In my QFT lecture the following was derived, but I have no idea how:
We consider a scalar field : $\phi(x^\mu)$.
$\phi(x^\mu)=\phi'(x'^\mu)$
Then:
$\phi'(x'^\mu)=\phi((\Lambda^{-1})^\mu_{\ \ \nu}x'^\nu)$
Then, he proceeds to say that we use an infinitesimal LT:
$\phi'(x'^\mu)=\phi(x'^\mu + \frac{i}{2}(L_{\alpha\beta})^\mu_{\ \ \nu}\omega_{\alpha\beta}x'^\nu)$
Then we say we perform LT and taylor expansion of $\omega_{\alpha\beta}$ (how is that possible when this is just an entry in the matrix):
$\phi'(x'^\mu)=\phi(x'^\mu)\hat L^{\alpha\beta}\omega_{\alpha\beta}\phi(x'^\mu)$.
where $\hat L^{\alpha\beta}=-(L^{\alpha\beta})^\mu_{\ \ \nu}x'^\nu{}\partial_{\mu}$
And he says that the following should be considered in the last step:
$\Lambda^\mu_{\ \ \rho}(\Lambda^{-1})^\rho_{\ \ \nu}=g^{\mu}_{\ \ \nu} + \omega^\mu_{\ \ \nu}-\omega^\mu_{\ \ \nu}$.
And clearly he is implies that: $\Lambda^\mu_{ \ \ \nu}=g^\mu_{ \ \ \nu} + \omega^\mu_{ \ \ \nu}$ which is known. But he says since we are considering infinitesimal transformations we can write: $(\Lambda^{-1})^\mu_{ \ \ \nu}=g^\mu_{ \ \ \nu} - \omega^\mu_{ \ \ \nu}$. I don't understand this and I am not sure the correct notes that I took are correct.
Anyway, I have no idea how he is able to derive the last expression.
Can someone show me how?
