Issue with Ramanujan Summation

[Isaac0427]

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, lim_n→∞(-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!

 

[JorisL]

See this post by micromass: https://www.physicsforums.com/threads/1-2-3-4-1-12-weirdness.817600/#post-5132436

I think it'll help you understand what the problem is.

 

[Isaac0427]

See this post by micromass: https://www.physicsforums.com/threads/1-2-3-4-1-12-weirdness.817600/#post-5132436

I think it'll help you understand what the problem is.

Thanks! That clears it up mostly, however I still don't know how this is used in string theory.

 

[JorisL]

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.

 

[Isaac0427]

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.

 

[JorisL]

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.

 

[Isaac0427]

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?

 

[JorisL]

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.