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One common idea behind many extensions to the Standard Model (such as String Theory or Kaluza-Klein Theory) are small or hidden "Extra-Dimensions", that are compactified.

According to my understanding of Quantum Physics, this would result in each particle's wavefunction having a component into the direction of these extra-dimensions, and only discrete energy-states would be allowed (similar to electrons in an atom).

Now imagine a photon, which is considered to be a particle without rest mass in the Standard Model. Its wavefunction would also have components into the direction of the extra-dimensions. Consequently, it would have to occupy one of these energy states. So there would be some energy consisting out of the photon's standing wave in the extra dimension, which - according to my understanding - would behave just like a finite rest mass of the photon.

So how can there be particles without any invariant mass in a theory with compact extra dimensions?

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    $\begingroup$ This question seems to be rather broad - any answer would have to be essentially an introductory text to compactifications in quantum field theory (and it appears from your use of "Quantum Physics" and "wavefunctions" that you don't know the framework of standard quantum field theory to begin with). $\endgroup$ – ACuriousMind Nov 7 '16 at 22:21
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    $\begingroup$ @ACuriousMind fair point, but remember that answers are for the general public, and not just for OP. If you (or anybody else) feels like they can contribute, they should do it regardless whether they think their answer would help OP or not. If the question requires high-level physics, then so be it. (Not saying that you don't know this, but I just wanted to point it out) $\endgroup$ – AccidentalFourierTransform Nov 7 '16 at 22:33
  • $\begingroup$ The two comments above are perfectly correct, in their own way, in my opinion. But as the first comment implies, in this particular case, there is simply not the space or the time to provide you with the background needed to answer your question in a rigorous manner. Statements such as According to my understanding of Quantum Physics, this would result in each particle's wavefunction having a component into the direction of these extra-dimensions, and only discrete energy-states would be allowed are questions in their own right. $\endgroup$ – user108787 Nov 7 '16 at 23:51
  • $\begingroup$ So perhaps you could ask this interesting question first, and then move on to your next question. But you would be expected to do as much research into your assumptions as possible yourself, which, from personal experiences, can take some time. $\endgroup$ – user108787 Nov 7 '16 at 23:53
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Let the speed of light $c=1$ be one for simplicity. Recall the mass-shell condition e.g. in 5D Kaluza-Klein theory,

$$ E^2~=~{\bf p}^2 + m_{4D}^2, \qquad m_{4D}^2 ~=~ p_5^2+ m_{5D}^2.$$

So taking OP's argument to its logical conclusion, we see that a massless particle $m_{4D}=0$ (from a 4D perspective) has no Kaluza-Klein momentum $p_5=0$, and no 5D mass $m_{5D}=0$.

[We can repeat this argument ad infinitum: A massless particle $m_{5D}=0$ (from a 5D perspective) has no Kaluza-Klein momentum $p_6=0$, and no 6D mass $m_{6D}=0$, and so forth.]

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The string tension Tstring is the energy per unit length of the string. If the string is wound w times around a circular dimension with radius R, then the energy Ew stored in the tension of the wound string is

tension

The mass of an excited string depends on the number of oscillator modes N and Ñ excited in the two directions of propagation around the closed string, minus the constant vacuum energy. Kaluza-Klein compactification adds the quantized momentum in the compact dimensions, and the tension energy from the string being wrapped w times around the compact dimension, so that the total squared mass becomes

string mass

As an experimentalist skimming over theory I am satisfied that zero masses can exist even with compactified dimensions, since there exists a minus sign in the mass formula. :).

This also helps:

The theory gains extra massless particles when the radius R of the compact dimension takes the minimum value possible given the above symmetry of T-duality

One will have to do the maths to really convince oneself of the above statements, but that takes time and effort.

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