Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere

@article{Dennis2004CanonicalRO,
  title={Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere},
  author={Mark R. Dennis},
  journal={Journal of Physics A},
  year={2004},
  volume={37},
  pages={9487-9500}
}
  • M. Dennis
  • Published 25 August 2004
  • Mathematics, Physics
  • Journal of Physics A
Any eigenfunction of the Laplacian on a sphere is given in terms of a unique set of directions: these are Maxwell's multipoles, their existence and uniqueness being known as Sylvester's theorem. Here, the theorem is proved by realizing the multipoles are pairs of opposite vectors in Majorana's sphere representation of quantum spins. The proof involves a physicist's standard tools of quantum angular momentum algebra, integral kernels and Gaussian integration. Various other proofs are compared… 
View PDF on arXiv
29 Citations
Background Citations
10
Methods Citations
4

29 Citations

Citation Type

Citation Type

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere
Using coherent-state techniques, we prove a sampling theorem for Majorana’s (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of N…
Correlations between Maxwell's multipoles for Gaussian random functions on the sphere
Maxwell's multipoles are a natural geometric characterization of real functions on the sphere (with fixed l). The correlations between multipoles for Gaussian random functions are calculated by…
On some orthogonality properties of Maxwell's multipole vectors
We determine the location of the expansion points with respect to which the two Maxwell's multipole vectors of the quadrupole moment and the dipole vector of a distribution of charge form an…
Probability density of the multipole vectors for a Gaussian cosmic microwave background
We review Maxwell's multipole vectors, and elucidate some of their mathematical properties, with emphasis on the application of this tool to the cosmic microwave background (CMB). In particular, for…
Orbifold construction of the modes of the Poincaré dodecahedral space
We provide a new construction of the modes of the Poincare dodecahedral space S3/I*. The construction uses the Hopf map, Maxwell's multipole vectors and orbifolds. In particular, the *235-orbifold…
Generalized Gaussian beams in terms of Jones vectors
Based on the operator formalism that arises from the underlying SU(2) group structure, a formula is derived that provides a description of the generalized Hermite-Laguerre Gauss modes in terms of a…
The multipole vectors of the Wilkinson Microwave Anisotropy Probe, and their frames and invariants
We investigate the statistical isotropy (SI) and Gaussianity of the cosmic microwave background fluctuations, using a set of multipole vector functions capable of separating these two issues. In…
Quantifying quantumness and the quest for Queens of Quantum
We introduce a measure of 'quantumness' for any quantum state in a finite-dimensional Hilbert space, based on the distance between the state and the convex set of classical states. The latter are…
Multipole vectors of completely random microwave skies for l≤50
The statistical cosmological principle states that observables on the celestial sphere are sampled from a rotationally invariant distribution. Previously certain large scale anomalies which violate…
Testing the statistical isotropy of large scale structure with multipole vectors
A fundamental assumption in cosmology is that of statistical isotropy - that the universe, on average, looks the same in every direction in the sky. Statistical isotropy has recently been tested…
...
...

References

SHOWING 1-10 OF 48 REFERENCES
Maxwell's multipole representation of traceless symmetric tensors and its application to functions of high-order tensors
  • W. Zou, Q. Zheng
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 2003
It is known from the theory of group representations that a general tensor can be expressed as a sum of traceless symmetric tensors. In this paper, based on Sylvester's theorem, it is shown that a…
A geometrical picture of anisotropic elastic tensors
A geometrical picture of fourth-order, three-dimensional elastic tensors in terms of Maxwell multipoles is developed and used to obtain the elastic tensors appropriate to various crystal symmetry…
Polarization singularities in isotropic random vector waves
  • M. Berry, M. Dennis
  • Physics
    Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 2001
Following Nye & Hajnal, we explore the geometry of complex vector waves by regarding them as a field of polarization ellipses. Singularities of this field are the C lines and L lines, where the…
Chaotic analytic zero points: exact statistics for those of a random spin state
A natural statistical ensemble of 2J points on the unit sphere can be associated, via the Majorana representation, with a random quantum state of spin J, and an exact expression is obtained here for…
The Majorana representation of polarization, and the Berry phase of light
A natural geometric representation of the polarization of light with fixed propagation direction is a dot on a sphere in an abstract space: the Poincare sphere. If the direction of propagation is…
Quantum Theory of Angular Momentum
Containing basic definitions and theorems as well as relations, tables of formulas and numerical tables which are essential for applications to many physical problems, the book is useful for…
LETTER TO THE EDITOR: The Berry phase for spin in the Majorana representation
A continuous cyclic sequence of quantum states has an associated geometric, or Berry, phase . For spin J, such a sequence is described by a cyclic change in the 2J+1 coefficients of the basis states…
Vortex lines of the electromagnetic field
Relativistic definition of the phase of the electromagnetic field, involving two Lorentz invariants, based on the Riemann-Silberstein vector is adopted to extend our previous study [I.…
Topological Singularities in Wave Fields
This thesis is a study of the natural geometric structures, arising through interference, in fields of complex waves (scalars, vectors or tensors), where certain parameters describing the wave are…
Spinor and twistor methods in space-time geometry
Preface Summary of volume 1 6. Twistors 7. Null congruences 8. Classification of curvature tensors 9. Conformal infinity Appendix References Subject and author index Index of symbols.
...
...