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edited Nov 8, 2019 at 14:48

For simplicity take the bosonic string theory in 26 dimensions. When you quantise the open and closed string you find excitations (states) of the string at any level $N$ with masses \begin{equation} M^2_\mathrm{open}=\frac{1}{\alpha '}\left(N-1\right),\qquad M^2_\mathrm{closed}=\frac{4}{\alpha '}\left(N-1\right). \end{equation} As $N$ takes on any non-negative value, you see that there is indeed an infinite tower of states. The tachyonic state (negative mass) at $N=0$ is a sickness of the purely bosonic string, which goes away for the superstring. At level $N=1$ you find the massless excitations (in 26 dimensions), and from the next level all states are massive. The mass is determined by the pre-factor. You can regard $\alpha '$ as the only free parameter of string theory, and it relates to the tension, $T$, of the string, which is expected to be set by the string scale, which is slightly below the Planck scale, so that \begin{equation} \frac{1}{\alpha '}= 2\pi T \lesssim M_\mathrm{Pl}^2 = \left(10^{19} \, \mathrm{GeV} \right)^2 . \end{equation} So in conclusion, the infinite tower of massive excited states have masses at the order of the Planck scale, which means they are unobservable, and there is only a finite number of massless excitations. The same thing goes for the superstring.

For simplicity take the bosonic string theory in 26 dimensions. When you quantise the open and closed string you find excitations (states) of the string at any level $N$ with masses \begin{equation} M^2_\mathrm{open}=\frac{1}{\alpha '}\left(N-1\right),\qquad M^2_\mathrm{closed}=\frac{4}{\alpha '}\left(N-1\right). \end{equation} As $N$ takes on any non-negative value, you see that there is indeed an infinite tower of states. The tachyonic state (negative mass) at $N=0$ is a sickness of the purely bosonic string, which goes away for the superstring. At level $N=1$ you find the massless excitations (in 26 dimensions), and from the next level all states are massive. The mass is determined by the pre-factor. You can regard $\alpha '$ as the only free parameter of string theory, and it relates to the tension, $T$, of the string, which is expected to be slightly below the Planck scale, so that \begin{equation} \frac{1}{\alpha '}= 2\pi T \lesssim M_\mathrm{Pl}^2 = \left(10^{19} \, \mathrm{GeV} \right)^2 . \end{equation} So in conclusion, the infinite tower of massive excited states have masses at the order of the Planck scale, which means they are unobservable, and there is only a finite number of massless excitations. The same thing goes for the superstring.

For simplicity take the bosonic string theory in 26 dimensions. When you quantise the open and closed string you find excitations (states) of the string at any level $N$ with masses \begin{equation} M^2_\mathrm{open}=\frac{1}{\alpha '}\left(N-1\right),\qquad M^2_\mathrm{closed}=\frac{4}{\alpha '}\left(N-1\right). \end{equation} As $N$ takes on any non-negative value, you see that there is indeed an infinite tower of states. The tachyonic state (negative mass) at $N=0$ is a sickness of the purely bosonic string, which goes away for the superstring. At level $N=1$ you find the massless excitations (in 26 dimensions), and from the next level all states are massive. The mass is determined by the pre-factor. You can regard $\alpha '$ as the only free parameter of string theory, and it relates to the tension, $T$, of the string, which is expected to be set by the string scale, which is slightly below the Planck scale \begin{equation} \frac{1}{\alpha '}= 2\pi T \lesssim M_\mathrm{Pl}^2 = \left(10^{19} \, \mathrm{GeV} \right)^2 . \end{equation} So in conclusion, the infinite tower of massive excited states have masses at the order of the Planck scale, which means they are unobservable, and there is only a finite number of massless excitations. The same thing goes for the superstring.

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created Nov 8, 2019 at 14:25

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For simplicity take the bosonic string theory in 26 dimensions. When you quantise the open and closed string you find excitations (states) of the string at any level $N$ with masses \begin{equation} M^2_\mathrm{open}=\frac{1}{\alpha '}\left(N-1\right),\qquad M^2_\mathrm{closed}=\frac{4}{\alpha '}\left(N-1\right). \end{equation} As $N$ takes on any non-negative value, you see that there is indeed an infinite tower of states. The tachyonic state (negative mass) at $N=0$ is a sickness of the purely bosonic string, which goes away for the superstring. At level $N=1$ you find the massless excitations (in 26 dimensions), and from the next level all states are massive. The mass is determined by the pre-factor. You can regard $\alpha '$ as the only free parameter of string theory, and it relates to the tension, $T$, of the string, which is expected to be slightly below the Planck scale, so that \begin{equation} \frac{1}{\alpha '}= 2\pi T \lesssim M_\mathrm{Pl}^2 = \left(10^{19} \, \mathrm{GeV} \right)^2 . \end{equation} So in conclusion, the infinite tower of massive excited states have masses at the order of the Planck scale, which means they are unobservable, and there is only a finite number of massless excitations. The same thing goes for the superstring.