Path Integrals and Oscillatory Integrals

Abstract

The time slicing approximation of a Feynman path integral does not converge absolutely. But it is expected to have a definite finite value, because the factor $\exp {i\nu S(\gamma _{\varDelta })(x_{J+1}, x_J,\dots , x_1, x_0)}$ oscillates rapidly and as a consequence there occurs a large scale of cancellation. Such an integral is commonly treated by oscillatory integral techniques and is given a definite value under some conditions. We give an example of a sufficient condition for that in Sect. 3.2. Furthermore, in such a case the stationary phase method, which is given by Theorem 3.5 in Sect. 3.3, gives the value of the oscillatory integral asymptotically as $\nu \rightarrow \infty$.