The complete Gaussian class of quasiprobabilities 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 quasiprobabilities, which can be obtained by convolutions of the Wigner quasiprobability with the complete class of normalized Gaussian functions, is represented by a three-dimensional complex vector parameter r = (r(1), r(2), r(3)) with the property of additivity when composing convolutions and meaning that the transition between two quasiprobabilities with the vector parameters r and s is given by the convolution with a Gaussian function belonging to the vector parameter r - s. The scalar product r(2) = r(1)(2) + r(2)(2) + r(3)(3) of r with itself is related to the determinant of the second-rank symmetric tensor of the quadratic 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 within this Gaussian class with the vector parameter r = (0, 0, 0). The class of s-ordered quasiprobabilities is described by the vector parameters r = (0, 0, r(3) = -s) with -1 less than or equal to r(3) less than or equal to 1 and its diagonalization is connected with displaced Fock states \alpha, n). The class of quasiprobabilities corresponding to the linear interpolation between standard and antistandard ordering belongs to the vector parameter r = (r(1), 0, 0) with -1 less than or equal to r(1) less than or equal to 1. Its diagonalization is connected with discrete series of excitations of the eigenstates of the canonical operators Q and P. The diagonalization of the complete Gaussian class 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 discrete excitations of squeezed coherent states with mutually orthogonal squeezing axes and properties of orthogonality and completeness and provides a possible generalization of the displaced Pock states. The diagonalization is basically different for real and complex vector parameters r. A generalization of the coherent-state quasiprobability, which uses squeezed coherent states instead of coherent states, belongs to a vector 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.