From Special Relativity to Feynman Diagrams.pdf

Applying this property to the squared norm v w 2 of

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Applying this property to the squared norm V + W 2 of the sum of two generic vectors V , W , one easily finds that the scalar product ( V , W ) = V T W is functionally invariant under R , namely that V T W = V T W = V T ( R T R ) W . From the arbitrari- ness of V and W , property (4.89) follows. Rotations about an axis and reflections in a plane are examples of orthogonal transformations in E 3 .Since ( 4.89 ) implies ( R T ) 1 = R , there is no distinction between the transformation properties of the covariant and contravariant components of a vector under orthogonal transforma- tions, as it is apparent from the fact that, being the metric δ i j invariant, the two kinds of components coincide V i = δ i j V j = V i in any Cartesian coordinate system. A simple example of orthogonal transformation is a rotation by an angle θ about the X axis, see Fig. 4.2 . 8 The relation between the new and the old basis reads u 1 = u 1 , u 2 = cos θ u 2 + sin θ u 3 , u 3 = − sin θ u 2 + cos θ u 3 . (4.91) Being u i = R x 1 j i u j , from ( 4.91 ) we can read the form of the inverse of the rotation matrix R x : R 1 x = ( R x 1 j i ) = 1 0 0 0 cos θ sin θ 0 sin θ cos θ , (4.92) 7 In what follows, when referring to Cartesian coordinate systems, the specification rectangular will be understood, unless explicitly stated, since we shall mainly restrict ourselves to coordinate systems of this kind. 8 In our conventions, the rotation angle θ , on any of the three mutually orthogonal planes XY, XZ, YZ , is positive if its orientation is related to that of the axis orthogonal to it (i.e. Z, Y, X ) by the right-hand rule.
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4.4 Rotations in Three-Dimensions 111 Fig.4.2 Rotation about the X axis by an angle θ . from which we derive: R x = ( R x j i ) = 1 0 0 0 cos θ sin θ 0 sin θ cos θ , (4.93) The new components V i of a vector are related to the old ones V i according to: V i = R x i j V j , that is V 1 = V 1 , V 2 = cos θ V 2 + sin θ V 3 , V 3 = − sin θ V 2 + cos θ V 3 . (4.94) The matrix R x in ( 4.93 ), which describes this rotation, depends on the continuous parameter θ : R x = R x (θ) . The reader can easily verify that ( 4.89 ) is satisfied by R x . Let us observe that det ( R x ) = 1. This is a common feature of all the rota- tion matrices and can be deduced by computing the determinant of both sides of ( 4.89 ) and using the known properties of the determinant: det ( A T ) = det ( A ), det ( AB ) = det ( A ) det ( B ) : det ( R ) det ( R T ) = det ( R ) 2 = 1 det ( R ) = ± 1 . (4.95) Orthogonal transformations with det ( R ) = + 1 are called proper rotations , or simply rotations , while those with det ( R ) = − 1 also involve reflections and are called A matrix R having this property is called improper rotations . A typical example of improper rotation is given by a pure reflection, that is a transformation changing the orientation of one or all the coordinate axes, e.g. 1 0 0 0 1 0 0 0 1 . (4.96)
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112 4 The Poincaré Group Letusnowperformtwoconsecutiverotations,representedbythematrices R 1 , R 2 .
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