I Metric tensor : raising/lowering indices

[tb87]

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

 

[zwierz]

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

 

[QuarkHead]

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