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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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### Regularization of the Casimir effect

For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have ...
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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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### 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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### What is the relation between renormalization in physics and divergent series in mathematics?

The theory of Divergent Series was developed by Hardy and other mathematicians in the first half of the past century, giving rigorous methods of summation to get unique and consistent results from ...
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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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### Critical Dimension of Bosonic Strings and Regularization of $\sum_{n=1}^\infty n$

If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at $$\frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0.$$ Now mathematically this is ...
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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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### Question about infinite sum in quantum field

I read from some books of number theory that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = -\frac{1}{12}\text{,when } s=-1.$$ Now is there such a result \sum_{n=1}^{\infty}\frac{1}{n^s} = \pi \text{,when } ...
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### Regularization and renomalization in the lightcone quantization of bosonic string

This question relates to this link. But I still don't understand it >_< In Polchinski's string theory vol I, p. 22, there is a divergence term (when $\epsilon \rightarrow 0$) in the zero point ...