University of Helsinki, Helsinki 2006
Mathematical Aspects of Functional Integration
Master's thesis, November 2000.
In the last fifty years funtional integration has become an important tool in physics and mathematics. Functionaal integration techniques are very versatile and can be applied to various types of physical problems. Functional integration is also more than just a calculational method; for example, the Feynman path integral is in fact an independent formulation of quantum mechanics.
Functional integration divides into two subclasses: the Wiener integral encountered in connection with diffusion and the Feynman path integral which is used to describe quantum phenomena. In this work we review the properties of the two integrals and compare them to each other. We find that even though they have almost the same mathematical structure the differences between them are profound.
Special emphasis is given to the Feynman path integral. We make the observation that it is not a true integral over a space of functions like the Wiener integral. It is only a shorthand notation for a limit of multiple integrals.
Ther have been several attempts to formulate a definition of Feynman path integrals that would be mathematically sound. Some of these are reviewed in this work. Even though they have had some success, none of them has achieved the intuitiveness of the original definition by Feynman. We also find them often too abstract to be useful. More research should thus be aimed at finding a proper mathematical definition for Feynman path integrals. This is prompted by their widespread use which often neglects the problems of the definition.
In addition to the discussion on the justification of functional integration we discuss some more specialized subjects, such as stochastic integration, discretization procedure of the Feynman path integral and the Feynman path integral on spaces with curvature.
We also present an extensive list of references on the subject.
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© University of Helsinki 2006
Last updated 08.08.2006