From Special Relativity to Feynman Diagrams.pdf

Let us now consider the decomposition 475 for tensors

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-invariant. Let us now consider the decomposition ( 4.75 ) for tensors transforming under O ( n ) . We have shown that the two vector spaces spanned by the symmetric F i j S and anti-symmetric F i j A components of rank 2 tensors F i j are invariant , in the sense that a symmetric (anti-symmetric) tensor is mapped by any element of GL ( n ) into a tensor with the same symmetry property. It is easy to show that, if we consider transformations of tensors with respect to O ( n ) , we can use the O ( n ) invariant tensor δ i j to decompose the symmetric component F i j S in ( 4.75 ) into a trace part δ i j F k k , where F k k δ i j F i j = δ i j F i j S , (4.104) and a traceless part ˜ F i j S defined as: ˜ F i j S = 1 2 ( F i j + F ji ) 1 n δ i j F k k . (4.105) 10 Complete antisymmetrization in the three indices μ, ν, ρ on a generic tensor U μνρ , is defined as follows: U [ μνρ ] = 1 3 ! ( U μνρ U μρν + U νρμ U νμρ + U ρμν U ρνμ ). It amounts to summing over the even permutations of μ, ν, ρ with a plus sign and over the odd ones with a minus sign, the result being normalized by dividing it by the total number 6 of permutations (see Chap.5 ).
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116 4 The Poincaré Group As the reader can easily verify from the definition of trace, ˜ F i j S is indeed a symmetric traceless tensor, namely: ˜ F i j S δ i j = 0. We can now decompose F i j as follows F i j = ˜ F i j S + D i j + F i j A , (4.106) where D i j = 1 n δ i j F k k , (4.107) is the trace part, while, as usual F i j A = 1 2 ( F i j F i j ), (4.108) is the anti-symmetric component. Let us show that the components ˜ F i j S and D i j of all the type (2,0) tensors span two invariant vector spaces with respect to O ( n ) . We need first to show that the O ( n ) -transformed of ˜ F i j S is still symmetric traceless: ˜ F i j S = R i k R j ˜ F k S =⇒ δ i j ˜ F i j S = δ i j R i k R j ˜ F k S = δ k ˜ F k S = 0 . (4.109) Finally the trace part D i j = 1 n δ i j δ k F k is also invariant being δ i j O ( n ) -invariant. 4.5.1 Lie Algebra of the SO(3) Group LetusconsidersomepropertiesoftherotationgroupSO(3).Thisgrouphas dimension 3, which means that the most general rotation in the three dimensional Euclidean space is parametrized by three angles, such as for instance the Euler angles defining the relative position of two Cartesian systems of orthogonal axes: R = R ( θ ) R 1 , θ 2 , θ 3 ) θ i ). (4.110) The Euler angles are often denoted by (θ, φ, ψ) and correspond to describing a generic rotation as a sequence of three elementary ones: a first rotation about the Z axis by an angle θ , followed by a rotation about the new Y axis by an angle φ , and a final rotation about the new Z axis by an angle ψ . The entries of the rotation matrix R ( θ ) are continuous functions of the three angles. In general the dependence of the group elements on their parameters θ i is contin- uous and the parameters are chosen so that R i 0 ) = 1 . (4.111) We also know that the product of two rotations is still a rotation and one can verify that the parameters defining the resulting rotation is an analytical function of those defining the first two:
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4.5 Groups of Transformations
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