Abstract
The spectral function (also known as the Plancherel measure), which gives the spectral distribution of the eigenvalues of the Laplace-Beltrami operator, is calculated for a field of arbitrary integer spin (i.e., for a symmetric traceless and divergence-free tensor field) on the N-dimensional real hyperbolic space (HN). In odd dimensions the spectral function μ(λ) is analytic in the complex λ plane, while in even dimensions it is a meromorphic function with simple poles on the imaginary axis, as in the scalar case. For N even a simple relation between the residues of μ(λ) at these poles and the (discrete) degeneracies of the Laplacian on the N sphere (SN) is established. A similar relation between μ(λ) at discrete imaginary values of λ and the degeneracies on SN is found to hold for N odd. These relations are generalizations of known results for the scalar field. The zeta functions for fields of integer spin on H N are written down. Then a relation between the integerspin zeta functions on HN and SN is obtained. Applications of the zeta functions presented here to quantum field theory of integer spin in anti-de Sitter space-time are pointed out.
| Original language | English |
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| Pages (from-to) | 4217-4246 |
| Number of pages | 30 |
| Journal | Journal of Mathematical Physics |
| Volume | 35 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - Aug 1994 |