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In chapter 2 (section 2.10) of Zwiebach's string theory book, there is a neat derivation of the energy eigenvalues for the two dimensional square well in which one dimension is compactified to a length $2\pi R$. One finds the energy eigenvalues to be $$ E_{nt} = \frac{\hbar^2}{2m}\left(\frac{n^2\pi^2}{a^2} + \frac{t^2}{R^2}\right), \tag{2.122} $$ where $n\in \mathbb{Z}_{>0}$, $t\in \mathbb{Z}_{\geq 0}$, and $a$ is the length of the square well along the non-compact dimension.

My question is how seriously one should take this result, especially in the context of string theory. I agree it's a tidy justification for why compact dimensions are difficult to probe (we may suppose, for instance, that $a \gg R$). But if compact dimensions do exist, will electrons with sufficient energy truly infiltrate them? Or is the notion of a compact dimension more complicated than a mere identification (e.g., $x\sim x + 2\pi R$), so much so that electrons (or other fermions for that matter) are forbidden from accessing them?

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I dont have this book, but the story of compactification is different. The Dirac operator splits in the sum of two.lower dimensional operators. The Eigenvalues on the compact one give the mass spectrum. Since Stringtheory 'works' at high energies, we.want a zero.Eigenvector. This condition leads directly to the Calabi-Yao manifolds for 6 dimensional.space (4+6=10). If you work in M Theroy this condition gives you.a manifold with exceptional (in the sense of Lie groups) holonomy.

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