THE COMPLETE GAUSSIAN CLASS OF QUASI-PROBABILITIES AND ITS RELATION TO SQUEEZED STATES AND THEIR DISCRETE EXCITATIONS

The requirements on the general structure of quasiprobabilities for a single boson mode are investigated. The complete Gaussian class of qua siprobabilities, which can be obtained by convolutions of the Wigner q uasiprobability with the complete class of normalized Gaussian functio ns, is represented by a three-dimensional complex vector parameter r = (r(1), r(2), r(3)) with the property of additivity when composing con volutions and meaning that the transition between two quasiprobabiliti es with the vector parameters r and s is given by the convolution with a Gaussian function belonging to the vector parameter r - s. The scal ar product r(2) = r(1)(2) + r(2)(2) + r(3)(3) of r with itself is rela ted to the determinant of the second-rank symmetric tensor of the quad ratic form in the exponent of the Gaussian functions. This is obtained by a mapping with the two symmetric Pauli spin matrices and the unity matrix. The Wigner quasiprobability takes on the central position wit hin this Gaussian class with the vector parameter r = (0, 0, 0). The c lass of s-ordered quasiprobabilities is described by the vector parame ters r = (0, 0, r(3) = -s) with -1 less than or equal to r(3) less tha n or equal to 1 and its diagonalization is connected with displaced Fo ck states \alpha, n). The class of quasiprobabilities corresponding to the linear interpolation between standard and antistandard ordering b elongs to the vector parameter r = (r(1), 0, 0) with -1 less than or e qual to r(1) less than or equal to 1. Its diagonalization is connected with discrete series of excitations of the eigenstates of the canonic al operators Q and P. The diagonalization of the complete Gaussian cla ss of quasiprobabilities with real vector parameter r = (r(1), r(2), r (3)) and r(2) less than or equal to 1 leads to dual systems of discret e excitations of squeezed coherent states with mutually orthogonal squ eezing axes and properties of orthogonality and completeness and provi des a possible generalization of the displaced Pock states. The diagon alization is basically different for real and complex vector parameter s r. A generalization of the coherent-state quasiprobability, which us es squeezed coherent states instead of coherent states, belongs to a v ector parameter r = (r(1), r(2), r(3)) with real r(3) and imaginary r( 1) and r(2) and with r(2) = 1. New representations of special classes of quasiprobabilities are derived.