The Feynman integral is considered as an intuitive representation
of quantum mechanics showing the complex quantum phenomena in a
language comprehensible at a classical level. It suggests that the
quantum transition amplitude arises from classical mechanics by an
average over various interfering paths. The classical picture
suggested by the Feynman integral may be illusory. By most
physicists the path integral is usually treated as a convenient
formal mathematical tool for a quick derivation of useful
approximations in quantum mechanics. Results obtained in the
formalism of Feynman integrals receive a mathematical justification
by means of other (usually much harder) methods. In such a case the
rigour is achieved at the cost of losing the intuitive classical
insight. The aim of this book is to formulate a mathematical theory
of the Feynman integral literally in the way it was expressed by
Feynman, at the cost of complexifying the configuration space. In
such a case the Feynman integral can be expressed by a probability
measure. The equations of quantum mechanics can be formulated as
equations of random classical mechanics on a complex configuration
space. The opportunity of computer simulations shows an immediate
advantage of such a formulation. A mathematical formulation of the
Feynman integral should not be considered solely as an academic
question of mathematical rigour in theoretical physics.
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