- #1
wafelosek
- 2
- 0
Hi everyone! 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^{μν}##, μ<ν, as the parameters on the Lorentz group in a vicinity of the unit matrix, then we can compute the corresponding Killing vectors as ##\xi ^μ_{αβ}=\frac{∂x′^μ}{∂ϵ^{αβ}}(\hat{ϵ}=0)## where ##x′^μ=L^μ_νx^ν##. Here is my problem: during the computations there is one line that I do not get, namely: ##\cfrac{∂L^{μν}}{∂ϵ^{αβ}}(\hat{ϵ}=0)=δ^μ_αδ^ν_β−δ^μ_βδ^ν_α##. The ## (\epsilon^{μν}) ## matrix is antisymmetric, so that is why we get the difference of the Kronecer delta?
Thank you in advance!
MW
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^{μν}##, μ<ν, as the parameters on the Lorentz group in a vicinity of the unit matrix, then we can compute the corresponding Killing vectors as ##\xi ^μ_{αβ}=\frac{∂x′^μ}{∂ϵ^{αβ}}(\hat{ϵ}=0)## where ##x′^μ=L^μ_νx^ν##. Here is my problem: during the computations there is one line that I do not get, namely: ##\cfrac{∂L^{μν}}{∂ϵ^{αβ}}(\hat{ϵ}=0)=δ^μ_αδ^ν_β−δ^μ_βδ^ν_α##. The ## (\epsilon^{μν}) ## matrix is antisymmetric, so that is why we get the difference of the Kronecer delta?
Thank you in advance!
MW