94,874 research outputs found
The Integrability of Pauli System in Lorentz Violating Background
We systematically analyze the integrability of a Pauli system in Lorentz
violating background at the non-relativistic level both in two- and
three-dimensions. We consider the non-relativistic limit of the Dirac equation
from the QED sector of the so-called Standard Model Extension by keeping only
two types of background couplings, the vector a_mu and the axial vector b_mu.
We show that the spin-orbit interaction comes as a higher order correction in
the non-relativistic limit of the Dirac equation. Such an interaction allows
the inclusion of spin degree non-trivially, and if Lorentz violating terms are
allowed, they might be comparable under special circumstances. By including all
possible first-order derivative terms and considering the cases a\ne 0, b\ne 0,
and b_0\ne 0 one at a time, we determine the possible forms of constants of
motion operator, and discuss the existence or continuity of integrability due
to Lorentz violating background.Comment: 19 page
Lie systems: theory, generalisations, and applications
Lie systems form a class of systems of first-order ordinary differential
equations whose general solutions can be described in terms of certain finite
families of particular solutions and a set of constants, by means of a
particular type of mapping: the so-called superposition rule. Apart from this
fundamental property, Lie systems enjoy many other geometrical features and
they appear in multiple branches of Mathematics and Physics, which strongly
motivates their study. These facts, together with the authors' recent findings
in the theory of Lie systems, led to the redaction of this essay, which aims to
describe such new achievements within a self-contained guide to the whole
theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure
On the use of projection operators in electrodynamics
In classical electrodynamics all the measurable quantities can be derived
from the gauge invariant Faraday tensor . Nevertheless, it is
often advantageous to work with gauge dependent variables. In [4],[2] and [8],
and in the present note too, the transformation of the vector potential in
Lorenz gauge to that in Coulomb gauge is considered. This transformation can be
done by applying a projection operator that extracts the transverse part of
spatial vectors. In many circumstances the proper projection operator is
replaced by a simplified transverse one. It is widely held that such a
replacement does not affect the result in the radiation zone. In this paper the
action of the proper and simplified transverse projections will be compared by
making use of specific examples of a moving point charge. It will be
demonstrated that whenever the interminable spatial motion of the source is
unbounded with respect to the reference frame of the observer the replacement
of the proper projection operator by the simplified transverse one yields, even
in the radiation zone, an erroneous result with error which is of the same
order as the proper Coulomb gauge vector potential itself.Comment: 15 pages, no figures, matched to the published versio
Quantized electric-flux-tube solutions to Yang-Mills theory
We suggest that long-distance Yang-Mills theory is more conveniently described in terms of electric rather than the customary magnetic vector potentials. On this basis we propose as an effective Lagrangian for this regime the most simple gauge-invariant (under the magnetic rather than electric gauge group) and Lorentz-invariant Lagrangian which yields a 1/q^4 gluon propagator in the Abelian limit. The resulting classical equations of motion have solutions corresponding to tubes of color electric flux quantized in units of e/2 (e is the Yang-Mills coupling constant). To exponential accuracy the electric color energy is contained in a cylinder of finite radius, showing that continuum Yang-Mills theory has excitations which are confined tubes of color electric flux. This is the criterion for electric confinement of color
Quasi-Lie schemes and Emden--Fowler equations
The recently developed theory of quasi-Lie schemes is studied and applied to
investigate several equations of Emden type and a scheme to deal with them and
some of their generalisations is given. As a first result we obtain t-dependent
constants of the motion for particular instances of Emden equations by means of
some of their particular solutions. Previously known results are recovered from
this new perspective. Finally some t-dependent constants of the motion for
equations of Emden type satisfying certain conditions are recovered
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