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

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).