From Special Relativity to Feynman Diagrams.pdf

Here we rewrite for the sake of completeness f i j f

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here we rewrite, for the sake of completeness: F i j = ˜ F i j S + F i j A + D i j , where ˜ F i j S = 1 2 ( F i j + F ji ) 1 n pq F pq i j , F i j A = 1 2 ( F i j F ji ), D i j = 1 n pq F pq i j . As it was shown in Chap.4 each of the three subspaces is invariant under O ( n ) transformations, elements of each subspace being transformed into elements of the same subspace. It follows that the n 2 -dimensional representation of O ( n ) is fully reducible into three irreducible representations D ( S ) , D ( A ) , D Tr = 1 of dimensions n ( n + 1 ) 2 1, n ( n 1 ) 2 and 1, respectively: D D = D ( S ) D ( A ) D Tr , (7.22) where D ( S ) act on symmetric traceless matrices and D Tr on the tensors proportional to δ i j (traces).
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7.2 Representations 189 The same decompositions hold if instead of the group O ( n ) we have a non-compact form like the Lorentz group SO ( 1 , 3 ) when n = 4. The only difference is that the one-dimensional subspace is now proportional to the Minkowski metric η μν . Let us now discuss a property in group theory which has important applications in physics. Schur’s Lemma: Let D be an irreducible n-dimensional representation of a group G . A matrix T which commutes with all matrices D (g) , for any g G , is proportional to the identity matrix 1 n . In formulas, if g G : TD ( g ) = D ( g ) T , (7.23) there exists a number λ such that: T = λ 1 n T i j = λδ i j , i , j = 1 , . . ., n . (7.24) To show this, let λ be an eigenvalue of T in V n (which always exists) and V the corresponding eigenvector: TV = λ V . (7.25) Let V λ = { V V n | TV = λ V } be the eigenspace of the matrix T corresponding to the eigenvalue λ . This space is non-empty since V V λ . It can be easily verified that V λ is invariant under the action of G . Indeed for any V V λ and g G , the vector D ( g ) V is still in V λ since: TD ( g ) V = D ( g ) TV = λ D ( g ) V , (7.26) where we have used the hypothesis (7.23) of Schur’s lemma that T commutes with the action of G on V n defined by the representation D . Since V λ is a non-empty invariant subspace of V n and being D an irreducible representation by assumption, V λ can only coincide with V n . We conclude that T acts on V n as λ times the identity matrix. An important consequence of Schur’s lemma is that, if D is a n -dimensional representation of a group G and if there exists a matrix T which commutes with all matrices D (g) , for any g G , and which is not proportional to the identity matrix 1 n , then D is reducible . This property provides us with a powerful criterion for telling if a representation is reducible and, in some cases, to determine its irreducible components: Suppose we find an operator T on V n which commutes with all the transformations D(g) representing the action of a group G on the same space. The matrix representation T of T will then have the form: T = c 1 1 k 1 0 .
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