Invariance of Dirac Lagrangian

[Gene Naden]

I am working through the first chapter of Lessons on Particle Physics by Luis Anchordoqui and Francis Halzen. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf

I am on page 22. Equation 1.5.61:
$L_{Dirac}=\psi \bar ( i\gamma^\mu \partial_\mu-m)\psi$
where $\psi bar = \psi^\dagger \gamma^0$

The authors state that this is invariant. I already proved the invariance of the mass term, but I don't see how to prove the invariance of the term involving $\partial_\mu$.

The authors seem to feel that the invariance of (1.5.61) follows directly from the transformation properties of $\psi \bar \gamma^\mu \psi$, which are:

$\psi \bar \prime \gamma^\mu \psi \prime = \Lambda^\mu_{\phantom \alpha} \psi \bar \gamma^\alpha \psi$

My question is how do I see the invariance of $L_{Dirac}$; how to see the invariance of the first term, which is proportional to $\psi bar \gamma^\mu \partial_\mu \psi$?

A related question: how to render $\psi bar$ in Tex. When I use \bar or \overline, the bar ends up too far to the right.

 

[Orodruin]

Are you aware of how partial derivatives transform under Lorentz transformations?

Also, the appropriate $\LaTeX$ code for $\bar\psi$ is \bar\psi.

 

[Gene Naden]

Thank you, when I get home I will look at the transformation of partial derivatives.

 

[Gene Naden]

So let's see, $\frac{\partial}{\partial x^\mu}$ is covariant and $\frac{\partial}{\partial x_\mu}$ is contravariant, right?

So $\frac{\partial}{\partial x\prime ^\mu}=\Lambda^\sigma_{\nu} \frac{\partial}{\partial x^\sigma}$

I don't know how to push the $\nu$ out to the second position...

 

[ChrisVer]

In general, any quantity that has its Lorentz-indices summed over is invariant under Lorentz transformations... (I suppose that's the invariance you are asking about, and not that of gauge symmetries)... That is all quantities written as minkowski products are invariant under Lorentz transfs, in the same way the quantities (in e.g. mechanics) that are written as vector dot procuts are invariant under Euclidean transfs.