The circle is a group because you can add angles. There's no way to make the sphere into a group. But the 3-sphere - the sphere in 4-dimensional space - is a group!
The 3-sphere is a group because rotations in 3d space form a group - you multiply them by doing first one rotation and then another. And the set of all these rotations is almost the 3-sphere!
But not quite. In fact two opposite points on the 3-sphere describe the same rotation. But that's actually a good thing, because when you try to rotate an object connected to its environment, like the box in this image, you'll discover that turning it around 360 degrees does not bring it back to where it was: you have to turn it around 720 degrees to do that!
This is especially important when your object is something like an electron, or proton, or neutron.
So there are two operations you can do to a physical object, that seem like the same rotation, but aren't. These two operations are described by opposite points on the 3-sphere.
For this reason, physicists love the 3-sphere. When physicists think of the 3-sphere as a group, they call it SU(2).
There's a way to think about hydrogen that makes it be all about the 3-sphere. Surprisingly, there are three basic ways to 'rotate' a hydrogen atom! But all of them use the 3-sphere, not ordinary rotations. This should be known already, but I've never seen anyone talk about it. So I wrote a paper about this fact, and its consequences.
(1/2)
The 3-sphere is a group because rotations in 3d space form a group - you multiply them by doing first one rotation and then another. And the set of all these rotations is almost the 3-sphere!
But not quite. In fact two opposite points on the 3-sphere describe the same rotation. But that's actually a good thing, because when you try to rotate an object connected to its environment, like the box in this image, you'll discover that turning it around 360 degrees does not bring it back to where it was: you have to turn it around 720 degrees to do that!
This is especially important when your object is something like an electron, or proton, or neutron.
So there are two operations you can do to a physical object, that seem like the same rotation, but aren't. These two operations are described by opposite points on the 3-sphere.
For this reason, physicists love the 3-sphere. When physicists think of the 3-sphere as a group, they call it SU(2).
There's a way to think about hydrogen that makes it be all about the 3-sphere. Surprisingly, there are three basic ways to 'rotate' a hydrogen atom! But all of them use the 3-sphere, not ordinary rotations. This should be known already, but I've never seen anyone talk about it. So I wrote a paper about this fact, and its consequences.
(1/2)
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