Trouble understanding spin-1 Pauli matrices

[gaze]

If we consider the spin-1/2 pauli matrices it makes sense that

$$[S_x,S^2] = [S_y,S^2] = [S_z,S^2] = 0$$

since $$S^2 = I$$... and this is supposed to be true in general, right?

Well, if I attempt to commute the spin-1 pauli matrices given on http://en.wikipedia.org/wiki/Pauli_matrices, with $$S^2$$, only $$S_z$$ appears to commute with $$S^2$$. Why would this be? At first observation, $$S_z$$ should obviously commute with $$S^2$$ since they'd appear to have the same eigenvectors, meaning they're simultaneously diagonalizable and therefore commute, but $$S_x$$ and $$S_y$$ obviously don't have the same eigenvectors, that wouldn't make much sense... I feel like I'm missing some really crucial detail and I can't seem to figure out what it is. Would someone mind shedding some light? Thanks!

 

[tom.stoer]

First of all you should write $$\sigma$$ instead of $$S$$ as one usually defines $$S_i = \sigma_i / s$$

Then I guess that instead of $$S^2$$ you mean $$\vec{\sigma}^2$$; you get a $$I$$ for each component.

And of course all 2*2 matrices do commute with $$I$$

 

[gaze]

Yes, but I'm interested in the spin-1 case, which has 3x3 matrices, and where $$S^2$$ is not proportional to $$I$$

 

[tom.stoer]

$$\vec{S}^2$$ is the Casimir operator of SU(2) and is proportional to $$I$$ in all represenations.

The commutation relations hold w/o restriction or modification in all representations. We start with the Pauli matrices (which are special for s=1/2)

$$[\sigma_i, \sigma_k] = 2i\epsilon_{ikl}\sigma_l$$

Now we define

$$S_i = \sigma_i / 2$$

We then have

$$[S_i, S_k] = [\sigma_i/2, \sigma_k/2] = 2i\epsilon_{ikl}\sigma_l/4 = i \left(\epsilon_{ikl}\right)S_l$$

1) The term in brackets represents the SU(2) structure constants which define SU(2) and which are valid in all representations. They define themselves one special representation, the so-called adjoint representation which in our case is just spin 1. You can check this by calculating the commutation relations of the 3*3 matrices

$$(M^i)_{kl} = \left(\epsilon_{ikl}\right)$$

2) The commutation relations

$$[S_i, S_k] = i \left(\epsilon_{ikl}\right)S_l$$

(constructed from the Pauli matrices) are now valid for all SU(2) representations (including the spin 1 case); so you can insert any n*n SU(2) matrix S.

3) Calculating $$\vec{S}^2$$ explicitly one finds for s=1/2

$$\vec{S}^2 = \vec{\sigma}^2/4 = \frac{3}{4} I = \frac{1}{2}\left(\frac{1}{2}+1\right) I = s(s+1) I$$

All reps of SU(2) are labelled by a value s which can be s = 1/2, 1, 3/2, ... The equation

$$\vec{S}^2 = s(s+1) I$$

is valid in all reps. This is the so-called Casimir operator of SU(2) and by construction it commutes with all generators

 

[gaze]

Haha I actually fudged my matrix multiplication. Really sorry to have wasted your time... but I did learn what a casimir operator is! Thanks a bunch for the help.

 

[tom.stoer]

Your welcome.

Last note: one can construct Casimir operators for all Lie algebras. One finds r independent Casimir operators where r is the rank of the algebra = the number of diagonal generators. For SU(2) this is just the 3rd matrix. Doing the same analysis for the Lorentz group i.e. SO(3,1) one finds two Casimir operators, the first one corresponding to M² (invariant mass), the second one to W² (which corresponds to spin). So SO(3,1) shows how spin is related to spacetime symmetry.