On Bargmann representations of the Wigner function

By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of the Wigner function (W). A non-integral representation is presented in terms of a quadratic form W ∼ V†FV, where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending, in a simple recursive way, on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of the Wigner function and the spatial uncertainty of the coherent state basis we use to represent it