Relation of quasiprobabilities to Bargmann representation of states

A relation of the Wigner quasiprobability to the Bargmann representation of pure and mixed states by convolution is derived and generalized to the mai n class of quasiprobabilities and its inversion is given. The derivation us es a realization of the abstract group SU(1, 1) by second-order differentia tion and multiplication operators for a pair of complex conjugated variable s and disentanglement of exponential functions of these operators by group- theoretical methods. Examples for the calculation of the Wigner quasiprobab ility via the Bargmann representation of states demonstrate the action of t his relation. A short collection of different basic representations of the Wigner quasiprobability is given. An appendix presents results for the dise ntanglement of SU(1, 1)-group operators by products of special operators in different ordering.