22,853 research outputs found
An Extension of the Faddeev-Jackiw Technique to Fields in Curved Spacetimes
The Legendre transformation on singular Lagrangians, e.g. Lagrangians
representing gauge theories, fails due to the presence of constraints. The
Faddeev-Jackiw technique, which offers an alternative to that of Dirac, is a
symplectic approach to calculating a Hamiltonian paired with a well-defined
initial value problem when working with a singular Lagrangian. This phase space
coordinate reduction was generalized by Barcelos-Neto and Wotzasek to simplify
its application. We present an extension of the Faddeev-Jackiw technique for
constraint reduction in gauge field theories and non-gauge field theories that
are coupled to a curved spacetime that is described by General Relativity. A
major difference from previous formulations is that we do not explicitly
construct the symplectic matrix, as that is not necessary. We find that the
technique is a useful tool that avoids some of the subtle complications of the
Dirac approach to constraints. We apply this formulation to the Ginzburg-Landau
action and provide a calculation of its Hamiltonian and Poisson brackets in a
curved spacetime.Comment: 30 pages, updated to reflect published versio
The deformation quantization mapping of Poisson- to associative structures in field theory
Let be a variational Poisson
bracket in a field model on an affine bundle over an affine base manifold
. Denote by the commutative associative multiplication in the
Poisson algebra of local functionals
that take field configurations to numbers. By applying
the techniques from geometry of iterated variations, we make well defined the
deformation quantization map
that produces a noncommutative -linear star-product in
.Comment: Proc. 50th Sophus Lie Seminar (26-30 September 2016, Bedlewo,
Poland), 8 figures, 24 page
The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform
This note reviews complex and real techniques in harmonic analysis. We
describe a common source of both approaches rooted in the covariant transform
generated by the affine group.
Keywords: wavelet, coherent state, covariant transform, reconstruction
formula, the affine group, ax+b-group, square integrable representations,
admissible vectors, Hardy space, fiducial operator, approximation of the
identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy
integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal
function, vertical maximal functions, non-tangential maximal functions,
intertwining operator, Cauchy-Riemann operator, Laplace operator, singular
integral operator, SIO, boundary behaviour, Carleson measure.Comment: 31 pages, AMS-LaTeX, no figures; v2: a major revision, sections on
representations of the ax+b group and transported norms are added; v3: major
revision: an outline section on complex and real variables techniques are
added, numerous smaller improvements; v4: minor correction
Poisson-noise induced escape from a metastable state
We provide a complete solution of the problems of the probability
distribution and the escape rate in Poisson-noise driven systems. It includes
both the exponents and the prefactors. The analysis refers to an overdamped
particle in a potential well. The results apply for an arbitrary average rate
of noise pulses, from slow pulse rates, where the noise acts on the system as
strongly non-Gaussian, to high pulse rates, where the noise acts as effectively
Gaussian
Hamilton-Jacobi quantization of singular Lagrangians with linear velocities
In this paper, constrained Hamiltonian systems with linear velocities are
investigated by using the Hamilton-Jacobi method. We shall consider the
integrablity conditions on the equations of motion and the action function as
well in order to obtain the path integral quantization of singular Lagrangians
with linear velocities.Comment: late
The Convergence of Particle-in-Cell Schemes for Cosmological Dark Matter Simulations
Particle methods are a ubiquitous tool for solving the Vlasov-Poisson
equation in comoving coordinates, which is used to model the gravitational
evolution of dark matter in an expanding universe. However, these methods are
known to produce poor results on idealized test problems, particularly at late
times, after the particle trajectories have crossed. To investigate this, we
have performed a series of one- and two-dimensional "Zel'dovich Pancake"
calculations using the popular Particle-in-Cell (PIC) method. We find that PIC
can indeed converge on these problems provided the following modifications are
made. The first modification is to regularize the singular initial distribution
function by introducing a small but finite artificial velocity dispersion. This
process is analogous to artificial viscosity in compressible gas dynamics, and,
as with artificial viscosity, the amount of regularization can be tailored so
that its effect outside of a well-defined region - in this case, the
high-density caustics - is small. The second modification is the introduction
of a particle remapping procedure that periodically re-expresses the dark
matter distribution function using a new set of particles. We describe a
remapping algorithm that is third-order accurate and adaptive in phase space.
This procedure prevents the accumulation of numerical errors in integrating the
particle trajectories from growing large enough to significantly degrade the
solution. Once both of these changes are made, PIC converges at second order on
the Zel'dovich Pancake problem, even at late times, after many caustics have
formed. Furthermore, the resulting scheme does not suffer from the unphysical,
small-scale "clumping" phenomenon known to occur on the Pancake problem when
the perturbation wave vector is not aligned with one of the Cartesian
coordinate axes.Comment: 29 pages, 29 figures. Accepted for publication in ApJ. The revised
version includes a discussion of energy conservation in the remapping
procedure, as well as some interpretive differences in the Conclusions made
in response to the referee report. Results themselves are unchange
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