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I am quite baffled at what the $n=0$ mean in compactification, why is this mode important? enter image description here

I mean if $n=0$ was applied here we'd just be left with enter image description here

I know that (39) holds from (37), but I can't figure out why (38) is of particular significance.

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  • $\begingroup$ Which textbook? $\endgroup$ – Qmechanic Dec 18 '15 at 14:55
  • $\begingroup$ I'm note too confident in this context, but generally speaking $n=0$ is the center-of-mass motion $\endgroup$ – Bort Dec 18 '15 at 15:24
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In the usual Kaluza-Klein reduction for scalar fields in $D$ dimension you got an infinite tower of scalar fields from the $D-1$ dimensional point of view. These scalars come with increasing mass labelled by an integer $n$.

The case $n=0$ is special, because is the massless case. For instance in String Theory, when you compactify on a torus, a consistent truncation of your theory is to consider only the massless field, that is looking at the Supergravity approximation.

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