# Slope parameter

Problem 14.5c4 | Counting states in heterotic SO(32) string theory

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c) Are there tachyonic states in heterotic string theory?

Write out the massless states of the theory (bosons and fermions) …

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What is the meaning of $\displaystyle{\alpha'}$?

How come $\displaystyle{\alpha' = \frac{1}{2}}$?

— Me@2018-12-07 10:43:10 PM

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If we consider a rigidly rotating open string, $\alpha'$ is the proportionality constant that relates the angular momentum $J$ of the string, measured in units of $\hbar$, to the square of its energy $E$. More explicitly,

$\displaystyle{ \frac{J}{\hbar} = \alpha' E^2 }$

— p.68

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$\displaystyle{ J = \frac{1}{ 2 \pi T_0 c} E^2 }$

As anticipated, the angular momentum is proportional to the square of the energy of the string. Comparing with equation (8.69) we deduce that

\displaystyle{ \begin{aligned} \alpha' &= \frac{1}{2 \pi T_0 \hbar c} \\ T_0 &= \frac{1}{2 \pi \alpha' \hbar c} \end{aligned}}

These equations relate the slope parameter $\alpha'$ to the string tension $T_0$.

— p.69

— A First Course in String Theory

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