By Breit G.

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Thus the structure constants deﬁne the Lie algebra, and the problem of ﬁnding all matrix representations of a Lie algebra amounts to the algebraic problem of ﬁnding all possible matrix solutions TRa of eq. 9). 2 Actually, the generators of a Lie group can even be deﬁned without making any reference to a speciﬁc representation. One makes use of the fact that a Lie group is also a manifold, parametrized by the coordinates θ a , and deﬁnes the generators as a basis of the tangent space at the origin.

81) with Lµν given in eq. 78). We write ΛL in the form ΛL = e− 2 ωµν S i µν . 82) Then eq. 83) J µν = Lµν + S µν . 84) with Comparing eq. 82) with eq. 85) σi . 86) 2 We recognize in eq. 84) the separation of the angular momentum into the orbital and the spin contributions. It is clear that this separation is completely general, and holds for any representation. 78) independently of the representation, while S µν depends on the speciﬁc representation used. For instance, for right-handed Weyl ﬁelds S i are still given by eq.

A representation in which all generators are equal to zero is trivially a solution of eq. 9), for any Lie group, and so it is called the trivial representation. The four-vector representation is more interesting. In this case i, j are themselves Lorentz indices, so each generator J µν is represented by a 4 × 4 matrix (J µν )ρ σ . The explicit form of this matrix is (J µν )ρ σ = i (η µρ δσν − η νρ δσµ ) . 23) This can be shown observing that, from eqs. 24) with (J µν )ρ σ given by eq. 23) (this solution for J µν is unique because we require the antisymmetry under µ ↔ ν).

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