Spherical Harmonics, the Weyl Transform and the Fourier Transform on the Heisenberg Group

Abstract

In the early days of quantum mechanics, Weyl asked the following question. Let λ be a non-zero real number, ℋ a separable Hilbert space. Given certain (unbounded) operators W 1 ,…, W n , W 1 + , …, W n + on ℋ satisfying (on a dense subspace D of ℋ ) with all other commutators vanishing. Given also a function where ζ ∈ C n . Let W = ( W 1 …, W n ) W + = ( W 1 + …, W n + ). How does one associate to f an operator f ( W , W + )? (Actually, Weyl phrased the question in terms of p = Re ζ, q = Im ζ, P = Re W , Q = Im W + which represent momentum and position. In this paper, however, we wish to exploit the unitary group on C n and so prefer complex notation.)