By the way, since many groups can have the same algebra, there is a small subtlety in gener-ating group representations from algebra representations. The group generated by exponenti-ating the Lie algebra of a given group is known as theuniversal coverof the given group. Forexample, exponentiating su(2)gives SU(2). Since SU(2)and SO(3)both have su(2)as theiralgebra, SU(2)is the universal cover of SO(3). The algebrao(1,3)generates SL(2,C), which isthe universal cover of the Lorentz group. We will revisit the distinction between SL(2,C)andthe Lorentz group more in section 6.1. For now, we will simply study su(2)×su(2).The names so(n)and su(n)stand forspecial orthogonalalgebra andspecial unitaryalgebra. A linear operator is calledorthogonalif it preserves a norm defined with transpose:VTVis invariant under SO(n). A linear operator is calledunitaryif it preserves a norm definedwith adjoint:V†Vis invariant under SU(n).Specialmeans the determinant of a matrix repre-sentation of that operator is 1.The decompositiono(1,3) =su(2)×su(2)makes studying the irreducible representations veryeasy. We already know from quantum mechanics what the representations of su(2)are, sincesu(2) =so(3)is the algebra of Pauli matrices, which generates the 3D rotation group SO(3). Therepresentations are characterized by a quantum numberj, and act on vector spaces with2j+ 1basis elements, labeled bym=jz. If this is unfamiliar to you, try Problem 1. It follows thatirreducible representations of the Lorentz group are characterized by two numbersAandB.The(A,B)representation has(2A+ 1)(2B+ 1)degrees of freedom.Representations of the Lorentz Group5
The regular rotation generators areJ=J++J-, where we use the vector superscript to callattention to the fact that the spins must be added vectorially, as you might remember fromstudying Clebsch-Gordan coefficients.Since the 3D rotation group SO(3)is a subgroup of theLorentz group, every representation of the Lorentz group will also be a representation of SO(3).In fact, finite-dimensional irreducible representations of the Lorentz group, which are character-ized by two half-integers(A,B), produce many representations of SO(3): with spinsJ=A+B,A+B−1, ,|A−B|. Example decompositions are shown in Table 1. Since SO(3)is the littlegroup of the Poincaré group (for massive particles) its representations determine the spins thatparticles have.rep of su(2)×su(2)spins(SO(3)representations)(A,B) = (0,0)J= 0(A,B) = (12,0)J=12(A,B) = (0,12)J=12(A,B) = (12,12)J= 1,0(A,B) = (1,0)J= 1(A,B) = (1,1)J= 2,1,0Table 1.Decomposition of irreducible representations of the Lorentz group into irreducible representa-tions of different spin, corresponding to the SO(3)subgroup comprising 3D rotations.