(adsbygoogle = window.adsbygoogle || []).push({}); From "Semiclassical" to Quantum Approach..

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)

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# From Semiclassical to Quantum Approach

**Physics Forums | Science Articles, Homework Help, Discussion**