Velocity dependence of operators in Inonu-Wigner contraction

[hideelo]

I'm reading Weinberg's QFT volume 1. At the end of section 2.4 he is deriving the Inonu-Wigner contraction where he reduces the Poincaré group to the Euclidean one by taking the low velocity limit. In analyzing how the operators depend on velocity there are some I understand and some I don't.

I understand why W ~ mv^2, P ~ mv, M ~ m. I don't understand why J ~ 1, why isn't it on the same order as P? They both have linear velocity dependence. Also why is K ~ 1/v that means that for smaller velocity the effect gets bigger. I would expect that in the limit of small velocity, boosts just aren't a thing.

 

[azntoon]

The answer to the first part of your question is that J (the angular momentum operator) is a vector quantity, while P (the linear momentum operator) is a scalar quantity. Therefore, J has a higher order of magnitude than P, because the components of J need to be multiplied together in order to obtain the magnitude of the vector.The answer to the second part of your question is that K (the boost operator) is related to the velocities of two reference frames with respect to one another. As the velocity of one frame approaches zero, the velocity of the other frame must increase in order to maintain a constant relative velocity. Therefore, as the velocity of one frame approaches zero, the effect of the boost operator increases.