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

Info icon This preview shows pages 139–141. Sign up to view the full content.

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).
Image of page 139

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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.
Image of page 140
Image of page 141
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern