In chapter 7 of superstring theory, it is written $$ g\langle0;k_1|\zeta\cdot\alpha_1V_0(k_2)\zeta_3\cdot\alpha_{-1}|0;k_3\rangle=g\langle0;k_1|\zeta\cdot\alpha_1e^{k_2\cdot\alpha_{-1}} e^{-k_2\cdot\alpha_{1}}\zeta_3\cdot\alpha_{-1}|0;k_3\rangle. \tag{7.1.56}$$ Why is it that only $\alpha_1$ and $\alpha_{-1}$ in $X^\mu$ are relevant to the calculation?


Here, $V_0(k_2)$ could have been replaced by $e^{k_2 \alpha_{-1}} e^{-k_2 \alpha_1}$ while the factors from $\alpha_{\pm n}$ for $n\gt 1$ could have been neglected because $\alpha_n$ annihilates everything that appears in the matrix element on the right side from $V_0$ (because it ultimately annihilates $|0\rangle$), and similarly for $\alpha_{-n}$ that annihilates the bra $\langle 0 |$.

The exponentials of these multiples of $\alpha_{\pm 2}$ etc. are therefore effectively equal to $\exp(0)=1$, in that situation. The oscillators $\alpha_{\pm 1}$ are the only exceptions because of the extra noncommuting $\alpha_{\pm 1}$ that appear in the matrix element.

The zero mode $x_0$ hiding on $X$ was ignored by Green-Schwarz-Witten but it's mostly because their treatment was a bit sloppy or heuristic. This $x_0$ factor of the calculation produces the delta-function $\delta(k_1+k_2+k_3)$ with the right signs and perhaps powers of $(2\pi)$ which they know to be there but they sometimes omit it when they talk about "the amplitude".

This zero-mode part is one that also appears in point-like particle field theories and they only focus on the factors that are "new" in string theory relatively to point-like QFTs.

  • $\begingroup$ Sorry for the late reply, the external Javascript had been blocked... Aside from the $x_0$ term, I'm also confused about the equality between the massless vector state $|1;k_3\rangle=:\zeta_3\cdot X e^{ik_3\cdot X}:|0;0\rangle$ and $\zeta_3\cdot\alpha_{-1}|0;k_3\rangle$. Why are they equal? $\endgroup$ – Xavier May 23 '16 at 13:13
  • $\begingroup$ It's how state-operator correspondence works, with some identities in the OPEs etc. It might be that Polchinski's book covers all these things much more systematically and accurately than GSW. $\endgroup$ – Luboš Motl May 23 '16 at 16:51

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