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For our story, the most important proposed expansion of the equations of physics is supersymmetry, often fondly called SUSY. Supersymmetry, as the name suggests, proposes that we should use equations with larger symmetry.
SUSY's new symmetry is related to the boost symmetry of special relativity. As you may recall, boost symmetry says that the basic equations don't change when you impart a common, constant velocity to all the components of the system you're describing. (Dirac had to modify Schrodinger's equation to give it that property.) Supersymmetry likewise says that the equations don't change when you impart a common motion to all the components of the system you're describing. But it's a very different kind of motion from what's involved in boost symmetry. Instead of motion through ordinary space with a constant velocity, supersymmetry involves motion into new dimensions!
Before you get carried away with visions of spirit worlds and wormholes through hyperspace, let me hasten to add that the new dimensions have a very different character from the familiar dimensions of space and time. They're quantum dimensions.
What happens to a body when it moves in the quantum dimensions isn't that it gets displace-there's no notion of distance out there-instead, its spin changes. The "superboosts" turn particles with a given amount of intrinsic spin into particles with a different amount of spin. Because the equations are supposed to stay the same, supersymmetry relates properties of particles with different spin. SUSY allows us to see them as the same particle, moving in different ways through the quantum dimensions of superspace.
You can visualize the quantum dimensions as new layers of the Grid. When a particle hops into these layers its spin changes, and so does its mass. Its charges-electric, color, and weak-stay the same.
SUSY might allow us to complete the work of unifying the Core. Unification of the different charges, using the symmetry SO(10), united all the gauge bosons into a common cluster, and all the quarks and leptons into a common cluster. But no ordinary symmetry is capable of joining those two clusters, for they describe particles with different spins. Sypersymmetry is the best idea we have for connecting them.
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Frank Wilczek (The Lightness of Being: Mass, Ether, and the Unification of Forces)