From Special Relativity to Feynman Diagrams.pdf

This implies that the scalar function will in general

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This implies that the scalar function will in general have a different dependence on the chosen coordinates, namely that, under a change of coordinates x i x i x i ( x ) it will be described by a new function f ( x ) related to f ( x ) as follows: f ( x ) = f ( x ). (4.70) If the functional dependence of f on the new and old coordinates does not change, that is if: f ( x ) = f ( x ), (4.71) the scalar function f is said to be invariant . 6 A tensor field is a tensor quantity which depends on the coordinates of a point P in space. A change in coordinates, besides transforming the tensor components, will also transform the coordinate dependence of the tensor, as we have shown for the vector and scalar fields. Take for instance a (2,1) tensor field described by a set of functions T i j k ( x ) in a given coordinate system. Under a coordinate transformation we have: T i j k ( x ) = D i D j m D 1 s k T m s ( x ). (4.72) Using the explicit form ( 4.36 ) of a Cartesian coordinate transformation, we find: T i j k ( r ) = D i D j m D 1 s k T m s ( D 1 r + D 1 r 0 ), (4.73) where, in the argument on the right-hand side, we have expressed the old coordinate vector r in terms of the new one r by inverting ( 4.37 ). The notion of invariance, which was given for scalar fields, can be extended to more general tensor fields. Let us still take, for the sake of simplicity, the type (2,1) tensor field T i j k ( r ) . We will say that T k i j ( x ) is invariant, if it transforms, under a coordinate transformation, as follows: 6 Of course ( 4.70 ) can be also written f ( x ) = f ( x ) , since x is a variable.
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4.3 Tensor Algebra 107 T i j k ( x ) = D i D j m D 1 s k T m s ( x ) T i j k ( x ). (4.74) The above invariance condition has an obvious generalization to tensors of type ( p, q ). An example of invariant tensor is the Kronecker symbol, as it was shown in the previous section. Let us define a (2,0)-tensor F i j symmetric if F i j = F ji and antisymmetric if F i j = − F ji . If we now consider a generic type (2,0) tensor F i j , it can be decom- posed into a symmetric and an anti-symmetric part, with respect to the exchange of the two indices, by writing the following trivial identity: F ik = 1 2 ( F ik + F ki ) + 1 2 ( F ik F ki ) . = F S ik + F A ik , (4.75) where F S ik = F S ki and F A ik = − F A ki define the symmetric and anti-symmetric parts of F i j . This decomposition does not depend on the coordinate basis we use, since under a coordinate transformation a symmetric tensor F S ik is mapped into a symmetric tensor and similarly for the anti-symmetric ones: F S i j = D i D j m F m S = D i D j m F m S = D j m D i F m S = F S ji , F A i j = D i D j m F m A = − D i D j m F m A = − D j m D i F m A = − F A ji . (4.76) We conclude that the vector space of type (2,0)-tensors can be decomposed into the direct sum of two disjoint subspaces spanned by symmetric and antisymmetric tensors. The same decomposition can be performed on the space of (0,2)-tensors, by writing a generic covariant rank 2 tensor F i j into the sum of its symmetric and anti-symmetric components: F i j = F Si j + F Ai j . It is straightforward to prove that the contraction over all indices of a type ( 2,0 ) and a type ( 0,2 ) tensors with opposite
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