# Do Lorentz boosts affect spin orientation?

1. Dec 14, 2012

### Zoot

Does a Lorentz boost change the "direction" of an electron's spin orientation?

For example, if an electron is in a state "spin-up" along x, and this electron is subjected to a large boost along another direction, say z, will the electron's spin remain "spin-up" along x?

What if the electron is instead boosted along the direction of its spin?

I know that Dirac 4-spinors transform under Lorentz boosts and rotations, for example as shown in Peskin/schroeder, however it is not clear to me how the transformation affects (if at all) the orientation of the electron's spin.

Please refer to my recent post a few hours ago: "Spin Direction and 4-spinor Components" for a more detailed and specific version of this question, including some calculations I did on boosting the spinors.

Thank you very much for any help on this subject!

2. Dec 15, 2012

### DrDu

The vector describing electron's spin is the Pauli Lyubarskii vector which is obtained by boosting the electron to it's rest frame. Now if an electron is already moving at constant speed v1 and you apply a second boost v2 , then, starting from the1st PL vector you have to apply 3 boosts to get the 2nd PL vector. Namely if the boost from 0 to v1 and than the boost v2 are not parallel, they will induce a net rotation. Hence the second PL vector will be rotated with respect to the first.

3. Dec 15, 2012

### Zoot

Thank you for replying DrDu. The Pauli-Lubansky vector is a 4-vector, which Lorentz transforms accordingly. The electron's 4-spinor transforms differently, since it is a spinor and not a vector. So what you are saying is that I should transform the DIRECTION of the electron's spin (which is described by a 4-vector) and NOT the spinor itself? Just wanted to clarify this. Thanks again.

4. Dec 15, 2012

### Bill_K

Zoot, In the rest frame of the particle, one can define its spin projection along any space axis. More generally if the particle has a 4-momentum pμ, one can use a 4-dimensional vector sμ for a spin polarization axis, provided that pμ and sμ are orthogonal.

Given sμ, one defines spin projection operators Σ±(s) = ½(1 ±γ5γμsμ), with eigenvalues ±1 for the two spin states. Σ does not commute with Lorentz boosts, so in general a spin-up state will not remain spin up.

5. Dec 15, 2012

### DrDu

It depends what you are behind. You were asking about the transformation of spin, not spinors, didn't you?

6. Dec 16, 2012

### andrien

using a boost,one can write for the transformation of spinor.but it is better not to confuse transformation of spinor under lorentz boost with something like helicity.