275 research outputs found
An introduction to the spectrum, symmetries, and dynamics of spin-1/2 Heisenberg chains
Quantum spin chains are prototype quantum many-body systems. They are
employed in the description of various complex physical phenomena. The goal of
this paper is to provide an introduction to the subject by focusing on the time
evolution of a Heisenberg spin-1/2 chain and interpreting the results based on
the analysis of the eigenvalues, eigenstates, and symmetries of the system. We
make available online all computer codes used to obtain our data.Comment: 8 pages, 3 figure
Noether symmetries for two-dimensional charged particle motion
We find the Noether point symmetries for non-relativistic two-dimensional
charged particle motion. These symmetries are composed of a quasi-invariance
transformation, a time-dependent rotation and a time-dependent spatial
translation. The associated electromagnetic field satisfy a system of
first-order linear partial differential equations. This system is solved
exactly, yielding three classes of electromagnetic fields compatible with
Noether point symmetries. The corresponding Noether invariants are derived and
interpreted
On the generalized Davenport constant and the Noether number
Known results on the generalized Davenport constant related to zero-sum
sequences over a finite abelian group are extended to the generalized Noether
number related to the rings of polynomial invariants of an arbitrary finite
group. An improved general upper bound is given on the degrees of polynomial
invariants of a non-cyclic finite group which cut out the zero vector.Comment: 14 page
The Lie derivative of spinor fields: theory and applications
Starting from the general concept of a Lie derivative of an arbitrary
differentiable map, we develop a systematic theory of Lie differentiation in
the framework of reductive G-structures P on a principal bundle Q. It is shown
that these structures admit a canonical decomposition of the pull-back vector
bundle i_P^*(TQ) = P\times_Q TQ over P. For classical G-structures, i.e.
reductive G-subbundles of the linear frame bundle, such a decomposition defines
an infinitesimal canonical lift. This lift extends to a prolongation
Gamma-structure on P. In this general geometric framework the concept of a Lie
derivative of spinor fields is reviewed. On specializing to the case of the
Kosmann lift, we recover Kosmann's original definition. We also show that in
the case of a reductive G-structure one can introduce a "reductive Lie
derivative" with respect to a certain class of generalized infinitesimal
automorphisms, and, as an interesting by-product, prove a result due to
Bourguignon and Gauduchon in a more general manner. Next, we give a new
characterization as well as a generalization of the Killing equation, and
propose a geometric reinterpretation of Penrose's Lie derivative of "spinor
fields". Finally, we present an important application of the theory of the Lie
derivative of spinor fields to the calculus of variations.Comment: 28 pages, 1 figur
Cole-Hopf Like Transformation for Schr\"odinger Equations Containing Complex Nonlinearities
We consider systems, which conserve the particle number and are described by
Schr\"odinger equations containing complex nonlinearities. In the case of
canonical systems, we study their main symmetries and conservation laws. We
introduce a Cole-Hopf like transformation both for canonical and noncanonical
systems, which changes the evolution equation into another one containing
purely real nonlinearities, and reduces the continuity equation to the standard
form of the linear theory. This approach allows us to treat, in a unifying
scheme, a wide variety of canonical and noncanonical nonlinear systems, some of
them already known in the literature. pacs{PACS number(s): 02.30.Jr, 03.50.-z,
03.65.-w, 05.45.-a, 11.30.Na, 11.40.DwComment: 26 pages, no figures, to be appear in J. Phys. A: Math. Gen. (2002
Noether's Theorem and time-dependent quantum invariants
The time dependent-integrals of motion, linear in position and momentum
operators, of a quantum system are extracted from Noether's theorem
prescription by means of special time-dependent variations of coordinates. For
the stationary case of the generalized two-dimensional harmonic oscillator, the
time-independent integrals of motion are shown to correspond to special
Bragg-type symmetry properties. A detailed study for the non-stationary case of
this quantum system is presented. The linear integrals of motion are
constructed explicitly for the case of varying mass and coupling strength. They
are obtained also from Noether's theorem. The general treatment for a
multi-dimensional quadratic system is indicated, and it is shown that the
time-dependent variations that give rise to the linear invariants, as conserved
quantities, satisfy the corresponding classical homogeneous equations of motion
for the coordinates.Comment: Plain TeX, 23 pages, preprint of Instituto de Ciencias Nucleares,
UNAM Departamento de F\ii sica and Matem\'aticas Aplicadas, No. 01 (1994
Symmetry, singularities and integrability in complex dynamics III: approximate symmetries and invariants
The different natures of approximate symmetries and their corresponding first
integrals/invariants are delineated in the contexts of both Lie symmetries of
ordinary differential equations and Noether symmetries of the Action Integral.
Particular note is taken of the effect of taking higher orders of the
perturbation parameter. Approximate symmetries of approximate first
integrals/invariants and the problems of calculating them using the Lie method
are considered
Hybrid exciton-polaritons in a bad microcavity containing the organic and inorganic quantum wells
We study the hybrid exciton-polaritons in a bad microcavity containing the
organic and inorganic quantum wells. The corresponding polariton states are
given. The analytical solution and the numerical result of the stationary
spectrum for the cavity field are finishedComment: 3 pages, 1 figure. appear in Communications in Theoretical Physic
Gauge transformations for higher-order lagrangians
Noether's symmetry transformations for higher-order lagrangians are studied.
A characterization of these transformations is presented, which is useful to
find gauge transformations for higher-order singular lagrangians. The case of
second-order lagrangians is studied in detail. Some examples that illustrate
our results are given; in particular, for the lagrangian of a relativistic
particle with curvature, lagrangian gauge transformations are obtained, though
there are no hamiltonian gauge generators for them.Comment: 22 pages, LaTe
On the classical central charge
In the canonical formulation of a classical field theory, symmetry properties
are encoded in the Poisson bracket algebra, which may have a central term.
Starting from this well understood canonical structure, we derive the related
Lagrangian form of the central term.Comment: 23 pages, RevTeX, no figures; introduction improved, a few references
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