Introduction

Feynman path integrals have been introduced by Feynman in his formulation of quantum mechanics [1].1 Since their inception they have occupied a somewhat ambiguous position in theoretical physics. On one hand they have been widely and profitably used in quantum mechanics, statistical mechanics and quantum field theory, because of their strong intuitive, heuristic and formal appeal. On the other hand most of their uses have not been supported by an adequate mathematical justification. Especially in view of the potentialities of Feynman's approach as an alternative formulation of quantum dynamics, the need for a mathematical foundation has been broadly felt and the mathematical study of Feynman path integrals repeatedly strongly advocated, see e.g. [4]. This is, roughly speaking, a study of oscillating integrals in infinitely many dimensions, hence closely connected with the development of the theory of integration in function spaces, see e.g. [5]. The present work intends to give a mathematical theory of Feynman path integrals and to yield applications to non relativistic quantum mechanics, statistical mechanics and quantum field theory. In order to establish connections with previous work, we shall give in this introduction a short historical sketch of the mathematical foundations of Feynman path integrals. For more details we refer to the references, in particular to the review papers [6].