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Playing with functions of positive type, classical and quantum

A function of positive type can be defined as a positive functional on a convolution algebra of a locally compact group. In the case where the group is abelian, by Bochner's theorem a function of positive type is, up to normalization, the Fourier transform of a probability measure. Therefore, considering the group of translations on phase space, a suitably normalized phase-space function of positive type can be regarded as a realization of a classical state. Thus, it may be called a function of classical positive type. Replacing the ordinary convolution on phase space with the twisted convolution, one obtains a noncommutative algebra of functions whose positive functionals we may call functions of quantum positive type. In fact, by a quantum version of Bochner's theorem, a continuous function of quantum positive type is, up to normalization, the (symplectic) Fourier transform of a Wigner quasi-probability distribution; hence, it can be regarded as a phase-space realization of a quantum state. Playing with functions of positive type, classical and quantum, one is led in a natural way to consider a class of semigroups of operators, the classical-quantum semigroups. The physical meaning of these mathematical objects is unveiled via quantization, so obtaining a class of quantum dynamical semigroups that, borrowing terminology from quantum information science, may be called classical-noise semigroups.