The Four-Dimensional Hyperbolic Spherical Harmonics

Abstract

In the four-dimensional hyperbolic space, we can derive Euler formula and establish the hyperbolic spherical polar coordinate and the hyperbolic spherical harmonics.

Relativistic quantum physics with hyperbolic numbers

“…Considerations of a non-relativistic hyperbolic Hilbert space with respect to the Born formula have been given by Kocik [18], Khrennikov [19,20], and Rochon and Tremblay [21]. Xuegang et al investigated the Dirac wave equation, Clifford algebraic spinors, a hyperbolic Hilbert space, and the hyperbolic spherical harmonics in hyperbolic spherical polar coordinates [22,23,24]. The hypercomplex numbers, and the geometries generated by these numbers, have been investigated by Catoni et al [25].…”

Hyperbolic Harmonic Function in Minkowski Space

“…Its basis set {e 1 , e 2 , e 3 , e 4 } for Clifford product and inner product of It is obvious that the R 3,1 is a four-dimensional Minkowski space for inner product of Clifford algebra Cl 3,1 , which abbreviates R 3,1 as M 4 .…”

The Poincaré mass operator in terms of a hyperbolic algebra

“…In this work Hucks also found that the operations of C, P, and T on Dirac spinors are closely related to the three types of complex conjugation that exist when both hyperbolic and ordinary imaginary units are present. Xuegang et al investigated in this context the Dirac wave equation, Clifford algebraic spinors, a hyperbolic Hilbert space, and the hyperbolic spherical harmonics in hyperbolic spherical polar coordinates [11,12,13].…”