What is spin? (advanced question)

[pellman]

I had thought that if under an infinitesimal rotation by $$\theta$$ around the z-axis a quantity transforms according to

$$\psi\rightarrow\left(1+i\left[S+\left(\frac{1}{i}x\frac{\partial}{\partial y}-\frac{1}{i}y\frac{\partial}{\partial x}\right)\right]\theta\right)\psi$$

then the operator S is by definition the z-component of the spin operator for that quantity.

However, the Dirac field operator transforms (in the appropriate representation) according to

$$\psi\rightarrow\left(1+i\left[\frac{1}{2}\left(\begin{array}{cc}\sigma_{z}&0\\ 0&\sigma_{z}\end{array}\right)+\left(\frac{1}{i}x\frac{\partial}{\partial y}-\frac{1}{i}y\frac{\partial}{\partial x}\right)\right)\theta\right]\right)\psi$$

yet https://www.amazon.com/gp/product/0521478146/?tag=pfamazon01-20 states that $$\tau_{z}=\frac{1}{2}\left(\begin{array}{cc}\sigma_{z}&0\\0&\sigma_{z}\end{array}\right)$$ cannot be the spin operator because it does not commute with $$\gamma^{\mu}\partial_{\mu}$$ The correct spin operator is--or has something to do with--the Pauli-Lubanski pseudovector.

My question is, first, what's wrong with not commuting with $$\gamma^{\mu}\partial_{\mu}$$? Why is that necessary for the a good spin operator?

And secondly, if $$\tau_{z}$$ is not a good spin operator, what about the fact that it generates the non-scalar part of a rotation? Is that not the definition of spin?

 

[masudr]

My question is, first, what's wrong with not commuting with $$\gamma^{\mu}\partial_{\mu}$$? Why is that necessary for the a good spin operator?

I don't know a definitive answer. However, isn't a part of $\gamma^{\mu}\partial_{\mu}$ the Hamiltonian? And (of course) if we want the eigenvalues of some operator to be constant in time, we require the operator to commute with the Hamiltonian.

 

[dextercioby]

The spin needn't be conserved. Actually for the Dirac Hamiltonian it isn't. It's only the total angular momentum that is conserved for a Hamiltonian generated time-evolution.

Actually, besides the total momentum and angular momentum, the Pauli-Lyubanski pseudovector also commutes with the Hamiltonian and therefore is conserved for all projective unitary irreds of the restricted Poincare' group.

Daniel.

 

[hellfire]

I have also problems to understand that section of Ryder's book. In the part where Wigner's little group is discussed it is shown in detail that the non-relativistic spin operator is the correct spin operator in the rest frame of a particle. Moreover, it is shown how the Pauli-Lubanski pseudovector reduces basically to the non-relativistic spin operator in the rest frame. But it seams to me that he does not spend a word to explain why the Pauli-Lubanski pseudovector should actually provide a generalized relativistic spin operator.