Completely divergent series

In summary, the conversation discusses a divergent series represented by a function with a coupling constant g and a parameter epsilon approaching 0. It is stated that this series is summable and can produce a correct result, S. However, it is also mentioned that S has a singularity at epsilon=0. The main question posed is how to remove this singularity using renormalization methods.
  • #1
eljose
492
0
let be the completely divergent series at [tex]\epsilon\rightarrow{0}[/tex] in the form of:

[tex]\sum_{n=0}^{\infty}\frac{a(n)g^{n}}{\epsilon^{n}}[/tex]

where g is the coupling constant of our theory..then let,s suppose this series is summable and that we can get the correct result S

[tex]S=S(g,\epsilon)[/tex] then let,s suppose that S have a singularity at
[tex]\epsilon=0[/tex] my question is how we could remove this singularity by renormalization methods...thanks.
 
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  • #2
I'm scooting this back out of the Homework section, because it's not an undergraduate level question, and I don't think that it is a textbook question anyway. Eljose tends to work on more open-ended stuff.
 
  • #3


Thank you for sharing this interesting concept with me. The concept of completely divergent series is a common one in theoretical physics, and it is often encountered when dealing with quantum field theories. It is a series that, as you mentioned, becomes infinite as the parameter \epsilon approaches zero.

In this case, the singularity at \epsilon=0 indicates that our theory is not well-defined at this point. This is a common problem in quantum field theories, as they often involve infinite quantities that need to be renormalized. Renormalization is a technique used to remove these infinities and make the theory well-defined.

To remove the singularity at \epsilon=0, we can use the renormalization methods of dimensional regularization or zeta function regularization. These methods involve introducing a new parameter, such as the number of dimensions or the zeta function, which allows us to manipulate the series and remove the singularity. This results in a finite, renormalized value for S.

Renormalization is an essential tool in theoretical physics, and it allows us to make meaningful predictions and calculations in quantum field theories. It is a complex and intricate process, and it requires a deep understanding of the underlying theory and its mathematical framework.

In summary, the singularity at \epsilon=0 in the completely divergent series can be removed through renormalization methods, which are techniques used to make quantum field theories well-defined and predictive. I hope this helps to clarify the concept of completely divergent series and the role of renormalization in theoretical physics.
 

1. What is a completely divergent series?

A completely divergent series is a mathematical series in which the sum of its terms diverges to positive or negative infinity. This means that the series does not have a finite sum and does not converge to a specific value.

2. How can you determine if a series is completely divergent?

A series can be determined to be completely divergent by using various convergence tests, such as the ratio test, root test, or comparison test. If these tests show that the series does not converge, then it can be concluded that the series is completely divergent.

3. What is the significance of completely divergent series?

Completely divergent series are important in mathematics because they can provide insights into the behavior of infinite series. They also have applications in different fields, such as physics and engineering.

4. Can completely divergent series be manipulated algebraically?

No, completely divergent series cannot be manipulated algebraically because they do not have a finite sum or a defined value. Any algebraic operations performed on completely divergent series would result in undefined or infinite values.

5. Are completely divergent series always considered "bad" or useless?

No, completely divergent series are not always considered "bad" or useless. In some cases, they can still provide useful information or insights into the behavior of a series. However, they are not typically used in practical applications due to their lack of convergence and defined value.

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