Nonuniqueness of the number-phase wigner function

Employing the extended phase states, we show that six conditions analogous to Wigner's original ones cannot lead uniquely to the number-phase Wigner function. To show this fact explicitly, we propose a new example of the Wigner function satisfying all these conditions. The nonuniqueness of the number-phase Wigner function results from the 2pi-periodicity of the phase. It is also shown that the two number-phase Wigner functions obtained by several authors have an endpoint problem, which leads to an ambiguous result. Their correct integral forms are derived from those defined in the extended Fock space with negative number states.