I What's the relation between spinor space and SO(3) vector space?

[joneall]

Is there a way to preview a post, or do I have to post it and then do an edit?

 

[PeterDonis]

Is there a way to preview a post

Click the Preview icon at the upper right corner of the post window.

 

[PeterDonis]

a rotation through an angle of 2π along the x, y or z axes returns the negative of the spinor (10).

Yes, this is correct.

And this is the difference between a 2-d spinor and a 3-d vector, their different properties under rotation.

As a general statement, this is correct, yes: the properties of spinors and vectors under rotation are different. But of course that statement by itself is not very informative.

what is the connection between a spinor and a vector under rotation?

And the answer to that question is, there isn't one. That is, you can't start from the properties of a vector under rotation and apply some simple rule to get the properties of a spinor under rotation, or vice versa. You have to go through a separate analysis of each case, as is done, for example, in Chapter 7 of Ballentine that I referred to before.

It's what I said in the lst paragraph.

No, it isn't. It may have been what you meant to say, but it's not what you said.

Can one deduce from this that a rotation on a system with a spinor and a vector will require conservation of the total angular momentum, spin plus orbital?

Not just from the spin properties alone, no. You have to work out the effects of the total angular momentum operator.

 

[joneall]

I have not given up on this subject. I have purchased and am now reading Ballentine. I will resist the temptation to continue the dialog until I have finished at least chapter 7. (Chapter 9 looks quite interesting too.)

Thanks to all who have contributed to raising my understanding, which clearly hasn't progressed far enough.