From Special Relativity to Feynman Diagrams.pdf

Suppose now the change of basis is such that the

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Suppose now the change of basis is such that the metric is invariant, that is g i j = g i j . We will then have: V · W = V i g i j W j = V i g i j W j , (4.82)
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4.4 Rotations in Three-Dimensions 109 that is the functional dependence of V · W over the old and new components of the two vectors is the same . Let us denote by R ( R i j ) = R 1 1 R 1 2 R 1 3 R 2 1 R 2 2 R 2 3 R 3 1 R 3 2 R 3 3 , (4.83) the matrix implementing such transformation: V i = R i j V j , W i = R i j W j (or, in matrix notation V = RV , W = RW ). Expressing in ( 4.82 ) the new components in terms of the old ones we find: V i g i j W j = V k R i k g i j R j W . (4.84) Requiring the above invariance to hold for any couple of vectors ( V i ) and ( W i ) , we conclude that: R i k g i j R j = g k . (4.85) In matrix notation ( 4.85 ) reads R T gR = g , (4.86) where g ( g i j ) is the matrix whose components are the entries of the metric tensor g i j . The above equation could have been obtained from Eq. (4.27), setting g = g and D = R . Recalling that g i j = u i · u j , the above relation is telling us that scalar products among the basis elements are invariant under R . It is now convenient to use an ortho-normal basis ( u i ) to start with: u i · u j = g i j δ i j , (4.87) since the ortho-normality property of a basis is clearly preserved by all the transfor- mations R which leave the metric invariant. In the ortho-normal basis the relations ( 4.85 ) and ( 4.86 ) become: R i k δ i j R j = n i = 1 R i k R i = δ k , (4.88) and, in matrix form: R T 1R = R T R = 1 , (4.89) where 1 i j ) = 1 0 0 0 1 0 0 0 1 . (4.90)
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110 4 The Poincaré Group Transformation matrices satisfying ( 4.88 ), or equivalently ( 4.89 ), are called ortho- gonal . Orthogonal transformations can be alternatively characterized as the most general Cartesian coordinate transformations in Euclidean space mapping two ortho-normal bases into one another, leaving the origin fixed, i.e. the most general homogeneous transformations between Cartesian rectangular coordinate systems. 7 Recalling from Eq. (4.13) that the distance squared between two points is defined as the squared norm of the relative position vector, an orthogonal transformation leaves its coordinate dependence invariant. Viceversa, if an affine transformation x i x i = R i j x j x i 0 of the Cartesian coordinates x i leaves the distance between any two points, as a function of their coordinates, invariant, its homogeneous part, described by the matrix R and defining the transformation of the relative position vector, is an invariance of the metric tensor. This means that, starting from an ortho- normal basis in which g i j = δ i j , R is an orthogonal matrix. To illustrate the above implication, note that the invariance of the coordinate dependence of the distance d ( A, B ) between any two points translates into the invariance of the norm of any vector as a function of its components. This latter property amounts to stating that, if V = ( V i ) and V = ( V i ) are the components of a same vector in the old and new bases, related by the transformation R , then V 2 = V T V = V 2 = V T V .
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