Symplectic geometry of the Wigner transform on noncompact symmetric spaces

摘要

We introduce a notion of Wigner transform on the symmetric spaces X = SO_0(1,n)/SO(n) which satisfies the usual marginality and covariance properties. Recalling the notion of the Helgason dual of X and denoting it by ≡ we show that there exists a natural and canonical SO_0(1,n)-invariant symplectic structure and briefly describe the corresponding geometric quantization, thus showing that the Wigner transform maps states in L~2(X) to functions on the phase space X * ≡, yielding an intermediate position-momentum representation of the quantum mechanics on X.