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I met some problem about the Virasoro operator in "old covariant quantization" in Polchinski's string theory vol I p 123.

It is given $$L_0^{\rm m}=\alpha' p^2 + \alpha_{-1} \cdot \alpha_1 + \cdots \tag{4.1.11a} $$ But on p 59, $$ L_0 = \frac{ \alpha' p^2 }{4} + \sum_{n=1}^{\infty} ( \alpha^{\mu}_{-n} \alpha_{\mu n} ) + a^X \tag{2.7.7} $$

Why there is a factor of $1/4$ difference in the first terms in Eqs. (2.7.7) and (4.1.11a)? Is that just a convention?

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$(2.7.7)$ is about the closed string, while $(4.1.11a)$ is about the open string.

You can compare the expansions of the closed string and the open string in $(2.7.4)$ and $(2.7.26)$.

You see that the term in front of $-ip^\mu \ln|z|^2$ is $\frac{\alpha'}{2}$ for the closed string, and $\alpha'$ for the open string.

When looking at the stress-energy tensor (see $2.4.4$), you have quadratic quantities of $\partial_a X^\mu$ divided by $\alpha'$, so you get a term $\frac{\alpha'}{4}$ for the closed strings and a term $\alpha'$ for the open string.

Now, the constant part of $L_o$ (the zero mode of the stress-energy tensor) ($\sim p^2$) is directly related to this term, so it explains the difference.

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