"Elegantly written, with obvious appreciation for fine points of higher mathematics...most notable is [the] author's effort to weave classical probability theory into [a] quantum framework." – The American Mathematical Monthly
"This is an excellent volume which will be a valuable companion both for those who are already active in the field and those who are new to it. Furthermore there are a large number of stimulating exercises scattered through the text which will be invaluable to students." – Mathematical Reviews
An Introduction to Quantum Stochastic Calculus aims to deepen our understanding of the dynamics of systems subject to the laws of chance both from the classical and the quantum points of view and stimulate further research in their unification. This is probably the first systematic attempt to weave classical probability theory into the quantum framework and provides a wealth of interesting features:
The origin of Ito's correction formulae for Brownian motion and the Poisson process can be traced to communication relations or, equivalently, the uncertainty principle.
Quantum stochastic interpretation enables the possibility of seeing new relationships between fermion and boson fields.
Quantum dynamical semigroups as well as classical Markov semigroups are realized through unitary operator evolutions.
The text is almost self-contained and requires only an elementary knowledge of operator theory and probability theory at the graduate level.
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 quantum dynamical semigroups and the Gorini, Kossakowski, Sudarshan, Lindblad Theorem.- References.- Author Index.
