Issue with Ramanujan Summation

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In summary: And these modes satisfy the commutation relations of the Poincaré group. By demanding this you find that the number of spacetime dimensions is fixed to be 26
  • #1
Isaac0427
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I feel like Ramanujan Summation is just very bizarre. How can 1+2+3+4...=-1/12? It all rests in the assumption that ∑n=0(-1)n=.5. However, in calculus, limn→∞(-1)n=undefined. The limit does not exist. It is not 0, the average of -1 and 1 which are the only values of the function (if the domain is only integers). Yet, there must be some sense in it as it is used in string theory. Can somebody please explain this. Thanks!
 
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  • #4
Do you have access to a copy of Becker,Becker and Schwarz's book (library, interlibrary loan perhaps)?

It's used in section 2.5 there for the Bosonic string.

It has to do with consistency of the theory. There are several approaches.
- In BBS they calculate the normal ordering constant in a certain generator of the Virasoro algebra.
- You want the generators of the Lorentz symmetry to satisfy the regular commutation relations. (A sketch can be found in section 12.5 of Zwiebach)

If you are comfortable with GR and have studied intro QFT I'd go for BBS.
The second approach is very clear to link to the physics (we want Lorentz invariance after all)

In conclusion it's all about consistency.
 
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  • #5
JorisL said:
Do you have access to a copy of Becker,Becker and Schwarz's book (library, interlibrary loan perhaps)?

It's used in section 2.5 there for the Bosonic string.

It has to do with consistency of the theory. There are several approaches.
- In BBS they calculate the normal ordering constant in a certain generator of the Virasoro algebra.
- You want the generators of the Lorentz symmetry to satisfy the regular commutation relations. (A sketch can be found in section 12.5 of Zwiebach)

If you are comfortable with GR and have studied intro QFT I'd go for BBS.
The second approach is very clear to link to the physics (we want Lorentz invariance after all)

In conclusion it's all about consistency.
Not quite ready for that book, still on Griffith's intro to quantum mechanics.
 
  • #6
Have you looked at commutators already?

We often know which symmetries we want the theory to have.
The symmetries satisfy some commutation relations we know beforehand.

The result of the summation comes about by demanding this commutator to hold. As a consequence you find that the bosonic string needs 26 spacetime dimensions.

This is a quick and dirty sketch of the way the result is used.
 
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  • #7
JorisL said:
Have you looked at commutators already?

We often know which symmetries we want the theory to have.
The symmetries satisfy some commutation relations we know beforehand.

The result of the summation comes about by demanding this commutator to hold. As a consequence you find that the bosonic string needs 26 spacetime dimensions.

This is a quick and dirty sketch of the way the result is used.
So by the symmetry, are you saying that the commutator must equal zero?
 
  • #8
Some of them are, you can look at the Lorentz group https://en.wikipedia.org/wiki/Lorentz_group

In fact the article on the Poincaré group is better https://en.wikipedia.org/wiki/Poincaré_group
You want to look at the bottom relation in the "details"-section. The link to the Lorentz group is made there as well.

All of this will probably be (way) over your head (at least the language used).
In string theory the generators are expanded in terms of ("vibrational") modes, very similar to the modes of a harmonic oscillator.
 

Related to Issue with Ramanujan Summation

1. What is Ramanujan Summation?

Ramanujan Summation is a method used to assign a finite value to divergent series, which are series that do not have a defined sum. It was developed by the Indian mathematician Srinivasa Ramanujan.

2. Why is there an issue with Ramanujan Summation?

The issue with Ramanujan Summation is that it can produce different results depending on the order in which the terms of a series are added. This is known as the order of summation problem.

3. How does the order of summation problem affect the accuracy of Ramanujan Summation?

The order of summation problem can lead to different values for the sum of a series, which can make it difficult to determine the true value of a series. This can also result in errors when using Ramanujan Summation to solve real-world problems.

4. Are there any methods to address the issue with Ramanujan Summation?

Yes, there are several methods that have been proposed to address the issue with Ramanujan Summation. These include the Abel Summation Method, Cesàro Summation Method, and Borel Summation Method. These methods aim to provide a more accurate and consistent value for divergent series.

5. Can Ramanujan Summation be used in all mathematical calculations?

No, Ramanujan Summation should only be used in specific cases where it has been proven to provide accurate results. It is not a general method for solving all mathematical problems and should be used with caution, especially in cases where the order of summation problem can significantly affect the outcome.

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