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If we use the semiclassical expansion of a "path integral" to quantize a theory we have the "Asymptotic series (in [tex] \hbar [/tex] )

[tex] I= \int D[\phi]e^{-S[\phi]/\hbar}=I_{WKB}(1+ \sum_{n=1}^{\infty} a(n,X)\hbar ^{n}) [/tex]

The problem is that the "series" involving [tex] \hbar [/tex] does only converges for h-->0 ( with a few terms) for example for h-->1 the series is divergent although if we apply "Borel resummation" we get:

[tex] \sum_{n=1}^{\infty}a(n,X)\hbar ^{n} \rightarrow \int_{0}^{\infty}duB(u,X)e^{-u/\hbar}(1/\hbar) [/tex]

With [tex] B(u,X)= \sum_{n=1}^{\infty} a(n,X)\frac{u^{n}}{n!} [/tex]

So, an asymptotic series can be "summed" for every value of the argument (big or small)

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# From Semiclassical to Quantum Approach

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