Hiero
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This question is beyond my level of understanding, nonetheless I feel it can’t be right. I have been studying Geometric algebra and was thinking about (6-component) bivectors in spacetime, (specifically the electromagnetic field and 4D-angular-momentum). The conventional perspective is to treat these bivectors as anti-symmetric second rank tensors. Regardless of perspective, (upon a change of reference frame,) these objects transform in a definite way.
I saw on Wikipedia though that the intrinsic spin is supposedly treated as a four-vector (arbitrarily taking the time component to be zero in the rest frame). This really bothers me because it seems clear that intrinsic spin should also be treated as a bivector/antisymmetric tensor. Spin is an axial vector which hints at its bivector nature. More importantly though, it should add with the orbital angular momentum (which is a bivector/antisymmetric tensor) and hence should transform the same way! Treating it like a four-vector just seems like non-sense!
In order to treat it as a bivector/antisymmetric tensor though we would need three time-like components which form a polar vector corresponding to the axial spin (in the same way that the electric field corresponds to the magnetic field or the “moment of mass” corresponds to the angular momentum). I have no idea what that corresponding polar vector might be.
So I ask you wise scientists; Is it not silly to treat the spin as a four-vector? (How would it combine with the orbital 4D-angular-momentum??) And if Wikipedia is mistaken and it should instead transform like the electromagnetic field, then what is the corresponding polar vector?
I saw on Wikipedia though that the intrinsic spin is supposedly treated as a four-vector (arbitrarily taking the time component to be zero in the rest frame). This really bothers me because it seems clear that intrinsic spin should also be treated as a bivector/antisymmetric tensor. Spin is an axial vector which hints at its bivector nature. More importantly though, it should add with the orbital angular momentum (which is a bivector/antisymmetric tensor) and hence should transform the same way! Treating it like a four-vector just seems like non-sense!
In order to treat it as a bivector/antisymmetric tensor though we would need three time-like components which form a polar vector corresponding to the axial spin (in the same way that the electric field corresponds to the magnetic field or the “moment of mass” corresponds to the angular momentum). I have no idea what that corresponding polar vector might be.
So I ask you wise scientists; Is it not silly to treat the spin as a four-vector? (How would it combine with the orbital 4D-angular-momentum??) And if Wikipedia is mistaken and it should instead transform like the electromagnetic field, then what is the corresponding polar vector?