A question about Kaluza-Klein mechanism

Question

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?

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user Wein Eld

Comments

...yes, a particle that is not localized on our 4D brane won't have probability 1 to be detectedon it. Again, so what?

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user ACuriousMind

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@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.

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user Wein Eld

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@ACuriousMind, My question is why didn't people propose the localization mechanism for KK theory. Why the localization mechanism was proposed only in the brane world scenario?

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user Wein Eld

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Could you please give an introductory reference (book preferably) where this is discussed?

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user magma

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@magma, I have added a link to Kaluza-Klein theory. But for the braneworld and localization mechanism, I am impressed by the scattered papers.

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user Wein Eld

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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?

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user ACuriousMind

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It isn't clear to me exactly what you're asking, but the particles aren't localised in the compact dimensions, and indeed would require of the order of the Planck energy to localise them in the compact dimensions.

This post imported from StackExchange Physics at 2016-07-26 09:17 (UTC), posted by SE-user John Rennie

Answer 1

I have answered my question at SE. Of course, any comments and answers are still welcome.

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.

This case changes in string theory. Beside the momentum along the extra small dimension, the string can wind around the compacted dimension, which becomes a quantum quantity after quantization. When the extra dimension becomes smaller and smaller, the excitation spectrum for the winding number becomes continuous. The gapless excitations emerge again. In that sense, the extra dimension actually are becoming bigger again.

Answer 2

My understanding is that Kaluza and Klein introduced a fourth compact scalar dimension (a small circle, or in other words a phase attached to every point), from which they derived both Einstein's equations of general relativity and Maxwell's equations of classical electrodynamics, but didn't go as far as to propose a specific localization mechanism. A localization mechanism would (I think) derive the circle and its size dynamically. Then the KK theory was mostly forgotten after the discovery of the weak and strong forces indicated that there is more to physics than gravitation and electrodynamics, and resurrected after decades as an inspiration for string theories.