Abstract In this talk I will try to give a survey of some aspects of interrelations between stochastic analysis and quantum theory, emphasizing particularly some recent developments. It will be, by necessity, a partial account mainly influenced by developments in which I participated and many important topics will be only briefly mentioned, fortunately some are being discussed in other talks, to which I hope mine can be in some sense complementary. I shall also try to supply omitted topics by some appropriate references. First of all I want to mention that there are two main ways probability, and more particularly stochastic analysis, enters quantum mechanics. The first is a basic, conceptual one, which goes back to the very origins of the probabilistic interpretation of the wave function (Schrödinger, Born,...), and has a modern extremely interesting version in Nelson's stochastic mechanics and the work around it, a very illuminating report on this has been given by Francesco Guerra at this Meeting, so I will only touch briefly these aspects (let me mention however an important paper by Nelson [1] which has appeared after this conference and will be published in the Proceedings of the 1. International Ascona-Como Meeting; incidentally these Proceedings [2] will contain also several other contributions of great relevance to this point). Besides this "conceptual" relation between probability and quantum mechanics, there is a more "technical one", in which probability theory, in particular stochastic analysis, is used as a mathematical tool, to obtain results about quantum mechanical objects. Viceversa, concepts and methods forged in connection with quantum mechanics influence new developments in stochastic analysis. In this talk I will mainly concentrate on these aspects. I like however to emphasize that in many instances the distinction between "conceptual" and "technical" is in this context somewhat artificial. E.g. the approach to quantum mechanics by Dirichlet forms, to be discussed below, is in many ways (connection with stochastic mechanics, alternative formulation of quantum field theory) more than a technical tool.
Some points of interaction between stochastic analysis and quantum theory
1986-01-01
26 pages
Aufsatz/Kapitel (Buch)
Elektronische Ressource
Englisch
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