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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?

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    $\begingroup$ 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

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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.

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