I Standard boost, particles with mass M > 0

[kent davidge]

Weinberg considers (p.68 QFT Vol. 1) particles with mass M > 0. The Little Group is SO(3). He wants to calculate the rotation
W(Λ,p) ≡ L^-1(Λp) Λ L(p). He says that for this we need to choose a standard boost L(p) which carries the four momentum from
k^μ = (0,0,0,M) to p^μ. He then shows the expressions for the components of L(p). What I don't understand is that because the spatial components of k^μ all vanish, i.e. kⁱ = 0, why does it matter to choose a specific set of expressions for (L(p))ⁱ _j?

 

[vanhees71]

You can choose any Lorentz transformation that maps $k^{\mu}$ into $p^{\mu}$. The most convenient (standard) choice is to use the uniquely defined rotation free Lorentz boost, i.e., a boost with velocity $\vec{p}/p^0=\vec{p}/\sqrt{M^2+\vec{p}^2}$.

For more details have a look at appendix B of my QFT manuscript:

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

 

[kent davidge]

Oh, your pdfs are incredible clear and helpful. I decided to read the whole manuscript instead of just appendix B.

 

[vanhees71]

Well, the English is a desaster. It was written when I just started to study QFT a long time ago...

 

[kent davidge]

Well, the English is a desaster. It was written when I just started to study QFT a long time ago...

Never mind about that. I even enjoy reading your manuscripts in german