Hilbert space. According to such results we
obtain two alternative descriptions of a quantum optical physical system, in
the realm of quaternionic quantum mechanics, while no alternative can exist in
complex quantum mechanics, and we discuss some differences between them.Comment: 16 page
Continuous approximation of binomial lattices
A systematic analysis of a continuous version of a binomial lattice,
containing a real parameter and covering the Toda field equation as
, is carried out in the framework of group theory. The
symmetry algebra of the equation is derived. Reductions by one-dimensional and
two-dimensional subalgebras of the symmetry algebra and their corresponding
subgroups, yield notable field equations in lower dimensions whose solutions
allow to find exact solutions to the original equation. Some reduced equations
turn out to be related to potentials of physical interest, such as the
Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like
approximate solution is also obtained which reproduces the Eguchi-Hanson
instanton configuration for . Furthermore, the equation under
consideration is extended to --dimensions. A spherically symmetric form
of this equation, studied by means of the symmetry approach, provides
conformally invariant classes of field equations comprising remarkable special
cases. One of these enables us to establish a connection with the
Euclidean Yang-Mills equations, another appears in the context of Differential
Geometry in relation to the socalled Yamabe problem. All the properties of the
reduced equations are shared by the spherically symmetric generalized field
equation.Comment: 30 pages, LaTeX, no figures. Submitted to Annals of Physic
Quasistationary quaternionic Hamiltonians and complex stochastic maps
We show that the complex projections of time-dependent -quasianti-Hermitian quaternionic Hamiltonian dynamics are complex stochastic
dynamics in the space of complex quasi-Hermitian density matrices if and only
if a quasistationarity condition is fulfilled, i. e., if and only if is
an Hermitian positive time-independent complex operator. An example is also
discussed.Comment: Submitted to J. Phys. A on October 25 200
Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries
We extend the definition of generalized parity , charge-conjugation
and time-reversal operators to nondiagonalizable pseudo-Hermitian
Hamiltonians, and we use these generalized operators to describe the full set
of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold
classification. In particular we show that and are the generators of
the antiunitary symmetries; moreover, a necessary and sufficient condition is
provided for a pseudo-Hermitian Hamiltonian to admit a -reflecting
symmetry which generates the -pseudounitary and the -pseudoantiunitary
symmetries. Finally, a physical example is considered and some hints on the
-unitary evolution of a physical system are also given.Comment: 20 page
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
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