From Special Relativity to Feynman Diagrams.pdf

5 groups of transformations 117 r θ 1 r θ 2 r θ 3

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4.5 Groups of Transformations 117 R ( θ 1 ) · R ( θ 2 ) = R ( θ 3 ), (4.112) where θ i 3 = θ i 3 ( θ 1 , θ 2 ) are analytical functions. In general a group of continuous transformations satisfying the above properties is called a Lie group . Since rotation matrices are continuous functions of angles, we can consider rota- tions which are infinitely close to the identity element. These transformations, called infinitesimal rotations, are defined by very small (i.e. infinitesimal) angles θ i . We can expand the entries of an infinitesimal rotation matrix R 1 , θ 2 , θ 3 ) in Taylor series with respect to its parameters and write, to first order in the angles: R 1 , θ 2 , θ 3 ) = 1 + R ∂θ i θ i = 0 θ i + O 2 ). (4.113) Introducing the matrices L i R ∂θ i | θ i = 0 , called infinitesimal generators of rotations, the above expansion, to first order, reads: R ( θ ) = 1 + θ i L i + O 2 ) 1 + θ i L i . (4.114) Let us consider, as an example, a rotation about the X axis, described by the matrix R x in ( 4.93 ), by an angle θ and let us expand it, for small θ , up to fist order in the angle: R x = 1 0 0 0 cos θ sin θ 0 sin θ cos θ = 1 0 0 0 1 0 0 0 1 + θ 0 0 0 0 0 1 0 1 0 + O 2 ) 1 + θ L 1 . (4.115) From this equation we can read the expression of the first infinitesimal generator L 1 , associated with rotations about the X axis: L 1 = 0 0 0 0 0 1 0 1 0 . (4.116) Similarly, expanding infinitesimal rotation matrices about the Y and Z axes we find: R y (θ) cos θ 0 sin θ 0 1 0 sin θ 0 cos θ 1 + θ L 2 , (4.117) R z (θ) = cos θ sin θ 0 sin θ cos θ 0 0 0 1 1 + θ L 3 , (4.118)
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118 4 The Poincaré Group from which we can derive the corresponding infinitesimal generators: L 2 = 0 0 1 0 0 0 1 0 0 , L 3 = 0 1 0 1 0 0 0 0 0 . (4.119) In a more compact notation we may write the three matrices L i as follows 11 : ( L i ) j k = i jk . (4.120) Since a generic rotation R i ) can be written as a sequence of consecutive rotations about the three axes: R 1 , θ 2 , θ 3 ) R z 3 ) R y 2 ) R x 1 ), (4.121) expanding the right-hand side for small θ i , up to the first order, we find R 1 , θ 2 , θ 3 ) 1 + θ 1 L 1 + θ 2 L 2 + θ 3 L 3 , (4.122) that is the infinitesimal generators of a generic rotation are expressed as a linear combination (whose parameters are the rotation angles) of the three matrices L i given in ( 4.116 ) and ( 4.119 ). In other words, any linear combination of infinitesimal generators is itself an infinitesimal generator, that is infinitesimal generators span a linear vector space , of which the matrices ( L i ) define a basis. From ( 4.120 ) it follows that the effect of an infinitesimal rotation R θ ) , by infinitesimal angles δθ i 0, can be described in terms of the following displacement of the coordinates: x i = x i i jk δθ j x k r = R θ ) r r δ θ × r , (4.123) where × denotestheexternalproductbetweentwovectors: δ θ × r ( i jk δθ j x k ) .The reader can easily verify the following commutation relation between the infinitesimal generators: L i , L j L i L j L j L i = C i j k L k , (4.124) where C i j k = − i jk . (4.125) In other words the commutator [ ,
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