Four Vectors in SR and QFT

Question

I'm covering both special relativity and quantum field theory in the summer. I'm currently using Spacetime Physics by Taylor and Wheeler to cover SR. Since I'm covering SR on the side with QFT, I'm having some conceptual troubles dealing with the following issues in SR:

1. I get how do four vectors, $x^{\mu}$, transform? I don't get how do four derivatives transform?
2. What's the difference between covariant and contrvariant derivatives and how to know when to use $\partial_{\mu}$ and/or $\partial^{\mu}$ while identifying an expression?
3. I'm having some trouble with dimensional analysis as well. For example, how does one identify that the first component of the four derivative is $\frac{1}{c}\frac{\partial}{\partial t}$, up to a plus or minus sign depending on the exact form of the Minkowski Metric one is using? Most books seem to use $c = 1$ so I don't get how such expressions hold?

Clearly, I'm having these troubles because I haven't reached the point in SR where one studies four vectors, especially the derivative four vectors. I don't think Taylor and Wheeler's book has this material so I'd probably want to go to another textbook to do four vectors in detail after I'm done with this book.

To that end, can anyone recommend some book where I can learn about these drills before I formally approach these topics in my study of SR. Reviewing these concepts right now will avoid the hurdles I get into while working out the details of QFT.

Answer 1

1. See this and links therein.
2. In general, $A_\mu B^\mu$ and $A^\mu B_\mu$ are the same thing, so you can use either $\partial_\mu$ or $\partial^\mu$, as long as the thing you're contracting it with has the index on the opposite position. That is, $\partial_\mu j^\mu$ is fine, $\partial^\mu j_\mu$ is fine though weird, and $\partial^\mu j^\mu$ is nonsense. By the time you get to QFT, this index matching is so second nature that many books write $\partial^\mu j^\mu$ just out of laziness.
3. Like normal vectors, the dimensions of all components of a 4-vector must match. $\partial / \partial x$ has dimensions of inverse length, but $\partial / \partial t$ has dimensions of inverse time, so we need to add a $c$.
4. I recommend Morin's chapter on it (it's a free sample in the link). It covers everything very carefully and completely, and points out all possible conceptual pitfalls.

Answer 2

1. A four vector transforms according to the way the position vector (mathematically a point in Minkowski space ($M_4$) and physically the position where the object of interest is). The transformation uses a Lorentz transformation $\Lambda$ i.e. $(\Lambda v,\Lambda w)=(v,w)$ for $v,w \in M_4$ : $(x^\prime)^\nu = \Lambda^\nu_\mu x^\mu$ so for some $v\in M_4$ we get $(v^\prime)^\nu = \Lambda^\nu_\mu v^\mu$.

2. $\partial_\mu$ is just a basis vector of the tangent space at some point. One can get the other form using some contraction with the metric tensor, i.e. $\partial^\mu = g^{\mu\nu}\partial_\nu$. Depending on the 'object' you are looking at you use the first or the latter form, there is no general rule for this.

3. c=1 just means that you express for example 1 second in meters and vice versa. In theoretical concepts try do not really care about units and just work through it and recover afterwards what the units were. And about the metric tensor it is just convention. It has to be explicitly stated. Otherwise if you know the equations for one convention you can simply see whether there is a extra -1 for example.