From Special Relativity to Feynman Diagrams.pdf

Μ q ρ p 1 σ 1 ν 1 1 σ q ν q t ρ 1 ρ p σ 1 σ

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μ q ρ p 1 σ 1 ν 1 . . . 1 σ q ν q T ρ 1 ...ρ p σ 1 ...σ q . (4.162) Tensors of the same type ( p , q ) form a linear vector space and the collection of all possible tensors form an algebra with respect to the tensor product operation and contraction. Anticipating some concepts which will be introduced and discussed in Chap. 7, tensors of a given type ( p , q ) form a basis of a representation of the Lorentz group , on which the group action is defined by ( 4.162 ). Such property means that the effect on a type ( p , q ) tensor of two consecutive Lorentz transformations 1 , 2 , is the transformation induced by the product of the two 2 , 1 . This follows from the definition of the Kronecker product of matrices and in particular from the property ( A B ) · ( C D ) = ( AC ) ( BD ) , see Sect. 4.6. This representation is in general 13 If the invariant metric were diagonal with entries ( + 1 , + 1 , 1 , 1 ) , the corresponding group would have been SO(2,2).
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128 4 The Poincaré Group reducible , that is the vector space spanned by type ( p , q ) tensors may decompose into the direct sum of orthogonal subspaces each of which are stable under the action of the Lorentz group, and therefore define themselves bases of representations of the group. As an example let us consider a Lorentz tensor with two contravariant indices F μν , transforming according to ( 4.162 ). Similarly to what happened in the case of the rotation group, see ( 4.104 ) and ( 4.109 ), we can decompose this tensor into three components which transform into themselves under the action of SO(1, 3). Let us define the trace operation: F ρ ρ η μν F μν , (4.163) and decompose F μν as follows F μν = ˜ F μν S + D μν + F μν A . (4.164) The first term within brackets denotes the symmetric traceless component of F μν : ˜ F μν S = 1 2 ( F μν + F μν ) 1 4 η μν F ρ ρ , ˜ F μν S η μν = 0 . The second term within brackets in (4.164) represents the trace part: D μν = 1 4 η μν F ρ ρ , and, finally, F μν A = 1 2 ( F μν F μν ). is the anti-symmetric component. With the above definitions the proof that each of these components, under a Lorentz transformation, is mapped into the corresponding component of the transformed tensor, is the same as the one given for the rotation group. We conclude that antisymmetric, symmetric traceless and the trace each span three orthogonal subspaces of the total space of type (2, 0) tensors, which are stable under the action of the Lorentz group. Since they cannot be further reduced, we say that they define the bases of three irreducible representations of SO (1, 3). The same result applies to (0, 2)-tensors as well. The importance of having a physical law written in a tensorial form with respect to the Lorentz group relies in the following property: If a physical law is written as an equality between Lorentz tensors of a same type, in a given RF, it will hold in any other RF connected to the original one by a Lorentz transformation.
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