Ok, the answer you’ve all been waiting for — the first half of it, anyway. Even though it is not the full story yet, you’ll find it is both self-contained and instructive.

Those of you who have been following my recent series of articles on extra dimensions of space — which include some articles on how to think about them (including some examples) and newer articles on how extra dimensions might reveal themselves to us — already know that for any type of particle that can travel in one or more extra dimensions that are unknown to us, nature will exhibit heavier versions of this particle, called Kaluza-Klein (KK) partner particles. But I haven’t yet told you why this is the case.

Today, Step 1: why the KK partner particles exist at all, and why they are heavier than the original one. But today’s argument is a bit too simple, and only partially correct: it gets the masses of the KK partners wrong. In Step 2 I’ll fix this problem by adding a little bit of quantum mechanics.

## 4 Responses

Matt said, “… already know that for any type of particle that can travel in one or more extra dimensions that are unknown to us, nature will exhibit heavier versions of this particle, called Kaluza-Klein (KK) partner particles. “

I am not sure that I understand the above statement exactly. I would like to ask a few questions to clarify my understanding of it.

For one extra dimension “e” which has a finite length L = 1/10,000 of a proton diameter. Then,

Question one: Can proton travel in this e-dimension?

Question two: Which particle in the entire known particle zoo can travel in this e-dimension?

Question three: Which particle in the entire fictitious particle zoo (such as s-particles, not yet discovered) can travel in this e-dimension?

The answer for these questions from your statement seems to be “we don’t know theoretically”, and the answer can only be obtained by a KK test. If my understanding of your statement is what you mean, I would like to make a comment on this issue with two points.

1. This KK test can kill the chance for us to understand the “reality” of extra dimensions, as the chance of discovering any KK partner might be nil.

2. The reality of the extra dimensions can be understood theoretically.

Discussing the concept of dimension without considering Georg Cantor’s theorem as the preemption controlling factor can always be misleading. Cantor’s theorem is well-known, and its proof is very intense. However, I will show a shorter proof here to show a point.

With Cantor’s theorem, there is one and only one dimension, the number line, and we can call it x-dimension. While this x-dimension has an infinite length, its thickness is zero.

For a plane, it is viewed to have two dimensions in physics, but it is, in fact, the x-dimension which acquires a width (y) for its dimension-line. And, this “width” is only a “trait” for this one-dimension of x. In essence, y is not a true dimension but is a “trait” of x. Perhaps, we can call it a “trait-dimension”.

For a physical three dimensions (x, y, z), it is again the result of the x-dimension which acquires two traits, the width (y) and the height (z).

With the above simple procedure, the Cantor’s theorem is proved. Every point in the three dimensional space is brought to a one-to-one correspondence to one-dimensional number line x.

Of course, there is no reason to re-prove Cantor’s theorem here. My point is trying to discuss the true meaning and the true essence on dimension. There is one and only one dimension, and any other dimensions are “trait-dimensions” and are extra-dimensions. If we already have three extra-dimensions [ y (width), z (height) and t (time)], why the nature prohibits a few more?

In a vector space, if a vector cannot be reduced to other vectors, it is a base-vector, a dimension for that vector space. Thus, if a “trait” in a system cannot be reduced to any known base-vectors, it itself is a base-vector.

In the known quark universe, there are a few traits which cannot be reduced to the spacetime dimensions, and they should be base-vectors of themselves. There are 7 of them.

1. Quark colors (R, Y, B), 3-base vectors.

2. Quark generations (G1, G2, G3), 3-base vectors.

3. A ceiling (W, colorless, separating the quark universe from hadrons), 1-base vector.

Thus, in addition to the 4 spacetime dimensions, there are 7 extra dimensions for the quark universe. These trait-dimensions might not have anything to do with the KK test, as there is no “movement” in those trait-dimensions.

Try to buy 100 gallons of oil and pay just for one arguing that there is one-to-one mapping from 100 gallon volume to 1 gallon and so the volumes are the same. Let me know the result.

… already know that for any type of particle that can travel in one or more extra dimensions that are unknown to us …But in what sense are the dimensions Real for propagation? Surely the nonperturbative foundations hinge on a non local (twistorial) picture, which dictates the particle spectrum via the localization factorization of N=8 supergravity.

Surely.