Calculating different "kinds" of variations

[Markus Kahn]

Homework Statement

Let $x$ and $x'$ be two points from the Minkowski space connected through a Poincare transformation such that $x'^\mu =\Lambda_{\nu}^\mu x^\nu+a^\mu$ and $u:\mathcal{M}\to \mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, $\mathcal{M}$ the Minkowski space. We define:
$$
\begin{align*} \Delta u(x) &~:=~ u^{\prime}(x^{\prime})-u(x) \qquad\text{ total infinitesimal variation}\cr \delta u(x) &~:=~ u^{\prime}(x)-u(x) \qquad \text{ local/vertical infinitesimal variation},\cr
\mathrm{d}u(x)&~:=~ u(x^{\prime})- u(x) \qquad\text{ differential/horizontal infinitesimal variation},
\end{align*}
$$
where and $u'$ is the function after the transformation. We further know that we can write
$$\Lambda_\nu^\mu = \delta_\nu^\mu - \frac{i}{2} \delta \omega^{\rho\sigma}(S_{\rho\sigma})^\mu_\nu,$$
with $\omega^{\mu\nu}+\omega^{\nu\mu}=0$, $S_{ij}=S_{ij}^\dagger$ and $S_{0k}-S_{0k}^\dagger$ (Spin antisymmetric tensor).

Now, show that:

1. $\delta x^\mu = \delta \omega^{\mu\nu}g_{\nu\rho}x^\rho +\delta \omega^\mu = \frac{1}{2}\delta\omega^{\rho\sigma}L_{\rho\sigma}x^\mu - i\delta\omega^{\rho}P_\rho x^\mu,$ where $P_\mu = i\partial_\mu$ and $L_{\mu\nu}\equiv x_\mu P_\nu - x_\nu P_\mu.$
   
2. For any relativistic covariant vector wave field $V'_\mu(x')=\Lambda_\nu^\mu V_\nu(x)$, we have $\Delta V_\mu (x)= -\frac{1}{2}i\delta \omega^{\rho\sigma}(S_{\rho\sigma})_\mu^\nu V_\nu (x).$

Homework Equations

I'm not sure if I really mentioned everything needed to complete the two exercises (since I honestly don't understand how to approach the problem in the first place), so everything that I summerized here can be found in the lec notes of R. Soldati on relativistic quantum field theory (pages 57-63, where I try to explicitly prove the result between eq. 2.16 and 2.17 as well as eq. 2.22), see here: http://robertosoldati.com/archivio/news/107/Campi1.pdf

The Attempt at a Solution

What I've tried:

1. $$\begin{align*}\delta x^\mu &= x'^\mu-x^\mu = \Lambda_\nu^\mu x^\nu + a^\mu -x^\mu = \left(\delta_\nu^\mu - \frac{i}{2} \delta \omega^{\rho\sigma}(S_{\rho\sigma})^\mu_\nu\right)x^\nu -x^\mu + a^\mu\\ &= \frac{i}{2} \delta \omega^{\rho\sigma}(S_{\rho\sigma})^\mu_\nu x^\nu + a^\mu\ \neq \delta \omega^{\mu\nu}g_{\nu\rho}x^\rho +\delta \omega^\mu. \end{align*}$$ Not sure how I'm supposed to get to the desired result... The last equality is a complete mystery to me. Here I can't even find a way to start calculating what I'm supposed to..
2. Similarly to above I just plugged in the definition: $$\begin{align*}\Delta V_\mu (x)&= V'_\mu (x') -V_\mu (x)=\Lambda_\mu^\nu V_\nu (x) - V_\mu(x)\\ &= - \frac{i}{2} \delta \omega^{\rho\sigma}(S_{\rho\sigma})^\nu_\mu V_\nu(x), \end{align*}$$ which leads to the desired result. I would still appreciate if someone could give a quick glance over it since I'm rather unsure about the indices...

 

[NateD]

So, can someone please explain me how to approach these kind of problems and how to calculate the desired result? Thanks in advance for any help!