From Special Relativity to Feynman Diagrams.pdf

10 here by operator we mean a linear mapping of the

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10 Here by operator we mean a linear mapping of the vector space of square-integrable functions on M n into itself, according to the definition given earlier. O g is actually a transformation and it is therefore invertible.
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7.4 Representation of a Group on a Field 197 x i g 1 −→ x i = R ( g 1 ) i j x j g 2 −→ x i = R ( g 2 ) i j R ( g 1 ) j k x k = R ( g 2 · g 1 ) i k x k or, expressing x i in terms of x k x i = [ R ( g 2 · g 1 ) 1 ] i k x k . Actually the operators O R give a homomorphic realization of the group G , where by realization we mean a homomorphic mapping on the function space. Indeed from ( r ) O g ( r ) = ( R ( g ) 1 r ), (7.54) using the short-hand notation R 1 R ( g 1 ) and R 2 R ( g 2 ) , it follows O g 2 · O g 1 ( r ) O g 2 ( O g 1 ( r ) ) = O g 2 ( R 1 1 r ) = ( R 1 1 R 1 2 r ) = (( R 2 R 1 ) 1 r ) = ( R ( g 2 · g 1 ) 1 r ). (7.55) However the same result is also obtained acting on with the operator O g 2 · g 1 cor- responding to the group element g 2 · g 1 : O g 2 · g 1 ( r ) = ( R ( g 2 · g 1 ) 1 r ). Therefore we conclude that O g 2 · g 1 = O g 2 · O g 1 . (7.56) O is thus a homomorphims of G into the group of linear transformations on the space of functions ( x ) on M n . It is easy to verify that O maps the unit element of G into the identity transformation I which maps a generic function ( x ) into itself. Moreover O 1 g = O g 1 . The mapping O : g G O g , has the same properties as a representation D . However the linear transformations O g are not implemented by matrices, since they affect the functional form of the field they act on. For this reason O should be referred to as a realization of G on fields rather than a representation. 7.4.1 Invariance of Fields The relation ( 7.53 ) is referred to general transformations of Cartesian coordinates (affine transformations), whose homogeneous part described a linear transformation on V n (i.e. belongs to the group GL ( n ) ). This relation is actually valid also for any any (invertible) coordinate transformation (thus including curvilinear coordinates) x i = f i ( x 1 , x 2 , . . ., x n ), (7.57) where f ( x ) ( f i ( x )) are differentiable functions which can be inverted to express the old coordinates ( x 1 , . . ., x n ) in terms of the new ones ( x 1 , . . ., x n ) :
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198 7 Group Representations and Lie Algebras x i = f 1 i ( x ) , or simply x = f 1 ( x ) . Also the effect of this coordinate transfor- mation on ( x ) can be represented by the action of an operator O f O f ( x ) = ( f 1 ( x )). (7.58) Only for linear coordinate transformations (among Cartesian coordinates) f i ( x ) reduces to: x i = R i j x j x i 0 . Let us now recall the definition of invariance of a function : If the functional form of does not change under a coordinate transformation ( 7.57 ), O f ( x ) = ( x ) then is invariant. From the relation ( 7.51 ) and the requirement of invariance we obtain ( x ) = ( f 1 ( x )). (7.59) From the active point of view this means that even if the geometric point is changed, the functional form remains the same.
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