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Mass and spin of the particle are used in classification of elementary particles. The mass is defined to be a Lorentz invariant quantity. On the other hand, the spin is a spacelike 4-vector and cannot be defined as an invariant quantity.

My question is, what would be a convenient definition of a relativistic spin 3-vector? As is known, 3-vectors get contracted/dilated under Lorentz transformations. So, is the 3-spin defined to be equal to the rest frame 3-spin, or do we allow it to contract/dilate? It seems to me that the first option is generally used, especially when considering Thomas precession and similar effects. Although, I haven't seen any relevant discussion in textbooks on this topic.

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  • $\begingroup$ the spin is a spacelike 4-vector Huh? Why do you say this? In relativity, the natural way to represent angular momentum is as a rank-2 tensor. My question is, what would be a convenient definition of a relativistic spin 3-vector? There is no such thing as a relativistic 3-vector. If it's a 3-vector, it's not relativistic. $\endgroup$ – Ben Crowell Sep 6 '14 at 21:00
  • $\begingroup$ @BenCrowell : there is two identical ways of relativistic consideration of the spin: through 4-vector (Pauli-Lubanski vector) or through spin tensor rank 2. I think that author means this statement. $\endgroup$ – Andrew McAddams Sep 6 '14 at 21:03
  • $\begingroup$ @BenCromwell From an experimentalists point of view, spin is, without a doubt, 3-vector. As are, e.g. electric and magnetic field. I am aware that it is advantageous to describe all this quantities in terms of various 4-tensors, but that doesn't mean that the frame-dependent 3-vector approach isn't useful sometimes. $\endgroup$ – user17116 Sep 6 '14 at 22:20

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