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I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame.

I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not looking to boost a 4-vector but a vector of length ##2j+1##, so I need a different representation.

On one hand, I know that formally the boost is of the form ##e^{-i \alpha \bar{p} \cdot \bar{K}} ##, where K is the generator for boosts, but it's not helpful as ##K## is never written down.

I'm probably somehow searching with wrong terminology as the representation I'm looking for is elusive.

Edit: I just realised that this might be possible by simply looking at the commutation relations ## K## have to satisfy, since the operators ## J## can be deduced for each ## j##, I think. In any case, There probably wouldn't be an easy form to write down how they operate on ##| j m \rangle## if I try that. Shouldn't it be possible to write the operation as some sort of rotation?

I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not looking to boost a 4-vector but a vector of length ##2j+1##, so I need a different representation.

On one hand, I know that formally the boost is of the form ##e^{-i \alpha \bar{p} \cdot \bar{K}} ##, where K is the generator for boosts, but it's not helpful as ##K## is never written down.

I'm probably somehow searching with wrong terminology as the representation I'm looking for is elusive.

Edit: I just realised that this might be possible by simply looking at the commutation relations ## K## have to satisfy, since the operators ## J## can be deduced for each ## j##, I think. In any case, There probably wouldn't be an easy form to write down how they operate on ##| j m \rangle## if I try that. Shouldn't it be possible to write the operation as some sort of rotation?

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