From Special Relativity to Feynman Diagrams.pdf

A we can write 184 7 group representations and lie

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184 7 Group Representations and Lie Algebras V = A ( V ) = V i A ( u i ) = V i A j i u j = V j u j . (7.7) Althoughwefindthesamerelation( 7.5 ),thequantitiesinvolvedhaveadifferentinter- pretation: V i and V i in the passive description are the components of the same vector in the new and old bases, while in the active representation they represent the compo- nents of the new and old vectors with respect to the same basis. We shall use the active description only when describing the effect of a coordinate transformation on the quantum states (which are vectors in a complex vector space). From now on we shall represent each vector by the array of its components V ( V i ) with respect to a given basis, so that the effect of a transformation A , in both the complementary descrip- tions, is then described by the same matrix relation ( 7.5 ): V V = A ( V ) AV . If we have two linear transformations A, B on V n , their product A · B is the linear transformation resulting from their consecutive action on each vector: If B maps V into V = B ( V ) = BV and A maps V into V = A ( V ) = AV , then A · B is the transformation which maps V into V = A ( B ( V )) = A ( BV ) = ( AB ) V . The product of two transformations is thus represented by the product of the matrices associated with each of them, in the same order. 3 The identity transformation I is the linear transformation which maps any vector into itself and it is represented by the identity n × n matrix 1 . For any linear transformation A we trivially have A · I = I · A = A . Finally, being a linear transformation invertible, we can define its inverse A 1 such that, if A maps V into V = A ( V ) , A 1 is the linear transformation mapping V into the unique vector V = A 1 ( V ) which corresponds to V through A . It follows that A 1 is represented by the inverse A 1 of the matrix A associated with A . Finally the product of linear transformations is associative, the argument being substantially the same as the one used for coordinate transformations in Sect.4.5 . Linear transfor- mations on vector spaces close therefore a group. Given the identification of linear transformations on V n with n × n non singular matrices, such group can be identified with the group GL ( n , C ) , if V n is complex, or GL ( n ) if V n is real (the symbol GL stands indeed for General Linear transformations ). An n-dimensional representation D (or representation of degree n ) consists in associating with each element g G a linear transformation D (g) on a linear vector space V n in such a way that: D ( g ) · D ( g ) = D ( g · g ). (7.8) Since linear transformations on V n are uniquely defined by n × n invertible matrices, with respect to a given basis, equation ( 7.8 ) characterizes a representation as a homo- 3 The action of a non-invertible operator A is also represented by a matrix A , its definition being analogous to the one given for transformations. Such matrix, however, is singular. The product of two generic operators A and B is defined as for transformations and is represented by the product of the corresponding matrices in the same order. Examples of non-invertible operators appear among
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