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We know by no-ghost theorem (Polchinski I, section 4.4) that the physical Hilbert space have no longitudinal excitations ($X^0, X^1, b, c$). This is obvious by light-cone gauge quantization and in fact BRST cohomology is isomorphic to this space (no-ghost theorem). Witten (beginning of chap. 2) in "Non-commutative geometry and string field theory" also argues that physical states should not contain ghosts "since the traditional covariant quantization did not involve ghosts at all". He then affirm that "Physical States of bosonic Open String has ghost number 1/2" (ghost number 1/2 = annihilated by all ghost (and anti-ghost) annihilation operators AND by zero mode antighost $b_0$). But in the most part of reviews - Ashoke Sen TASI lecture 2003- about STF the statement is that "physical states has ghost number 1". I don't understant this difference on notation. I most agree with Witten argument, but in practicle is very convenient to work with $|\psi\rangle = \int d^{26}k(T(k)c_1 + C(k)c_0 + A_\mu(k)\alpha_{-1}^\mu c_1 + \cdots)|0;k\rangle$, which explicitly have ghost number 1. I know that these ideas are equivalent but I don't know how. I think I don't understant Witten's statement. Can someone help me to understand this equivalence of these different notations? Thanks

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  • $\begingroup$ This also is related to the fact that zero mode state is of the form $c_1\alpha_{-1}|0\rangle$, or by Vertex Operator, by S-O correspondence, $c\partial X$, in fact $c\partial X(0)|0\rangle = c_1\alpha_{-1}|0\rangle$. In general vertex operator are of the form $cW$ for open string. Why the need for the $c$ ghost in front of $W$ a one dim. primary field? $\endgroup$ Jul 10, 2020 at 0:35

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This is just a question of convention: Polchinski, Witten and most older papers quote the ghost number on the cylinder, which is half-integer and such that the ghost vacua have $\pm 1/2$. On the other hand, most recent papers quote the ghost number on the plane, which is an integer and such that the ghost vacua have ghost numbers $1$ and $2$.

[A small caveat about my last explanation: since the ghost number is an additive number, you can always choose to shift all of them by a constant, which explains why some papers may say that ghost numbers on the plane are half-integers: to obtain this, they shift the ghost number. There is a second point to take care of: when using $A_\infty$ homotopy algebra to describe the open string field theory, it is quite common to not work with the ghost number but with a related number called the degree (they are related by what is called a suspension). One has $\mathrm{degree} = N_{gh} - 1$, in which case the $\mathrm{U}(1)$ charge of the classical string field is $0$ instead of $1$. Details are not important, just to remember that it's another possible source of language mismatch.]

The reason is that the ghost current is not primary, which means that it does not transform covariantly under changes of coordinates: $$ j(z) = \frac{dw}{dz} \, j'(w) + \frac{q}{2} \, \frac{d}{dz} \, \ln \frac{dw}{dz}, $$ and as a consequence the ghost number is not invariant. Going from the cylinder with coordinates $w$ to the plane with coordinates $z = \mathrm{e}^w$, we find that the ghost number varies as: $$ N_{\text{gh}}^{\text{plane}} = N_{\text{gh}}^{\text{cyl}} - \frac{q}{2}. $$ Here, $q$ is the background charge and measures the violation of the ghost charge conservation. We have $q = - 3$ for the ghosts.

You can find more details in the draft of my book, sections 7.2.3 and 7.2.6 (see in particular remark 7.6).

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