Tachyon vertex operator (Polchinski's book)

Question

• I would like to know how does Polchinski in his book "derive" what is the "tachyon vertex operator" (..as say stated in equation 3.6.25, 6.2.11..) I can't locate a "derivation" of the fact that $:e^{ikX}:$ is the tachyon vertex operator.

  (..I understand that it follows from some application of the state-operator map but I can't put it together..)

• And then what is the meaning of the ``higher vertex operators" - which are of the form of arbitrary number of either operators of the above kind or the derivatives of $X$ w.r.t either $z$ or $\bar{z}$. (..like in equation 6.2.18..)

Answer 1

Firstly, I would like to figure out this: $$\left[\hat{p}^{\mu},\exp(ikx)\right]=\left[\hat{p}^{\mu},\sum_{n=0}^{\infty}\frac{1}{n!}(ik\hat{x})^{n}\right]=\sum_{n=0}^{\infty}\frac{1}{n!}\left[\hat{p}^{\mu},(ik\hat{x})^{n}\right]$$ As we can see, for $n=0$, $\left[\hat{p}^{\mu},(ik\hat{x})^{n}\right]$ gives zero. So in the following cases I'll solve the problem for the case in which n is not zero. $$\left[\hat{p}^{\mu},(ik\hat{x})\right]=ik_{\nu}\left[\hat{p}^{\mu},\hat{x}^{\nu}\right]=k^{\mu}$$ $$\left[\hat{p}^{\mu},(ik\hat{x})^{n}\right]=nk^{\mu}(ik\hat{x})^{n-1}$$ Now, replacing these results in the first equation: $$\left[\hat{p}^{\mu},\exp(ikx)\right]=\sum_{n=1}^{\infty}\frac{1}{(n-1)!}k^{\mu}(ik\hat{x})^{n-1}=k^{\mu}\exp(ik\hat{x})$$ A second point it's the following: $$\hat{p}^{\mu}\exp(ik\hat{x})|0,0>=\exp({ik\hat{x}})\hat{p}^{\mu}|0,0>+k^{\mu}\exp({ik\hat{x}})|0,0>$$ $$\hat{p}^{\mu}\exp(ik\hat{x})|0,0>=k^{\mu}\exp({ik\hat{x}})|0,0>$$ Now, I would like to identify the state $\exp(ik\hat{x})|0,0>$ as $|0,k>$, by obvious reasons.

To end, Polchinski wrote in his book the following:

Any state can be obtained from $|0,0>$ by acting with the operators $\alpha^{\mu}_{-m},\tilde{\alpha}^{\mu}_{-m},x^{\mu}_{0}$. The operator corresponding to this state is then given by the : : normal-ordered product of the corresponding local operators.
The corresponding operators are on the book, what you only need to know to solve this problem is the corresponding operator for $x^{\mu}_{0}$ is $X^{\mu}(0,0)$.

Therefore, the corresponding operator to the state $|0,k>$ is $:e^{ikX(0,0)}:$

Answer 2

Polchinski explains the state-operator correspondence in section 2.8, in particular equations 2.8.3, 2.8.4, and 2.8.9.

What you call "higher vertex operators" create multiple particles (if there are multiple exponential vertex operators) with higher spin (if there are extra derivatives multiplying the exponentials).