Monday, December 14, 2015

special relativity - Adjoint representation of the Lorentz group


Is it possible to construct an adjoint representation for the Lorentz group?




Answer



I suspect the impeccable WetSavannaAnimal answer is too abstract and general to satisfy the OP. Let me vulgarize it a bit. The Lorentz group has 6 generators, so your adjoint rep will be a 6-dimensional rep, a set of 6x6 matrices, which satisfy the same Lie algebra (commutators) as the conventional 4x4 matrices transforming the fundamental (x,y,z,t) 4-vector. As an aside, note the curvature form (parity inv. 2-form) of the Lorentz group is also 6-dimensional.


Recall the Lorentz group may be written in a pretty form, that is rotations JiϵimnMmn/2 and boosts, KiM0i, so that [Jm,Jn]=iϵmnkJk , [Jm,Kn]=iϵmnkKk , [Km,Kn]=iϵmnkJk.


Might further note the important simplification [Jm+iKm,JniKn], which permits reduction of the Lorentz algebra to su(2)su(2) and efficient treatment of its associated representations.


Now consider the six 6x6 matrices with the 3 J+iKs on the upper left 3x3 subspace of these 6x6 ones, and the 3 J-iKs on the lower right block, spanned by the 4,5,6 indices. Keep the indices of the upper left block to be the usual 1,2,3; and rename the indices of the lower right block from 1,2,3 to 4,5,6. The commutation relations are then manifest, and the structure constants fmnl are sparse, basically ϵijk for indices (1,2,3) or (4,5,6), and zero otherwise.


So, then, as you probably learned from SU(3), these very structure constants f provide your 6x6 matrices in the adjoint, with one of the 3 indices (taking 6 values) specifying the particular generator of the Lorentz group represented.


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