An introduction to the tomographic picture of quantum mechanics

29 pages, no figures.-- PACS codes: 03.65.Wj, 03.65.Ca, 03.67.Lx, 03.65.Fd, 42.50.Dv, 03.65.Ud.-- ArXiv pre-print available at: http://arxiv.org/abs/0904.4439v1Starting from the famous Pauli problem on the possibility of associating quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc is clarified and a connection with abstract Hilbert space properties is established. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two-qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.VIM thanks Dipartimento di Scienze Fisiche dell’Università ‘Federico II’ of Naples and INFN, Sezione di Napoli and Departamento de Matemáticas, Universidad Carlos III de Madrid for kind hospitality and the Russian Foundation for Basic Research for partial support under project nos 07-02-00598 and 09-02-00142