Symmetric tensor spherical harmonics on the Nsphere and their application to the de Sitter group SO(N,1)
Abstract
The symmetric tensor spherical harmonics (STSH's) on the Nsphere (S^{N}), which are defined as the totally symmetric, traceless, and divergencefree tensor eigenfunctions of the LaplaceBeltrami (LB) operator on S^{N}, are studied. Specifically, their construction is shown recursively starting from the lowerdimensional ones. The symmetric traceless tensors induced by STSH's are introduced. These play a crucial role in the recursive construction of STSH's. The normalization factors for STSH's are determined by using their transformation properties under SO(N+1). Then the symmetric, traceless, and divergencefree tensor eigenfunctions of the LB operator in the Ndimensional de Sitter spacetime which are obtained by the analytic continuation of the STSH's on S^{N} are studied. Specifically, the allowed eigenvalues of the LB operator under the restriction of unitarity are determined. Our analysis gives a grouptheoretical explanation of the forbidden mass range observed earlier for the spin2 field theory in de Sitter spacetime.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 July 1987
 DOI:
 10.1063/1.527513
 Bibcode:
 1987JMP....28.1553H