44,743 research outputs found
Addition theorems for spin spherical harmonics. II Results
Based on the results of part I, we obtain the general form of the addition
theorem for spin spherical harmonics and give explicit results in the cases
involving one spin- and one spin- spherical harmonics with ,
1, 3/2, and , 1. We obtain also a fully general addition theorem for
one scalar and one tensor spherical harmonic of arbitrary rank. A variety of
bilocal sums of ordinary and spin spherical harmonics are given in explicit
form, including a general explicit expression for bilocal spherical harmonics
Fermion Quasi-Spherical Harmonics
Spherical Harmonics, , are derived and presented (in a
Table) for half-odd-integer values of and . These functions are
eigenfunctions of and written as differential operators in the
spherical-polar angles, and . The Fermion Spherical Harmonics
are a new, scalar and angular-coordinate-dependent representation of fermion
spin angular momentum. They have symmetry in the angle , and hence
are not single-valued functions on the Euclidean unit sphere; they are
double-valued functions on the sphere, or alternatively are interpreted as
having a double-sphere as their domain.Comment: 16 pages, 2 Tables. Submitted to J.Phys.
Monopole Vector Spherical Harmonics
Eigenfunctions of total angular momentum for a charged vector field
interacting with a magnetic monopole are constructed and their properties
studied. In general, these eigenfunctions can be obtained by applying vector
operators to the monopole spherical harmonics in a manner similar to that often
used for the construction of the ordinary vector spherical harmonics. This
construction fails for the harmonics with the minimum allowed angular momentum.
These latter form a set of vector fields with vanishing covariant curl and
covariant divergence, whose number can be determined by an index theorem.Comment: 21 pages, CU-TP-60
A fuzzy bipolar celestial sphere
We introduce a non-commutative deformation of the algebra of bipolar
spherical harmonics supporting the action of the full Lorentz algebra. Our
construction is close in spirit to the one of the non-commutative spherical
harmonics associated to the fuzzy sphere and, as such, it leads to a maximal
value of the angular momentum. We derive the action of Lorentz boost generators
on such non-commutative spherical harmonics and show that it is compatible with
the existence of a maximal angular momentum.Comment: 15 pages, 4 figures; v2: typos corrected, references added; v3 title
slightly changed, minor adjustments in the presentation, results unchanged.
References added, matches published versio
Smooth approximation of data on the sphere with splines
A computable function, defined over the sphere, is constructed, which is of classC1 at least and which approximates a given set of data. The construction is based upon tensor product spline basisfunctions, while at the poles of the spherical system of coordinates modified basisfunctions, suggested by the spherical harmonics expansion, are introduced to recover the continuity order at these points. Convergence experiments, refining the grid, are performed and results are compared with similar results available in literature.\ud
\ud
The approximation accuracy is compared with that of the expansion in terms of spherical harmonics. The use of piecewise approximation, with locally supported basisfunctions, versus approximation with spherical harmonics is discussed
Obtaining Potential Field Solution with Spherical Harmonics and Finite Differences
Potential magnetic field solutions can be obtained based on the synoptic
magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of
the magnetogram is used to construct the current and divergence free magnetic
field solution. This method works reasonably well when the order of spherical
harmonics is limited to be small relative to the resolution of the magnetogram,
although some artifacts, such as ringing, can arise around sharp features. When
the number of spherical harmonics is increased, however, using the raw
magnetogram data given on a grid that is uniform in the sine of the latitude
coordinate can result in inaccurate and unreliable results, especially in the
polar regions close to the Sun.
We discuss here two approaches that can mitigate or completely avoid these
problems: i) Remeshing the magnetogram onto a grid with uniform resolution in
latitude, and limiting the highest order of the spherical harmonics to the
anti-alias limit; ii) Using an iterative finite difference algorithm to solve
for the potential field. The naive and the improved numerical solutions are
compared for actual magnetograms, and the differences are found to be rather
dramatic.
We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a
publically available code, so that other researchers can also use it as an
alternative to the spherical harmonics approach.Comment: This paper describes the publicly available Finite Difference
Iterative Potential field Solver (FDIPS). The code can be obtained from
http://csem.engin.umich.edu/FDIP
Odd-Parity Bipolar Spherical Harmonics
Bipolar spherical harmonics (BiPoSHs) provide a general formalism for
quantifying departures in the cosmic microwave background (CMB) from
statistical isotropy (SI) and from Gaussianity. However, prior work has focused
only on BiPoSHs with even parity. Here we show that there is another set of
BiPoSHs with odd parity, and we explore their cosmological applications. We
describe systematic artifacts in a CMB map that could be sought by measurement
of these odd-parity BiPoSH modes. These BiPoSH modes may also be produced
cosmologically through lensing by gravitational waves (GWs), among other
sources. We derive expressions for the BiPoSH modes induced by the weak lensing
of both scalar and tensor perturbations. We then investigate the possibility of
detecting parity-breaking physics, such as chiral GWs, by cross-correlating
opposite parity BiPoSH modes with multipole moments of the CMB polarization. We
find that the expected signal-to-noise of such a detection is modest.Comment: 19 pages, 4 figures, Accepted to PR
- …