From Special Relativity to Feynman Diagrams.pdf

Let us now come back to the case in which g is a

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Let us now come back to the case in which G is a transformation group acting on the space-time RF. The simplest instance of field is the scalar field in which D is the trivial representation 1 , defining a type-(0,0) tensor, with p = 1 that is V p = R (real scalar field) or C (complex scalar field). A complex scalar field ( r ) can be 8 Note that the analogous of the Poincaré group in the three dimensional Euclidean space E 3 is the known group of congruences of Euclidean geometry, acting on the space coordinates as in ( 4.102 ).

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196 7 Group Representations and Lie Algebras described as a couple of real scalar fields 1 ( r ), 2 ( r ) , defined at each point r by the real and imaginary parts of ( r ) : ( r ) = 1 ( r ) + i 2 ( r ) ( 1 ( r ), 2 ( r )) . (7.50) The transformation law ( 7.45 ) reduces, for a scalar field, to ( r ) = ( r ( r )) = ( R 1 ( r + r 0 )). (7.51) If M n = M 4 and G is the Poincaré group, r is the space-time coordinate vector ( x μ ) and R = = ( μ ν ) SO ( 1 , 3 ) . In general, as discussed in Sect.4.3 , the coordinates r = ( x i ) and r = ( x i ) refer to the same point P of M n , therefore the numerical value of the scalar field must be the same, even if, when substituting x i = x i ( x j ) = ( R 1 ) i j ( x j + x j 0 ) the functional form changes from to . Writing ( r ) = ( r ) we are considering the transformation r = Rr r 0 from a passive point of view since space-points are considered fixed while only the coordinate frame is changed. However the same transformation can be also considered from a different point of view, namely as a change in the functional form of ( r ) ( r ) ( r ), (7.52) with ( r ) = ( R 1 ( r r 0 )) . In this case we consider the transformation as an active transformation, since the emphasis is on the functional change of . The given change of coordinate in this case is thought of as due to a change of the geometric point. 9 When considering the change in the functional form from an active point of view it is sometimes convenient to denote the new functional form taken by as consequence of the coordinate change induced by an element g G , as the action of an operator O g on . 10 Equation ( 7.51 ) takes the following form: O g ( r ) = ( R 1 ( r r 0 )), (7.53) where, as usual, R = R ( g ) and r 0 = r 0 ( g ) . Consider, for the sake of simplicity, a group G acting in a homogeneous way on the coordinates (i.e. r 0 0 ) and apply in succession two transformations g 1 , g 2 G , the resulting transformation corresponding to the product g 2 · g 1 G . We have: 9 Note that in the discussion of the vector and tensor calculus in Sect.4.1 the emphasis was on the passive point of view since the reference frame was changed by the transformations. Therefore the whole of the vector and tensor calculus was developed taking this point of view. The active point, as previously mentioned, will be actually adopted in Chap.9 when discussing the action of a group on the Hilbert space of states in quantum mechanics.
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