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### Quantum expressions for the Virasoro constraints

I am trying to derive the quantum form of the Virasoro constraints. $$L_{m} = \frac{1}{2} \sum_{n} :\alpha_{m-n}.\alpha_{n}:$$ :...: meaans normal ordering. Using the common commutator between ...
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### Applications of $1+2+3+… = -1/12$

The equation $$1 + 2 + 3 + \dots = -1/12$$ is quite famous. From the point of view of mathematics, I have no problem with it. My (probably naive) understanding is that there are certain "sums'' ...
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### Is the fact that the sum of all natural numbers $\sum_{n=1}^\infty n = -\frac{1}{12}$ essential to the understanding of the Casimir Force In QED?

Apparently this result is used in many areas physics including the extra dimensions of string theory, which is not the scope of the question. The result is apparently also used to understand the ...
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### What is the reason/significance of using $\sum\limits_{n=1}^{\infty}n\rightarrow-\frac{1}{12}$?

What is the reason/significance of using a trick equation in the Volume I - String Theory - Joseph Polchinsky? I have no doubts at all that the author knows extremely well the subject and that this ...
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### How does the sum of natural numbers arise in the derivation of critical string dimensions?

In the standard treatment of bosonic string theory the “heuristic” argument for the critical dimension goes as follows (see Ref. 1-4). Upon quantization the mass-squared operator becomes normal ...
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### Have experiments ever suggested two different values to the same divergent series?

I believe to have understood that some physical experiments suggest finite values to divergent series (please correct me if I'm wrong, my understanding of these matters is limited). I heard, for ...