On the Weyl algebra for a particle on a sphere

Abstract.

We study the Weyl algebra A pertaining to a particle constrained on a sphere, which is generated by the coordinates n and by the angular momentum J . A is the algebra E ₃ of the Euclidean group in space. We find its irreducible representations by a novel approach, by showing that they are the irreducible representations (l ₀, 0) of so(3, 1) , with l₀ or -l₀ being equal to the Casimir operator J ^. n . Any integer or half-integer l₀ is allowed. The Hilbert space of a particle of spin S hosts 2S + 1 such representations. J can be analyzed into the sum L + S , i.e. pure spin states can be identified, provided 2S + 1 irreducible representations of A are glued together. These results apply to any surface which is diffeomorphic to S₂.