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- #101
Lemme see, SO3 and SU2 are algebraic groups, their names come from the "group catalog" (universal classification of groups, which I already mentioned is about 20 years old - even though, physicists have been using these names for over 100 years).
SO3 stands for "special orthogonal group in 3 dimensions", it's special because all the determinants are 1. Basically it's the group of rotations in 3 dimensions, so the matrices (rotation operators) might look something like this:
| Cos t, Sin t, 0 |
| - Sin t, Cos t, 0 |
| 0, 0, 1|
where t is an angle, usually written as theta.
We apply this to a 3d vector (x,y,z) like this:
rotated result = MV
where M is the rotation operator (matrix), and V is our vector. In this example we're rotating in the XY plane and Z is left alone (hence the 1 at the bottom right of the matrix).
However we can write our 3d vector as a 2x2 Pauli vector, like this:
| z, x + yi |
| x - yi, z |
and apply the SU2 rotations to it (SU2 is also a Lie group, it's the special unitary group in 2 dimensions, unitary meaning that the inverse equals the Hermitian conjugate). In this case, our vector transforms in a different way - we have to use a double sided transformation with half the angle. Like this:
V' = U V U°
where U° is the inverse (Hermitian conjugate) of U, and U looks like this:
| e^(-it/2), 0 |
| 0, e^(it/2) |
In this case the angle t (theta) is halved, and each side of the double sided transformation performs half the rotation.
This construction is also how we get spinors from vectors, the SU2 matrices are "spin operators". But you'll notice that
U V U° = (-U) V (-U)°
so there are two SU2 matrices that perform the exact same rotation, and this is why we say that SU2 is a "double" cover of SO3.
So this is how we understand 1/2-spin particles in relation to 1-spin particles. For a 1/2-spin particles the SU2 transformation is 1-sided, whereas for a full spin particles it's 2-sided.
Very logical, right? Simple, easy, nothing more than matrix math. But we're using groups, and geometric rotations, and we're using them to understand particle physics.
This is a great example of Scruffy's math program for kids. We already used the exact same math to understand the quantum spin states that underlie quantum computing, and now we're using it to understand particle physics, and it's nothing more than rotation and matrix multiplication.
Similarly, we want to introduce kids to "connections" before they graduate high school. Probably 12th grade, would be a great time for it. When we do rotations of a point on a sphere, we want to show the kids the tangent plane associated with each point. Then we can introduce them to "tangent bundles" and show them the Levi-Civita connection for parallel transport
SO3 stands for "special orthogonal group in 3 dimensions", it's special because all the determinants are 1. Basically it's the group of rotations in 3 dimensions, so the matrices (rotation operators) might look something like this:
| Cos t, Sin t, 0 |
| - Sin t, Cos t, 0 |
| 0, 0, 1|
where t is an angle, usually written as theta.
We apply this to a 3d vector (x,y,z) like this:
rotated result = MV
where M is the rotation operator (matrix), and V is our vector. In this example we're rotating in the XY plane and Z is left alone (hence the 1 at the bottom right of the matrix).
However we can write our 3d vector as a 2x2 Pauli vector, like this:
| z, x + yi |
| x - yi, z |
and apply the SU2 rotations to it (SU2 is also a Lie group, it's the special unitary group in 2 dimensions, unitary meaning that the inverse equals the Hermitian conjugate). In this case, our vector transforms in a different way - we have to use a double sided transformation with half the angle. Like this:
V' = U V U°
where U° is the inverse (Hermitian conjugate) of U, and U looks like this:
| e^(-it/2), 0 |
| 0, e^(it/2) |
In this case the angle t (theta) is halved, and each side of the double sided transformation performs half the rotation.
This construction is also how we get spinors from vectors, the SU2 matrices are "spin operators". But you'll notice that
U V U° = (-U) V (-U)°
so there are two SU2 matrices that perform the exact same rotation, and this is why we say that SU2 is a "double" cover of SO3.
So this is how we understand 1/2-spin particles in relation to 1-spin particles. For a 1/2-spin particles the SU2 transformation is 1-sided, whereas for a full spin particles it's 2-sided.
Very logical, right? Simple, easy, nothing more than matrix math. But we're using groups, and geometric rotations, and we're using them to understand particle physics.
This is a great example of Scruffy's math program for kids. We already used the exact same math to understand the quantum spin states that underlie quantum computing, and now we're using it to understand particle physics, and it's nothing more than rotation and matrix multiplication.
Similarly, we want to introduce kids to "connections" before they graduate high school. Probably 12th grade, would be a great time for it. When we do rotations of a point on a sphere, we want to show the kids the tangent plane associated with each point. Then we can introduce them to "tangent bundles" and show them the Levi-Civita connection for parallel transport
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