# The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth

## Abstract

Recently N. Kumano-go [<b>15</b>] succeeded in proving that piecewise linear time slicing approximation to Feynman path integral<br>\int F(\gamma)e^{i\nu S(\gamma)}\,\mathscr{D}[\gamma]<br>actually converges to the limit as the mesh of division of time goes to 0 if the functional <i>F</i>(γ) of paths γ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form;<br>$$\label{stieltjesint}%1F(\gamma) = \int_0^T f(t,\gamma(t)) \rho(dt), \tag{1}$$<br>where ρ(<i>t</i>) is a function of bounded variation and <i>f</i>(<i>t</i>, <i>x</i>) is a sufficiently smooth function with polynomial growth as |<i>x</i>| → ∞. Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties (see also [<b>10</b>]).<br>The present paper has two aims. The first aim is to show that a large part of discussion in [<b>15</b>] becomes much simpler and clearer if one uses piecewise classical paths in place of piecewise linear paths.<br>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 [<b>15</b>]. If <i>F</i>(γ) ≡ 1, this second term coincides with the one given by G. D. Birkhoff [<b>1</b>].

## Journal

• Tokyo Sugaku Kaisya Zasshi

Tokyo Sugaku Kaisya Zasshi 58(3), 837-867, 2006-07-01

The Mathematical Society of Japan

## References:  17

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## Codes

• NII Article ID (NAID)
10018381149
• NII NACSIS-CAT ID (NCID)
AA0070177X
• Text Lang
ENG
• Article Type
ART
• ISSN
00255645
• NDL Article ID
7987167
• NDL Source Classification
ZM31(科学技術--数学)
• NDL Call No.
Z53-A209
• Data Source
CJP  NDL  J-STAGE

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