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String Theory-Virasoro Generators -- show commutator relation

  1. Apr 17, 2017 #1
    1. The problem statement, all variables and given/known data

    (I have dropped the hats on the ##\alpha_{n}^{u}## operators and ##L_{m}##)

    ##[\alpha_{n}^u, \alpha_m^v]=n\delta_{n+m}\eta^{uv}##
    ##L_m=\frac{1}{2}\sum\limits_{n=-\infty}^{\infty} : \alpha_{m-n}^u\alpha_{n}^v: \eta_{uv}-\delta_{m,0}##

    where : denotes normal-ordered.

    Show that : ##[\alpha_{m}^u,L_n]=m\alpha_{m+n}^u##

    2. Relevant equations

    see above
    3. The attempt at a solution

    For a given ##n## we are looking at the following commutator: ##[\alpha_m,\alpha_{n-m}\alpha_m]##

    to use commutator relation:

    ##[a,bc]=-a[b,c]-[a,c]b##

    ##a= \alpha_m##
    ##b= \alpha_{n-m}##
    ##c= \alpha_m ##

    ##[a,c]=0##
    ##[b,c]=(n-m)\delta_{n=0}\eta^{uv}## using (1)

    ##\implies [\alpha_{m}^u,L_n]=\alpha_m^u(m)\eta^{uv} ## which is wrong...

    thanks in advance
     
  2. jcsd
  3. Apr 19, 2017 #2
    bump please. many thanks in advance, very grateful.
     
  4. Apr 20, 2017 #3

    king vitamin

    User Avatar
    Gold Member

    The commutator relation you've written down is wrong.
     
  5. Oct 8, 2017 #4
    Sorry to re-bump an old thread but I thought it would be a waste to start a new one.
    I think I understood this at the time but right now I am not getting it.
    Why is the commutator relation wrong?

    Let me instead use:

    ##[a,bc]=[a,b]c+b[a,c]##

    ##a=\alpha_m##
    ##b=\alpha_{n-m}##
    ##c=\alpha_m##

    ##[\alpha_m, \alpha_{n-m}]\alpha_m + \alpha_{m-n}[\alpha_m,\alpha_m] = m\delta_n \eta^{uv}\alpha_m + m\delta_{2m}\eta^{uv}##

    which is wrong..
     
  6. Oct 8, 2017 #5

    king vitamin

    User Avatar
    Gold Member

    You shouldn't use the same index [itex]m[/itex] for both the [itex]\alpha_m[/itex] and the summed index in the definition of [itex]L_n[/itex].

    So using the (now correct) relation

    [tex] [a,bc] = [a,b]c + b[a,c][/tex]

    with

    [tex]
    a = \alpha_m^u \\

    b = \alpha_{n-k}^{\nu} \\

    c = \alpha_k^{\alpha}
    [/tex]

    you should get the correct answer after contracting with [itex]\eta_{\nu \alpha}[/itex].
     
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