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Adjoint of commutator

  1. Sep 16, 2009 #1
    Hi all.

    I found the following identity in a textbook on second quantization:

    [tex]([a_1^{\dagger},a_2^{\dagger}]_{\mp})^{\dagger}=[a_1,a_2]_{\mp}[/tex]

    but why?

    [tex]([a_1^{\dagger},a_2^{\dagger}]_{\mp})^{\dagger}=(a_1^{\dagger}a_2^{\dagger}\mp a_2^{\dagger}a_1^{\dagger})^{\dagger}=a_2a_1\mp a_1a_2[/tex]

    and in the case of the commutator (and not the anticommutator) this isn't the result mentioned in the book.

    i would be glad if someone can explain. thanks.
     
  2. jcsd
  3. Sep 16, 2009 #2
    This looks fine to me
     
  4. Sep 17, 2009 #3

    haushofer

    User Avatar
    Science Advisor


    Well, my best guess is that

    [tex]
    [A,B]^{\dagger} = (AB - BA)^{\dagger} = B^{\dagger}A^{\dagger}-A^{\dagger}B^{\dagger} = [B^{\dagger},A^{\dagger}] = -[A^{\dagger},B^{\dagger}]
    [/tex]

    So

    [tex]
    [A^{\dagger},B^{\dagger}]^{\dagger} = - [A,B]
    [/tex]

    what you also wrote down. Which textbook are you refering to?
     
  5. Sep 17, 2009 #4
    Thanks for your replies.

    I found the problem. The identity only holds for the special case of creation-/annihilation-operators, where the (anti-)commutator for fermions or bosons resplectively is zero.

    thanks and greetings.
     
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