I saw you answered someone else about the Pauli-Lubanski 4-vector properties. I was doing some simple analysis of polarized leptons, started with the canonical convention of the spin operator wich is just $s=(0,\vec{s})$ and made the simple boost. However I am confused because I have found in the literature (such as Greiner's Gauge theory of Weak Interactions) a different boosted vector which is just what you mentioned in your answer,

$$\mathbf{w} = m \mathbf{s} + \frac{ \mathbf{p}\cdot\mathbf{s}}{p_0+m}\mathbf{p}$$

If this is an approximation of a simple Lorentz Boost I cant seem to obtain it, could you show me explicitly where does this former expression comes from?

  • 1
    $\begingroup$ Comment to the question (v3): Which Phys.SE post are you referring to? $\endgroup$
    – Qmechanic
    Sep 11, 2013 at 18:02
  • $\begingroup$ Don't forget about tensor nature of the spin $\mathbf s$ operator. $\endgroup$
    – user8817
    Sep 11, 2013 at 19:26

1 Answer 1


Check the correctness of my expressions, as I'm inconsiderate and could be mistaken.

You only need the expressions for the full momentum tensor:

$$ S^{\mu \nu} = (\hat {\mathbf S} , \hat {\mathbf K}_{1}), \quad \hat {\mathbf K}_{1} = \frac{[\hat {\mathbf p} \times \hat {\mathbf S}]}{p_{0} + m}, $$ $$ L^{\mu \nu} = (\hat {\mathbf L} , \hat {\mathbf K}_{2}), \quad \hat {\mathbf K}_{2} = \hat {E}\hat {\mathbf r} - t \hat {\mathbf p} $$

After that, using formal expression for 3-part of Pauli-Lubanski vector,

$$ \hat {\mathbf W} = p^{0}\hat {\mathbf J} - [\hat {\mathbf K} \times \hat {\mathbf p}], \quad \hat {\mathbf J} = \hat {\mathbf S} + \hat {\mathbf L} , \quad \hat {\mathbf K} = \hat {\mathbf K}_{1} + \hat {\mathbf K}_{2}. $$ From these equation it's easy to express $\hat {\mathbf S}$ via $\hat {\mathbf W}$ or to get your expression.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.