JO URNAL OF G EO METRY :ND ELSEWIER Journal of Geometry and Physics 20 ( 1996) l-1 8 PHYSIC S On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces Roberto Camporesi a,*, Atsushi Higuchi b a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino. Ital) b lnstitutfiir theoretische Physik, Universitdt Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Received 28 April 1995; revised 19 July 1995 zyxwvutsrqponmlkjihgfedcbaZYXWV Abstract The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are then used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra’s formula for the radial part of the Casimir operator. Subj. Class.: Complex differential geometry; Quantum mechanics 1991 M SC: 81QO581R25 Keywords: Dirac operator; Eigenfunctions 1. Introduction The N-dimensional sphere (sN) and the real hyperbolic space (HN), which are “dual” to each other as symmetric spaces [lo], are maximally symmetric. This high degree of symmetry allows one to compute explicitly the eigenfunctions of the Laplacian for various fields on these spaces. These eigenfunctions can be used in studying field theory in de Sitter and anti-de Sitter space-times since S4 and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ H4 are Euclidean sections of these space-times. Also S3 and H3 appear as the spatial sections of cosmological models, and various field equations and their solutions on these spaces have physical applications in this context. In addition to these applications, fields on SN and HN provide concrete examples for the theory * Corresponding author. E-mail: camporesi@polito.it. 0393~0440/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. SSDI 0393.0440(95)00042-9