Quantum Theory and Its Stochastic Limit

ISBN-13: 9783540419280 / Angielski / Twarda / 2002 / 474 str.

Luigi Accardi; Lu Yun Gang; Igor Volovich

Opis

Nowadays it is becoming clearer and clearer that, in the description of natural phenomena, the triadic scheme - microseopie, mesoscopic, macroscopic - is only a rough approximation and that there are many levels of description, probably an infinite hierarchy, in which the specific properties of a given level express some kind of cumulative or collective behaviour of properties or sys tems corresponding to the lower levels. One of the most interesting challenges for contemporary natural sciences is the comprehension of the connections among these different levels of description of reality and the deduction of the laws of higher levels in this hierarchy from basic laws corresponding to lower levels. Since these cumulative or collective phenomena are, typically, nonlin ear effects, the transition from this general program to concrete scientific achievements requires the developement of techniques which allow physical information to be extracted from nonlinear quantum systems. Explicitly in tegrable examples of such systems are rare, and the most interesting physical phenomena are not captured by them. Even in the case of linear systems the fact that an explicit solution is formally available is often useless, since it is impossible to interpret interesting physical phenomena from it."

Recenzje

"The authors have produced a very important and detailed book on the subject of the stochastic limit of quantum field theory. In particular, they formulate the stochastic limit in the framework of an algebraic central limit theory using the sort of scaling limits encountered in quantum transport phenomena. [...] The book is well written and covers many related problems [...] The audience will be mathematical and theoretical physicists. The authors are to be congratulated for showing the power and range of application of quantum probability to quantum physics." (Mathematical Reviews 2003h)

Spis treści

1. Notations and Statement of the Problem.- 2. Quantum Fields.- 3. Those Kinds of Fields We Call Noises.- 4. Open Systems.- 5. Spin—Boson Systems.- 6. Measurements and Filtering Theory.- 7. Idea of the Proof and Causal Normal Order.- 8. Chronological Product Approach to the Stochastic Limit.- 9. Functional Integral Approach to the Stochastic Limit.- 10. Low-Density Limit: The Basic Idea.- 11. Six Basic Principles of the Stochastic Limit.- 12. Particles Interacting with a Boson Field.- 13. The Anderson Model.- 14. Field—Field Interactions.- 15. Analytical Theory of Feynman Diagrams.- 16. Term-by-Term Convergence.- References.

Blurb

The subject of this book is a new mathematical technique, the stochastic limit developed for solving nonlinear problems in quantum theory involving systems with infinitely many degrees of freedom (typically quantum fields or gases in the thermodynamic limit). This technique is condensed into some easily applied rules (called "stochastic golden rules") which allow to single out the dominating contributions to the dynamical evolution of systems in regimes involving long times and small effects. In the stochastic limit the original Hamiltonian theory is approximated using a new Hamiltonian theory which is singular. These singular Hamiltonians still define a unitary evolution and the new equations give much more insight into the relevant physical phenomena than the original Hamiltonian equations. Especially, one can explicitly compute multi-time correlations (e.g. photon statistics) or coherent vectors, which are beyond the reach of typical asymptotic techniques as well as deduce in the Hamiltonian framework the widely used stochastic Schrödinger equation and the master equation. This monograph is well suited as a textbook in the emerging field of stochastic limit techniques in quantum theory.