Finding Representations
According to general representation theory of Lie groups, one first looks for the representations of the complexification of the Lie algebra of the Lorentz group. A convenient basis for the Lie algebra of the Lorentz group is given by the three generators of rotations Jk = εijkLij and the three generators of boosts Ki = L0i where i, j, and k run over the three spatial coordinates and ε is the Levi-Civita symbol for a three dimensional spatial slice of Minkowski space. Note that the three generators of rotations transform like components of a pseudovector J and the three generators of boosts transform like components of a vector K under the adjoint action of the spatial rotation subgroup.
This motivates the following construction: first complexify, and then change basis to the components of A = (J + i K)/2 and B = (J – i K)/2. In this basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra su2 and moreover that they commute with each other. In other words, one has the isomorphism
The utility of this isomorphism comes from the fact that su2 is the complexification of the rotation algebra, and so its irreducible representations correspond to the well-known representations of the spatial rotation group; for each j in ½Z, one has the (2j + 1)-dimensional spin-j representation spanned by the spherical harmonics with j as the highest weight. Thus the finite dimensional irreducible representations of the Lorentz group are simply given by an ordered pair of half-integers (m, n) which fix representations of the subalgebra spanned by the components of A and B respectively.
Read more about this topic: Representation Theory Of The Lorentz Group
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