An elementary equation manipulation in CFT

[Lapidus]

A presumably basic introductory equation manipulation in 2-d conformal field theory. How does from

(when the metric is Euclidean) follow

The right equation is clear (the metric is zero for different indices). But how do i get to the first equation on the left?

thank you

 

[fresh_42]

set μ = ν = 0 and calculate, then μ = ν = 1 and compare the results

 

[Lapidus]

set μ = ν = 0 and calculate, then μ = ν = 1 and compare the results

Thanks for the reply! But I don't understand what you mean. If I set zero or one, then RHS and LHS do not add up, in either case.

∂₀ε₀ + ∂₀ε₀ = ∂₀ε₀

??

 

[fresh_42]

I do not know for sure how $η$ is defined. Since you did not explain it and I'm too lazy to search what you might have meant, I supposed $η$ to be the Kronecker symbol according to your calculation above.
If I then first compute $δ_0 ε_0 = ...$ and next $δ_1 ε_1 = ...$ then I get the same result, i.e. they are equal.

 

[fzero]

Thanks for the reply! But I don't understand what you mean. If I set zero or one, then RHS and LHS do not add up, in either case.

∂₀ε₀ + ∂₀ε₀ = ∂₀ε₀

??

Since presumably $\partial\cdot \epsilon = \partial_0 \epsilon_0+\partial_1 \epsilon_1$ in your conventions, you should have found $2\partial_0 \epsilon_0 = \partial_0 \epsilon_0+\partial_1 \epsilon_1$ for the 00 component.

 

[Lapidus]

Ahaa. I did not understand that ∂⋅ε is a sum. (or an inner product)

Thanks everybody!