Multipole expansions in four-dimensional hyperspherical harmonics

A. V. Meremianin∗ Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187, Dresden, Germany and REC-010, Voronezh State University, 394006, Voronezh, Russia (Dated: October 24, 2005) Abstract The technique of vector differentiation is applied to the problem of the derivation of multipole expansions in four-dimensional space. Explicit expressions for the multipole expansion of the function rCj(r̂) with r = r1 + r2 are given in terms of tensor products of two hyperspherical harmonics depending on the unit vectors r̂1 and r̂2. The multipole decomposition of the function (r1 · r2) is also derived. The proposed method can be easily generalised to the case of the space with dimensionality larger than four. Several explicit expressions for the four-dimensional ClebschGordan coefficients with particular values of parameters are presented in the closed form.