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An elementary equation manipulation in CFT

  1. Nov 13, 2015 #1
    A presumably basic introductory equation manipulation in 2-d conformal field theory. How does from
    upload_2015-11-13_21-52-51.png

    (when the metric is Euclidean) follow
    upload_2015-11-13_21-53-14.png

    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
     
  2. jcsd
  3. Nov 13, 2015 #2

    fresh_42

    Staff: Mentor

    set μ = ν = 0 and calculate, then μ = ν = 1 and compare the results
     
  4. Nov 14, 2015 #3
    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

    ??
     
  5. Nov 14, 2015 #4

    fresh_42

    Staff: Mentor

    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.
     
  6. Nov 14, 2015 #5

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
     
  7. Nov 15, 2015 #6
    Ahaa. I did not understand that ∂⋅ε is a sum. (or an inner product)

    Thanks everybody!
     
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