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I Metric tensor : raising/lowering indices

  1. Apr 20, 2017 #1
    Hi everyone,

    I'm currently studying Griffith's Intro to Elementary Particles and in chapter 7 about QED, there's one part of an operation on tensors I don't follow in applying Feynman's rules to electron-muon scattering :

    ## \gamma^\mu g_{\mu\nu} \gamma^\nu = \gamma^\mu \gamma_\mu##

    My teacher spent very little time on tensors and I'm really not sure 1) what's the difference between ##\gamma^{\mu\nu}## and ##\gamma_{\mu\nu}##, 2) why is the second ##\gamma##'s indice switched from ##\nu## to ##\mu## (and was also lowered).

    Besides, I'm still strugling to understand the general difference between lowered and raised indices (e.g. : why I'm never seeing ##\gamma_\mu^\nu##, but other tensors are written that way). In Griffith's electrodynamics, the author says that in ##\Lambda_\mu^\nu##, it is ##\mu## that represents lines and ##\nu## that represents columns. However, this isn't consistent with everything I'm seeing on tensors.

    Thanks!
    Alex
     
  2. jcsd
  3. Apr 20, 2017 #2
    it is very bad idea to study tensor calculus by physics textbooks. I am sure participants of PF soon prompt you good textbook in differential geometry or in tensor calculus
     
  4. Apr 27, 2017 #3
    Yes, this notation is not good. As a general guidance, consider the following.....

    Suppose ##V## an arbitrary finite-dimensional vector space. Then there will always exist a dual space ##V^*:V \to \mathbb{R}## such that for any ##\varphi \in V^*## and any ##v \in V## that ##\varphi(v) = \alpha \in \mathbb{R}##.

    And if there exists a bilinear form on ##V##, that is ##b:V \times V \to \mathbb{R},\,\,\,b(v,w) \in \mathbb{R}## ie. an inner product(or norm), then there will always be some particular ##\varphi_v \in V^*## such that ##\varphi_v(w)=b(v,w)## for any ##w \in V##

    Now if I fix some ##v \in V## and write, say ##b(v,\,\cdot) \in \mathbb{R}## then I may have the equality ##b(v,\,\cdot)= \varphi_v## That is, ##b(v,\,\cdot)## acts as though it is a dual vector.

    Just carry out the obvious notational substitutions, i.e ##b \mapsto g_{\mu\nu}##, and ##v \mapsto\gamma^{\mu},\,\varphi_v \mapsto \gamma_\nu## and you have your answer
     
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