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We know that there is a kind of compactification mechanism named Kaluza-Klein theory which states that the extra dimensions can be compacted dimensions such as $T^n$ or $S^n$ and so on. To make the extra dimensions to be consistent with our observed world, we need to ask for that the extra dimensions are very small so that they are invisible. But I still have a question about the KK theory.

Let $x$ denotes the extended dimensions and $y$ denotes the compacted extra dimensions. Then we can consider a particle state $|x',p_y\rangle$ representing the position eigenstate for the $x$ and momentum eigenstate for the $y$. The wave function then is $$\psi(x,y)=\langle x, y|x',p_y\rangle\sim\delta(x-x')e^{ip_yy}.$$ Then probability of detecting such a particle in the extended spacetime is $$\frac{\int dx |\psi(x,y=y_0)|^2}{\int dxdy |\psi(x,y)|^2}\sim 0,\tag{1}$$ where I have assumed that our extended world is located at $y_0$ of the extra dimensions. Eq.(1) is true no matter how small the extra dimensions are because the measure of our extended world is zero in the total spacetime. So why can people accept KK mechanism if there are no some localization mechanisms? (note that the localization mechanism was only proposed for the brane world). Put it another way, why didn't people propose the localization mechanism for KK theory? Why the localization mechanism was proposed only in the brane world scenario?

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    $\begingroup$ Um...the probability to detect a single particle at a set of measure zero is always zero, that's nothing to do with the KK-mechanism. The probability to detect a particle moving in one dimensions at a point is zero, the probability to detect a particle moving in two dimensions at a line is zero, etc. I'm not sure why you seem to think this has something to do with the KK mechanism. Also, note that the KK mechanism is a toy model for compactification, no one "accepts" it as a true model of the world as far as I know. $\endgroup$ – ACuriousMind Jul 25 '16 at 14:02
  • $\begingroup$ @ACuriousMind, Since if there is no a localization mechanism, then Eq.(1) indicates that the matter is nonlocal in the extended spacetime. Further, it means that the total probability for detecting a specific particle in our Minkowski spacetime is not 1. $\endgroup$ – Wein Eld Jul 25 '16 at 14:05
  • $\begingroup$ ...yes, a particle that is not localized on our 4D brane won't have probability 1 to be detectedon it. Again, so what? $\endgroup$ – ACuriousMind Jul 25 '16 at 14:06
  • $\begingroup$ @ACuriousMind, But our observed world indicates the energy-momentum conservation and particularly, that the total probability is always one. So we always need a localization mechanism to constrain the SM matter fields (particles) on our 4D world. $\endgroup$ – Wein Eld Jul 25 '16 at 14:08
  • $\begingroup$ Yes. The KK model (nowadays) is a toy model for compactification, not supposed to be a true QFT model of our world. What exactly is your question about that? $\endgroup$ – ACuriousMind Jul 25 '16 at 14:09
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I think I have a misunderstanding for the KK theory. In the KK theory, we are living in, say, a 5-dimensional spacetime with one dimension compactified. What's different from the brane-world theory is that, in brane-world theory, we are living on a 4-dimensional brane which is embedded in the 5-dimensional spacetime. So in the post, I can not assume that the extended world is at $y=y_0$. Actually, in KK theory, the particles can be in principle everywhere. There is no a 4-dimensional subset which can be identified with our observed 4-dimensional spacetime. But since we observe the world by exchanging momenta and energy with the objects and also the compactified dimension is very small, all the low energy particles are frozen in the extra small dimension so that there is no exchanges of momenta in the extra dimension. In that case, the particles can not feel the existence of the small extra dimension.

Note since the small extra dimension is compactified, the minimal momentum for a moving particle in the extra dimension can be obtained according $$e^{ipL}\rightarrow p=\frac{2\pi n}{L}.$$ So if $L$ is very small, the first excitation energy $\frac{2\pi}{L}$ to move the particle in the extra dimension is very large. Equivalently, the low energy particles are frozen at that direction and can not feel the extra dimension. In a word, the momentum excitation is gapless in extended dimensional space while not in compactified dimensional space.

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