相关论文: Canonical representation of spherical functions: S…
Maxwell's multipoles are a natural geometric characterisation of real functions on the sphere (with fixed $\ell$). The correlations between multipoles for gaussian random functions are calculated, by mapping the spherical functions to…
A geometric theory of the irreducible tensor operators of quantum spin systems. It is based upon the Maxwell-Sylvester geometric representation of the multipolar electrostatic potential. In the latter, an order-$\ell$ multipolar potential…
Across many areas of physics, multipole expansions are used to simplify problems, solve differential equations, calculate integrals, and process experimental data. Spherical harmonics are the commonly used basis functions for a multipole…
The Majorana representation of spin-$\frac{n}{2}$ quantum states by sets of points on a sphere allows a realization of SU(n) acting on such states, and thus a natural action on the two-dimensional sphere $S^2$. This action is discussed in…
Pure quantum spin-$s$ states can be represented by $2s$ points on the sphere, as shown by Majorana in 1932 --- the description has proven particularly useful in the study of rotational symmetries of the states, and a host of other…
The electric and magnetic fields of a pole-dipole singularity attributed to a point-electron-singularity in the Maxwell field are expressed in a Colombeau algebra of generalized functions. This enables one to calculate dynamical quantities…
Multipolar expansions are a foundational tool for describing basis functions in quantum mechanics, many-body polarization, and other distributions on the unit sphere. Progress on these topics is often held back by complicated and competing…
In "Classical Electrodynamics" (Jackson) a theorem is proved on the average of an electrostatic or magnetostatic field over a spherical volume. The proof of the theorem is based on an expansion in spherical harmonics and it is useful for…
I discuss the relation between harmonic polynomials and invariant theory and show that homogeneous, harmonic polynomials correspond to ternary forms that are apolar to a base conic (the absolute). The calculation of Schlesinger that…
In the previous paper [25], Stolarsky's invariance principle, known for point distributions on the Euclidean spheres [27], has been extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane.…
Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…
In contrast to elementary Majorana particles, emergent Majorana fermions (MFs) in condensed-matter systems may have electromagnetic multipoles. We developed a general theory of magnetic multipoles for surface helical MFs on…
A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…
For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the…
The representation theory underlying the infinite-component relativistic wave equation written by Majorana is revisited from a modern perspective. On the one hand, the massless solutions of this equation are shown to form a supermultiplet…
The possibility that our space is multi - rather than singly - connected has gained a renewed interest after the discovery of the low power for the first multipoles of the CMB by WMAP. To test the possibility that our space is a…
Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in…
This work is developed in the context of Lorentzian spin-foams with space- and time-like boundaries. It is argued that the equations describing the semiclassical regime of the various spin-foam amplitudes admit a common biquaternionic…
The representation theory of the quantum group su$_q(2)$ is used to introduce $q$-analogues of the Wigner rotation matrices, spherical functions, and Legendre polynomials. The method amounts to an extension of variable separation from…
The first part of this article is devoted to a brief review of the results about representation theory of the spin group Spin(m) from the point of view of Clifford analysis. In the second part we are interested in Clifford-valued functions…