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My question originates from what is done in the book on Quantum Field Theory book by Mark Srednicki on page 21 (if anyone has it).

So say you have an inertial frame that is represented in the coordinates $x^{\mu}$. Any other coordinates $\bar{x}^{\mu}$ will also be represented by an inertial frame if they are related in the following way.

\begin{equation} \bar{x}^{\mu}=\Lambda^{\mu}{}_{\nu}x^{\nu}+a^{\mu} \end{equation}

Where $\Lambda^{\mu}{}_{\nu}$ is a Lorentz transformation matrix and $a^{\mu}$ is a translation vector.

This is all well and good and I understand it, it is just a transformation on the coordinates. What I don't fully quite grasp is the notation of the matrix $\Lambda^{\mu}{}_{\nu}$. I know that something like $x^{\mu}$ represents a contravariant vector and something like $x_{\mu}$ represents a covariant vector, and both of these are rank 1 tensors. When it comes to matrices though I get a bit confused.

I know the basic rule of raising and lowering indices, but I guess I don't know it in a very robust manner. For example, I know that $\Lambda^{\mu}{}_{\nu}x^{\nu}$ MUST yield a rank 1 tensor $y^{\mu}$ since the $\nu$ indices "cancel." But I don't know the difference between the tensors $\Lambda^{\mu}{}_{\nu}$, $\Lambda_{\nu\mu}$ and $\Lambda^{\nu\mu}$. (I am aware that for this example, only the first one makes sense but I don't understand the conceptual differences between these three terms.)

I hope all this makes sense!

Qmechanic
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user41178
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    "I know the basic rule of raising and lowering indices" If you do, why don't you know the difference between $\Lambda_{\mu\nu}$,${\Lambda_\mu}^\nu$ and $\Lambda^{\mu\nu}$? – ACuriousMind Mar 01 '16 at 20:19
  • may I suggest this answer of mine, where I explain the basics of Lorentz transformations and raising/lowering indices? – AccidentalFourierTransform Mar 01 '16 at 20:21
  • ACuriousMind, What I meant by that is, I know of only the rule, not the origin of the rule. My knowledge is limited in this subject.

    @AccidentalFourierTransform I will look at that right now! Thanks!

     – user41178 Mar 01 '16 at 20:25
  • Related: http://physics.stackexchange.com/q/158309/2451 , http://physics.stackexchange.com/q/169762/2451 , http://physics.stackexchange.com/q/237270/2451 and links therein. – Qmechanic Mar 01 '16 at 21:01
  • I read through your post @AccidentalFourierTransform and I think I understand much better. So say we have a matrix $A$, and we multiply it by a vector $x^{\mu}$, if this matrix transforms the vector from contravariant to covariant, then we denote the matrix $A$ as $A_{\mu\nu}$, but if the vector simply transforms the vector to another frame, but keeps it contravariant, we denote it $A^{\nu}_{\mu}$, so the result would be $x^{\nu}$. Is this correct? It is simply a way of denoting what kind of tensor $A$ is? – user41178 Mar 01 '16 at 21:02
  • @user41178 yes, you can put it that way. If you really want to understand what's going on, you will have to read a book on differential geometry sooner on later. For now, your interpretation is fine. Strictly speaking, index placement is more that just notation: it specifies the rank of the tensor. But all this can only be clear in the context of manifolds. If you have time, I'd suggest the first part of the book "The Geometry of Physics", by T. Frankel. IMHO, it is a must-read for physicists. But until then, what you said is correct. – AccidentalFourierTransform Mar 01 '16 at 21:14
  • Thanks so much! I will see if I can get my hands on that book and take a look sometime in the future! – user41178 Mar 01 '16 at 21:16
  • One really important point is that the $\left{\Lambda^\mu{}_\nu\right}$ are not the components of a tensor, because the two indices refer to different bases: instead they are just the components of whatever nonsingular matrix transforms from one basis to the other. –  Mar 01 '16 at 21:47

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