Realization of the Noncommutative Seiberg-Witten Gauge Theory by Fields in Phase Space

Representations of the Poincar'{e} symmetry are studied by using a Hilbert space with a phase space content. The states are described by wave functions ( quasi amplitudes of probability) associated with Wigner functions (quasi probability density). The gauge symmetry analysis provides a realization of the Seiberg-Witten gauge theory for noncommutative fields.