From Semiclassical to Quantum Approach

[lokofer]

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 $$\hbar$$ )

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

The problem is that the "series" involving $$\hbar$$ 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:

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

With $$B(u,X)= \sum_{n=1}^{\infty} a(n,X)\frac{u^{n}}{n!}$$

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

 

[Haelfix]

Thats correct, but note that Borel summation is limited by its own axioms, you cannot use it for every asymptotic series, just some nice ones (and the ones where you can that have physical relevance are typically famous and rare examples)