Feynman integral with harmonic analysis and related topics

05/20/2015 - 05/22/2015
Feynman introduced a famous heuristic formula which is known as "Feynman path integral" for the evolution of a non-relativistic quantum system. The integral has been approached from many different point of view by mathematicians and physicists with varied background and interests. Wiener process also plays an important role in stochastic calculus, applied and financial calculus and in theoretical physics such as quantum physics, statistical physics and condensed matter physics, etc. In particular, the Wiener measure can be widely applicable to the mathematical physics including quantum mechanics. The followings are shortcomings of many of the mathematical theories of the Feynman integral which are often pointed out. (1) The existence theories are not sufficiently general. In particular, many of the standard real valued time independent potentials which are used in modeling quantum systems are singular (ex. the attractive Coulomb potential) and do not fit within the theories.) (2) No much information is given about how the various approaches to the Feynman integral are related to one another or to the unitary group which gives the evolution of the quantum systems in the standard approach to quantum dynamics. (3) There is a shortage of satisfactory limiting theorems. Indeed, in some cases, no such theorems are available, while in others, the results do not seem natural from a physical point of view. The goal of this workshop is that we discuss quite satisfactory responses to all three objections, especially for the approaches to the Feynman integral which are developed using Wiener integral or path integral without measures.