A coordinate independent formulation of the Weyl–Wigner transform theory

We present the Weyl–Wigner (WW) transform theory in a much more compact way than usual, by introducing the Δ basis in an intrinsic form. This permits the derivation of new identities and also leads to generalizations, like the inclusion of finite-dimensional systems in the WW theory, which is also discussed. We show, in this case, some striking differences in the structure of finite phase space depending on the underlying dimension of quantum space being an even or odd integer.