Warped Passages - Lisa Randall [175]
Unfortunately, Athena heightened their suspicions by telling them that K. Square and his relatives couldn’t possibly stay long in any case, since they were all very unstable and the K. Square family could visit only during the commotion of energetic gatherings. Recognizing her unfortunate choice of words, Athena reassuringly added that the foreigners would stick to local laws during their brief and exciting visits. Convinced, her neighbors then joined Athena in welcoming the K. Square clan.
Earlier in this book I explained how extra dimensions might be hidden. They could be rolled up or hemmed in by branes so as to be imperceptibly small. But can an extra-dimensional universe really hide its nature so completely that none of its features distinguishes it from a four-dimensional world? That would be hard to believe. Even if compactified dimensions are so small that we could be lulled into believing that the world is four-dimensional, a higher-dimensional world must contain some new elements that distinguish it from a truly four-dimensional one.
If there are extra dimensions, such fingerprints of extra dimensions are sure to exist. Such fingerprints are particles called Kaluza-Klein (KK) particles.* KK particles are the additional ingredients of an extra-dimensional universe. They are the four-dimensional imprint of the higher-dimensional world.
Should KK particles exist and be sufficiently light, high-energy colliders will produce them and they will leave their mark in experimental data. The extra-dimensional detectives—the experimenters—will piece together these clues, transforming data into forensic evidence of a higher-dimensional world. This chapter is about these Kaluza-Klein particles, and why, in a higher-dimensional world, you can be confident of their existence.
Kaluza-Klein Particles
Even if a bulk particle travels in higher-dimensional space, we still should be able to describe its properties and interactions in four-dimensional terms. After all, we don’t see extra dimensions directly, so everything should appear to us as if it is four-dimensional. Just as Flatlanders, who see only two spatial dimensions, could observe only two-dimensional disks when a three-dimensional sphere passed through their world, we can see only particles that look like they travel in three spatial dimensions, even if those particles originated in higher-dimensional space. These new particles that originate in extra dimensions, but appear to us as extra particles in our four-dimensional spacetime,* are Kaluza-Klein (KK) particles. If we could measure and study all their properties, they would tell us everything there is to know about the higher-dimensional space.
Kaluza-Klein particles are the manifestation of a higher-dimensional particle in four dimensions. Just as you can reproduce any sound a violin string could make by the superposition of many resonant modes, you can reproduce a higher-dimensional particle’s behavior by replacing it with appropriate KK particles. The KK particles fully characterize higher-dimensional particles and the higher-dimensional geometry in which they travel.
In order to mimic the behavior of higher-dimensional particles, KK particles would have to carry extra-dimensional momentum. Every bulk particle that travels through the higher-dimensional space gets replaced in our effective four-dimensional description by KK particles that have the correct momenta and interactions to mimic that particular higher-dimensional particle.31 A higher-dimensional universe hosts both familiar particles and their KK relatives that carry extra-dimensional momenta that are determined by the detailed properties of the curled-up space.
However, a four-dimensional description doesn’t include information about extra-dimensional position or momentum. Therefore, the extra-dimensional momentum of the KK particles must be called something else when viewed