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Following Joe Polchinski’s Little Book of String, page 12, he use the sum $$1+2+3+...=-1/12$$ to find the zero point energy of the bosonic string (and later used the result to argue that we must have 26 dimensions).

When we use such a normalization for example in QFT it is said that the specific value that we obtain is not important and that we only care for the differences between the difference results. (For example sometime this difference does not depend on high energies beyond our theory).

But it seems to me that such argument can only work providing that we admit that our theory is not complete and that we cannot calculate some values using it.

This doesn't seems to go along with the fact that string theory try to be a complete theory.

  1. Do I get something wrong about the using of such renormalization in string theory?

  2. Is there a more clear method to argue about the number of dimension in string theory?

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    $\begingroup$ See "this is the same result we got in section 2.2.2 but this time with no funny business where we throw out infinities" in damtp.cam.ac.uk/user/tong/string/string4.pdf $\endgroup$
    – Connor Behan
    Commented Jan 28, 2023 at 23:42
  • $\begingroup$ Can you explain why you say "But it seems to me that such argument can only work providing that we admit that our theory is not complete and that we cannot calculate some values using it."? Renormalization has nothing to do with incompleteness. $\endgroup$
    – Prahar
    Commented Feb 2, 2023 at 10:34
  • $\begingroup$ Because without the renormalization require you to add arbitrary axiom that let you to get the final answer $\endgroup$
    – ziv
    Commented Feb 2, 2023 at 14:15

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