Recently N. Kumano-go [15] succeeded in proving that piecewise linear time slicing approximation to Feynman path integral $$\int F(\gamma)e^{i\nu S(\gamma)}\mathscr{D}[\gamma]$$ actually converges to the limit as the mesh of division of time goes to $0$ if the functional $F(\gamma)$ of paths $\gamma$ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form;
$$F(\gamma) = \int_0^T f(t,\gamma(t)) \rho(dt), \qquad (1)$$
where $\rho(t)$ is a function of bounded variation and $f(t,x)$ is a sufficiently smooth function with polynomial growth as $|x| \to \infty$ . Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties (see also [10]).
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The present paper has two aims. The first aim is to show that a large part of discussion in [15] becomes much simpler and clearer if one uses piecewise classical paths in place of piecewise linear paths.
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The second aim is to explain that the use of piecewise classical paths naturally leads us to an analytic formula for the second term of the semi-classical asymptotic expansion of the Feynman path integrals under a little stronger assumptions than that in [15]. If $F(\gamma)\equiv 1$ , this second term coincides with the one given by G. D. Birkhoff [1].