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)

 

[Entropia]

using "Borel resummation". However, this approach is only valid in the semiclassical limit, where \hbar is small. In order to fully understand the quantum behavior of a system, we need to take into account the full quantum approach, where \hbar cannot be ignored.

This means that we need to consider the full quantum mechanical equations and principles, such as the Schrödinger equation and the Heisenberg uncertainty principle, in order to accurately describe and predict the behavior of a system at the quantum level. The semiclassical approach is a useful tool for approximating certain systems, but it is limited in its ability to fully capture the complexity of quantum behavior.

As scientists, we must constantly strive to improve our understanding and methods in order to accurately describe and predict the behavior of the natural world. The transition from the semiclassical to the quantum approach is a reflection of this ongoing effort to refine our understanding of the universe. By incorporating the principles of quantum mechanics, we can gain a deeper understanding of the fundamental nature of reality and make more accurate predictions about the behavior of physical systems.