Warped Passages - Lisa Randall [176]
The KK particles’ masses are determined by the higher-dimensional geometry. However, their charges are the same as those of known four-dimensional particles. That is because if known particles originate from higher-dimensional spacetime, higher-dimensional particles have to carry the same charges as known particles. That’s also true for the KK particles that mimic the higher-dimensional particles’ behavior. So for each particle we know about, there should be many KK particles with the same charge, each with different mass. For example, if an electron travels in higher dimensions, it would have KK partners that have the same negative charge. And if a quark travels in higher dimensions, it would have KK relatives that, like the quark, experience the strong force. KK partners have identical charges to the particles we know, but masses that are determined by extra dimensions.
Determining Kaluza-Klein Masses
Understanding the origin and masses of KK particles requires taking a step beyond the intuitive picture of invisible curled-up dimensions that we looked at earlier on. For simplicity, we’ll first consider a universe without branes, in which every particle is fundamentally higher-dimensional and is free to move in all directions—including any additional ones. To be concrete, we’ll imagine a space with only one additional dimension which has been rolled up into a circle and elementary particles that travel inside that space.
Had we lived in a world where classical Newtonian physics was the final word, Kaluza-Klein particles could have had any amount of extra-dimensional momentum and therefore any mass. But because we live in a quantum mechanical universe, this is not the case. Quantum mechanics tells us that, just as only the resonant violin modes contribute to the sounds the violin strings can make, only quantized extra-dimensional momenta contribute when KK particles reproduce the motion and interactions of a higher-dimensional particle. And just as the notes of a violin string depend on its length, the quantized extra-dimensional momenta of the KK particles depend on the extra dimensions’ sizes and shapes.
The extra-dimensional momenta that the KK particles carry would appear to us in our apparently four-dimensional world as a distinctive pattern of KK particle masses. If physicists discover KK particles, these masses will tell us about the geometry of the extra dimensions. For example, if there is a single extra dimension that is curled into a circle, these masses would tell us the extra dimension’s size.
The procedure for finding the allowed momenta (and hence masses) for KK particles in a universe with a curled-up dimension is very similar to the method you use to mathematically determine resonant violin modes, and also to the method that Bohr used to determine quantized electron orbits in an atom. Quantum mechanics associates all particles with waves, and only those waves that can oscillate an integer number of times over the extra-dimensional circle are allowed. We determine the allowed waves, and then use quantum mechanics to relate wavelength to momentum. And the extra-dimensional momenta tell us the allowed KK particles’ masses, which is what we want to know.
The constant wave—the one that doesn’t oscillate at all—is always allowed. This “wave” is like the surface of a perfectly still pond, without any visible ripples, or a violin string that has not yet been plucked. This probability wave has the same value everywhere in the extra dimension. Because of the constant value of this flat probability wave, the KK particle associated with this wave doesn’t favor any particular extra-dimensional location over any other. According to quantum mechanics, this particle carries no extra-dimensional