From Special Relativity to Feynman Diagrams.pdf

Examples of non invertible operators appear among the

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the corresponding matrices in the same order. Examples of non-invertible operators appear among the hermitian operators representing observables in quantum mechanics, V n being in this case the infinite dimensional vector space of quantum states. An other example of not necessarily invertible operators are the infinitesimal generators of continuous transformations, to be introduced below, which are indeed related, as we shall discover in the next chapters, to observables in quantum mechanics.
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7.2 Representations 185 morphic map of G into the set (group) of n × n invertible matrices. 4 Introducing a basis ( u i ), i = 1 , . . ., n , on V n , D(g) acts as an n × n matrix D ( g ) ( D ( g ) i j ) on the components of a vector V = ( V i , . . ., V n ) according to the law V i = D ( g ) i j V j V = D ( g ) V , We shall denote by the bold symbol D the representation of a group in terms of matri- ces. The vector space V n is called the carrier of the representation or representation space . In the case of the rotation group, for example, the three dimensional Euclidean space V 3 is the carrier of the representation studied in Chap.4 : g 1 , θ 1 , θ 2 ) SO ( 3 ) D −→ D ( g ) i j = R 1 , θ 1 , θ 2 ) i j , i , j = 1 , 2 , 3 . For a general representation the matrix D (g) is an element of GL ( n , C ) or of GL ( n , R ) , depending on whether the base space is a complex or real vector space. In the active picture, for any g G , D (g) maps vectors into vectors, all represented with respect to a same basis ( u i ) . On replacing the original basis ( u i ) by a new one ( u i ) , related to it through a non singular matrix A , as in ( 7.4 ), the matrix D (g) gets replaced by the matrix D ( g ) = AD ( g ) A 1 which represents the action of D(g) in the new basis. This is easily shown starting from the matrix relation between the components of a vector V 1 and its transformed V 2 in the old basis: V 2 = D ( g ) V 1 . Being the components V 1 and V 2 of the two-vectors in the new basis given by V 1 = AV 1 , V 2 = AV 2 , we find: V 2 = AV 2 = AD ( g ) V 1 = AD ( g ) A 1 V 1 = D ( g ) V 1 . (7.9) It is easily verified that the mapping D of a generic group element g into D ( g ) is still a representation, also denoted by D = ADA 1 . The representations ADA 1 and D are then said equivalent , and we write: D ADA 1 . (7.10) If the homomorphic mapping: g −→ D ( g ), (7.11) is isomorphic , namely it is one-to-one and onto, then the representation is faithful , otherwise it is unfaithful . A trivial, but important, representation is obtained by the mapping: g −→ 1 , g G , (7.12) and is called the identity, or trivial representation , simply denoted by 1 . 4 It is obvious that the identity g 0 = e element of the group is represented by the unit n-dimensional matrix that we will denote by 1 or else by I .
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186 7 Group Representations and Lie Algebras Coming back to the general case, let us assume that there exists a subspace V m V n of dimensions m < n such that every element of the subspace V m is transformed into an element of the same subspace under all the transformations of the group G : g G D ( g ) : V m V m .
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