Representation Theory of The Lorentz Group - Finding Representations

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 = εijkL_ij and the three generators of boosts Ki = L_0i 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.