# Understanding Borel transform (sum) in QM

In summary, the conversation is discussing the use of Borel transform in quantum mechanics to make divergent perturbation series convergent. The Borel transform is computed using the formula B(x)=\sum_{n=0}^{\infty}\frac{a(n)x^{n}}{n!}, but it can only be applied to certain cases where a(n)=(-1)^{n} or a(n)=(-1)^{n} n!. Without following a known pattern, the Borel transform is considered useless.
Hello, i have heard tat you can use Borel transform in QM so the divergent perturbation series become convergent.. i am a bit confused since if you must compute the Borel transform:

$$B(x)=\sum_{n=0}^{\infty}\frac{a(n)x^{n}}{n!}$$

But this can only be made for a few cases $$a(n)=(-1)^{n}$$ and
$$a(n)=(-1)^{n} n!$$

But if the a(n) don't follow a known pattern Borel Transform is useless..

sorry the first "image" (i really hate this f... Latex ) should be:

$$\sum_{n=0}^{\infty}(a(n) x^{n})/n!$$

-the second: $$a(n)= (-1)^{n} n!$$

- And the third $$a(n)= (-1)^{n}$$

Hello, thank you for your question. The Borel transform is a mathematical tool that can be used in quantum mechanics to help make divergent perturbation series converge. This is because the Borel transform allows us to resum the perturbation series, which means combining all the terms in the series to get a more accurate result.

You are correct that the Borel transform can only be computed for certain cases, such as when the a(n) values follow a known pattern like (-1)^n or (-1)^n n!. However, this does not mean that the Borel transform is useless for cases where a(n) does not follow a known pattern. In fact, there are techniques that can be used to approximate the Borel transform for these cases.

Furthermore, the Borel transform is just one tool that can be used to make perturbation series converge in quantum mechanics. There are other methods, such as renormalization, that can also be used. It is important for scientists to understand and use a variety of tools and techniques in their research, rather than relying on just one method.

I hope this helps clarify the role of the Borel transform in quantum mechanics. Keep exploring and learning about different mathematical tools and their applications in science!

## 1. What is the Borel transform in quantum mechanics?

The Borel transform in quantum mechanics is a mathematical operation used to convert a given function from its original domain (such as time or energy) to a new domain (such as frequency or momentum). It is commonly used in quantum field theory to analyze the behavior of particles and their interactions.

## 2. How is the Borel transform related to the sum over histories in quantum mechanics?

The Borel transform is closely related to the sum over histories formulation of quantum mechanics, which states that the probability amplitude for a particle to transition from one state to another is equal to the sum of all possible paths that the particle could take. The Borel transform allows for a mathematical representation of this sum over histories, making it a useful tool in calculating probabilities and analyzing quantum systems.

## 3. What are the applications of the Borel transform in quantum mechanics?

The Borel transform has various applications in quantum mechanics, including in the study of quantum field theory, quantum chromodynamics, and quantum electrodynamics. It is also used in the calculation of scattering amplitudes and cross-sections in particle physics experiments.

## 4. How does the Borel transform help with divergent series in quantum mechanics?

In quantum mechanics, some calculations can result in infinite series that do not have a finite sum. These divergent series can be difficult to work with, but the Borel transform offers a method for resumming them into a well-behaved function that can be used for further analysis. This allows for more accurate and meaningful results in quantum calculations.

## 5. Are there any limitations to the use of the Borel transform in quantum mechanics?

While the Borel transform is a powerful tool in quantum mechanics, it does have some limitations. It is not always possible to perform the Borel transform on a given function, and even when it is possible, the resulting transformed function may not always have a physical interpretation. Additionally, the Borel transform may introduce new mathematical complexities that can be challenging to handle.

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