Quantum Stochastic Analysis – an Introduction

Kalyanapuram Rangachari Parthasarathy

Kalyanapuram Rangachari Parthasarathy

R. L. Hudson

Rolando Rebolledo

The interpretation provided by Schrödinger of quantum mechanics through classical stochastic processes is the root of Euclidean quantum mechanics founded on Bernstein processes. Open quantum systems have been extensively investigated during the last three decades and have motivated new probabilistic interpretations for quantum dynamics. This article, mainly addressed to classical probabilists, gives a short introduction to quantum dynamical semigroups and quantum Markov flows as a natural extension of Schrödinger ideas in noncommutative stochastic analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nicola Cufaro Petroni, S. De Martino, Silvio De Siena +1 more

: We perform a detailed analysis of the non stationary solutions of the evolution (Fokker-Planck) equations associated to either stationary or non stationary quantum states by the stochastic mechanics. For the excited stationary states of quantum systems with singular velocity fields we explicitely discuss the exact solutions for the HO case. Moreover the possibility of modifying the original potentials in order to implement arbitrary evolutions ruled by these equations is discussed with respect to both possible models for quantum measurements and applications to the control of particle beams in accelerators. 1. Introduction In a few papers [1] the analogy between diffusive classical systems and quantum systems has been reconsidered from the standpoint of the stochastic mechanics (SM) [2], [3], and particular attention was devoted there to the evolution of the classical systems associated to a quantum wave function when the conditions imposed by the stochastic variational principle are...

Wladyslaw Adam Majewski

We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed.