A measure theoretical approach to quantum stochastic processes (Q384801)

The monograph under review presents an approach to quantum stochastic and white noise type analysis due to the author, based on one hand on the mixture of combinatorial methods of the theory of kernels and measure-theoretic techniques, and on the other on tools of the semigroup theory. The special focus, making it a very characteristic aspect of the text in comparison to other introductions to quantum stochastics, is put on the study of several concrete quantum physical examples, which are analysed in detail both from the point of view of the assumptions and approximations leading to a concrete mathematical model and from the point of view of the consequences and interpretations of the solutions obtained. The detailed plan of the book is as follows: Chapter 1 contains purely algebraic preliminaries, such as the construction of the Weyl algebra and identification of its basis, discussion of the Wick ordering theorem and some of its consequences and the definition of creation and annihilation operators associated to a finite set. In Chapter 2 creation and annihilation operators are studied in the continuous setting; here the sum-integral theorem for kernels is phrased and proved for measures and shown to lead to a natural notion of Fock space white noise operators. Chapter 3 contains a short introduction to one-parameter semigroup theory, with special focus on unitary evolutions, resolvent methods and some distributional techniques. These are applied in Chapter 4 to the analysis of four physical models, mainly related to simple, `finite' quantum systems interacting with a quantum field of a certain type. Here also the singular coupling procedure is described, leading to the explicit description of the associated quantum evolutions. Chapters 5-7 concern a more advanced white noise theory; here Wick theorem is revisited in the continuous setting, certain iterated integrals defined, quantum stochastic processes introduced and the quantum version of Itô's theorem formulated in the framework developed by the author. In Chapter 8 these are used to study certain concrete quantum stochastic differential equations à la Hudson-Parthasarathy, related to models of Chapter 4; in particular the solution of the relevant equation is shown to arise from integrals described earlier, and proved to be a unitary cocycle for the time-shift. The last fact allows the author to associate with it a natural unitary semigroup and characterize explicitly its generator, playing the role of the Hamiltonian of the system. Another model, that of an amplified oscillator, is discussed in Chapter 9; the main difference there is that this time the equation is driven by unbounded coefficients. The last chapter contains a brief analysis of the approximation of the solutions via so-called coloured noises. The mathematical background required of the reader seems to be somewhat varied: for example no prior knowledge of \(C_0\)-semigroup theory is assumed, but certain familiarity with the theory of distributions is clearly a prerequisite, as is the knowledge of basics of quantum mechanics.