By the way since many groups can have the same

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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 the universal cover of the given group. For example, exponentiating su (2) gives SU (2) . Since SU (2) and SO (3) both have su (2) as their algebra, SU (2) is the universal cover of SO (3) . The algebra o(1 , 3) generates SL (2 , C ) , which is the universal cover of the Lorentz group. We will revisit the distinction between SL (2 , C ) and the 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 for special orthogonal algebra and special unitary algebra. A linear operator is called orthogonal if it preserves a norm defined with transpose: V T V is invariant under SO ( n ) . A linear operator is called unitary if it preserves a norm defined with adjoint: V V is invariant under SU ( n ) . Special means the determinant of a matrix repre- sentation of that operator is 1. The decomposition o(1 , 3) = su (2) × su (2) makes studying the irreducible representations very easy. We already know from quantum mechanics what the representations of su (2) are, since su (2) = so (3) is the algebra of Pauli matrices, which generates the 3D rotation group SO(3). The representations are characterized by a quantum number j , and act on vector spaces with 2 j + 1 basis elements, labeled by m = j z . If this is unfamiliar to you, try Problem 1. It follows that irreducible representations of the Lorentz group are characterized by two numbers A and B . The ( A,B ) representation has (2 A + 1)(2 B + 1) degrees of freedom. Representations of the Lorentz Group 5
The regular rotation generators are J = J + + J - , where we use the vector superscript to call attention to the fact that the spins must be added vectorially, as you might remember from studying Clebsch-Gordan coefficients. Since the 3D rotation group SO (3) is a subgroup of the Lorentz 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 spins J = A + B, A + B 1 , , | A B | . Example decompositions are shown in Table 1. Since SO (3) is the little group of the Poincaré group (for massive particles) its representations determine the spins that particles have. rep of su (2) × su (2) spins ( SO (3) representations ) ( A,B ) = (0 , 0) J = 0 ( A,B ) = ( 1 2 , 0) J = 1 2 ( A,B ) = (0 , 1 2 ) J = 1 2 ( A,B ) = ( 1 2 , 1 2 ) J = 1 , 0 ( A,B ) = (1 , 0) J = 1 ( A,B ) = (1 , 1) J = 2 , 1 , 0 Table 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.

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