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B Issue with Ramanujan Summation

  1. Mar 12, 2016 #1
    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!
     
  2. jcsd
  3. Mar 12, 2016 #2
  4. Mar 13, 2016 #3
  5. Mar 13, 2016 #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.
     
  6. Mar 13, 2016 #5
    Not quite ready for that book, still on Griffith's intro to quantum mechanics.
     
  7. Mar 13, 2016 #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.
     
  8. Mar 13, 2016 #7
    So by the symmetry, are you saying that the commutator must equal zero?
     
  9. Mar 13, 2016 #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.
     
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