I have a problem with one thing.

Let's consider the Lorentz group and the vicinity of the unit matrix. For each $\hat{L}$ from such vicinity one can prove that there exists only one matrix $\hat{\epsilon}$ such that $\hat{L}=exp[\hat{\epsilon}]$. If we take $\epsilon^{\mu \nu},\ \mu < \nu$, as the parameters on the Lorentz group in a vicinity of the unit matrix, then we can compute the corresponding Killing vectors as $\xi_{\alpha \beta}^{\mu}=\frac{\partial x^{'\mu}}{\partial \epsilon^{\alpha \beta}}\left(\hat{\epsilon}=0 \right) $ where $x^{'\mu}=L^{\mu}_{\nu}x^{\nu}$. Here is my problem: during the computations there is one line that I do not get, namely: $\cfrac{\partial L^{\mu \nu}}{\partial \epsilon^{\alpha \beta}}\left(\hat{\epsilon}=0 \right)=\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}-\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha} $. The $(\epsilon^{\mu \nu})$ matrix is antisymmetric, so that is why we get the difference of the Kronecer delta?

Thank you in advance!



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.