# Expansion in solid spherical harmonics on the lattice

I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like the expansion of a plane wave in spherical harmonics on the lattice? (I mean in discrete space modeled by a three dimensional, finite, cubic lattice.)

So I am looking for an orthonormal basis for complex valued functions on a finite lattice, where the angular and radial variables would be (approximately?) separated, as this is the case for the solid harmonics.

Thank you for your help!

• Do you think of some expansion as in real space Wannier functions? But unfortunately these functions can not be used explicitely as I remember... In addition you will need the interaction Hamiltonian to describe the scattering in the correct shape. Please be more specific what is scattered on the lattice sites!? – strpeter Mar 6 '13 at 23:12
• To be quite specific: I would like to express, on the position basis, the Coulomb scattering process calculated in chapter 10 of Quantum Ethics. For the time being, it is expressed on the momentum basis, and I would like to evaluate the angular distribution using a development on the position basis; in continuous space-time, I would develop the result on the spherical harmonics, since they separate the angular and radial variables, but is there something similar on the lattice, where no spherical symmetry is granted? – Sébastien Fauvel Mar 7 '13 at 7:49
• Hi user21566, I'm afraid such a thing won't exist: spherical harmonics in $d$ dimensions arise directly because of the $SO(d)$ symmetry around a point, which is broken to a discrete group on the lattice. But good luck finding an alternative! – Vibert Mar 7 '13 at 9:21
• I urge you to look at this paper: arxiv.org/pdf/hep-th/9303048v2.pdf in which the author uses spherical harmonics, but discretizes only the radial coordinate. Apparently, this lattice regularization method gives finite answers in d<=3. – Srivatsan Balakrishnan Dec 7 '14 at 16:49
• Thank you Srivatsan for this hint! It looks like Mark Srednicki would have needed something similar in his paper, but as you can see, he hasn't written down this lattice version of the spherical harmonics, he's only using them - without even knowing if they exist, I guess. So the question remains open... – Sébastien Fauvel Dec 8 '14 at 9:18