Friday, March 30, 2018

lie algebra - Spin matrix for various spacetime fields


Let Vμ be a vector field defined in a Minkowski spacetime and suppose it transforms under a Lorentz transformation Vμ=ΛμνVν. We can write this like Vμ=(eiω)μνVν I think where ω denotes a rotation in some plane spanned by indices {ρσ}, say. In 2D Euclidean space time, we can write the matrix representation of Λ as (cosωsinωsinωcosω) and in Minkowski space this changes to the 'hyperbolic' rotation. Linearising the above yields (1ωω1)=Id+(0ωω0)=Id+ω(0110)


Now compare with the more general treatment: Vμ=ΛμνVν(δμν+ωμν)Vν, where ωμν(ωρσSρσ)μν In 2D, the spin matrix S when acting on vectors in 2D Euclidean space time is therefore the matrix multiplying ω above, which agrees with the single generator of the SO(2) group.


If we continue with the general analysis, we obtain VμVμ=ωμνVν=ημρωρνVν=ωρσδμρησνVν Now use the antisymmetry of ωρσ gives 2(VμVμ)=ωρσ(δμρησνδμσηρν) from which we can identify S to be δμρησνδμσηρν. I am wondering how this agrees with the matrix I obtained above.


Many thanks.




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