An Introduction to Quantum Stochastic Calculus

I Events, Observables and States. - 1 From classical to quantum probability. - 2 Notational preliminaries. - 3 Finite dimensional quantum probability spaces. - 4 Observables in a simple quantum probability space. - 5 Variance and covariance. - 6 Dynamics in finite dimensional quantum probability spaces. - 7 Observables with infinite number of values and the Hahn-Hellinger Theorem. - 8 Probability distributions on? (?)and Gleason s Theorem. - 9 Trace class operators and Schatten s Theorem. - 10 Spectral integration and Stone s Theorem on the unitary representations of ? pk. - 11 Basic notions of the theory of unbounded operators. - 12 Spectral integration of unbounded functions and von Neumann`s Spectral Theorem. - 13 Stone generators, characteristic functions and moments. - 14 Wigner s Theorem on the automorphisms of ? (?). - II Observables and States in Tensor Products of Hubert Spaces. - 15 Positive definite kernels and tensor products of Hilbert Spaces. - 16 Operators in tensor products of Hilbert Spaces. - 17 Symmetric and antisymmetric tensor products. - 18 Examples of discrete time quantum stochastic flows. - 19 The Fock Spaces. - 20 The Weyl Representation. - 21 Weyl Representation and infinitely divisible distributions. - 22 The symplectic group of ? and Shale s Theorem. - 23 Creation, conservation and annihilation operators in ? a(?). - III Stochastic Integration and Quantum Ito s Formula. - 24 Adapted processes. - 25 Stochastic integration with respect to creation, conservation and annihilation processes. - 26 A class of quantum stochastic differential equations. - 27 Stochastic differential equations with infinite degrees of freedom. - 28 Evans-Hudson Flows. - 29 A digression on completely positive linear maps and Stinespring s Theorem. - 30 Generators of quantumdynamical semigroups and the Gorini, Kossakowski, Sudarshan, Lindblad Theorem. - References. - Author Index.