Path integration at the crossroad of stochastic and differential calculus

Laura M. Morato

Horacio S. Wio

伸明 尾畑, 卓 松井, 彰人 洞

Mathematical Theory of Quantum Particles Interacting with a Quantum Field (A Arai) H-P Quantum Stochastic Differential Equations (F Fagnola) Quantum White Noise Calculus (U C Ji & N Obata) Can Quantumness Be an Origin of Dissipation? (T Arimitsu) What is Stochastic Independence? (U Franz) Creation-Annihilation Processes on Cellar Complecies (Y Hashimoto) Fock Space and Representation of Some Infinite-Dimensional Groups (T Matsui & Y Shimada) Free Product Actions and Their Applications (Y Ueda) Remarks on the s-Free Convolution (H Yoshida) and other papers.

Werner Fischer, Hajo Leschke, Peter Müller

In a reasonably self-contained presentation mathematical rigour is supplied to the important ideas of solving certain non-Gaussian path integrals by changes of dimension and/or path-dependent time transformations. The resulting genuine path-integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations. The power of the calculus is illustrated by two standard quantum-physics applications. First, the calculation of the time-dependent propagator corresponding to a particle on the half-line in a harmonic plus inverse-square potential is shown to be a simple exercise. Second, the first rigorous derivation of the energy-dependent Green function of the one-dimensional Morse system is given.

Michio Masujima

Zhiyuan Huang, Shunlong Luo

A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated. As applications, quantum stochastic integrals and quantum stochastic differential equations are treated from the viewpoint of Wick calculus.