There are different ways in which the extra dimensions required by string theory may possibly manifest and whether the extra dimensions are really necessary. In Gates’s approach, he essentially trades dimensions for charges. This creates a sort of dual approach that’s mathematically similar to the approach in extra space dimensions, but doesn’t actually require the extra space dimensions nor require guessing at compactification techniques to eliminate the extra dimensions.

This idea dates back to a 1938 proposal by British physicist Nicolas Kemmer. Kemmer proposed that the quantum mechanical properties of charge and spin were different manifestations of the same thing. Specifically, he said that the neutron and proton were identical, except that they rotated differently in some extra dimension, which resulted in a charge on the proton and no charge on the neutron.

The resulting mathematics, which analyzes the physical properties of these particles, is called an isotopic charge space (originally developed by Werner Heisenberg and Wolfgang Pauli, then used by Kemmer). Though this is an “imaginary space” (meaning that the coordinates are unobservable in the usual sense), the resulting mathematics describes properties of protons and neutrons, and is at the foundation of the current Standard Model.

Gates’s approach was to take Kemmer’s idea in the opposite direction: If you wanted to get rid of extra dimensions, perhaps you could view them as imaginary and get charges. (The word “charge” in this sense means a new property to be tracked, like “color charge” in QCD.) The result is to take vibrational dimensions of the heterotic string and view them as “left charge” and “right charge.”

When Gates applied this concept to the heterotic string, the trading didn’t come out even — to give up six space dimensions, he ended up gaining more than 496 right charges!

In fact, together with Siegel, Gates was able to find a version of heterotic string theory that matched these 496 right charges. Furthermore, their solution showed that the left charges would correspond to the family number. (There are three known generations, or families, of leptons — the electron, muon, and tau families. The family number indicates which generation the particle belongs to.)

This may explain why there are multiple families of particles in the Standard Model of particle physics. Based on these results, a string theory in four dimensions could require extra particle families! In fact, it would require many more particle families than the three that physicists have seen. These extra families (if they exist) could include particles that could make up the unseen dark matter in our universe.