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Questions on GSW 2.2.3

  1. Mar 31, 2004 #1
    Hello,

    I stumbled about two things in GSW section 2.2.3 on Vertexoperators,
    that I don't really understand.

    The first one is GSW's statement just before 2.2.54, p. 88, that the
    L_m's of the Virasoro algebra generate transformations like

    tau -> tau -ie^{im tau}

    .... From what was said on p. 65 of the generators of the residual
    symmetry and on p. 72 from the Virasoro generators I know that these
    should be conserved charges, thus generating some transformations with
    f(sigma^+) = e^imsigma^+ and f(sigma^-) = e^imsigma^-, which is at
    sigma = 0 just e^imtau. But how to get from this result to the above
    transformation law?

    The second one is the statement on p. 92 that for the two conditions
    k_mu zeta^munu = 0 and tr zeta = 0 the tensor zeta should be a
    symmetric traceless tensor. Tracelessness is clear, but how to show
    that under this condition the tensor should be symmetric?

    I hope that these questions are not too elementary, but as I am new
    with the string stuff, many elementary things bother me most,
    sometimes.

    René.

    --
    René Meyer
    Student of Physics & Mathematics
    Zhejiang University, Hangzhou, China
     
  2. jcsd
  3. Mar 31, 2004 #2
    On Wed, 31 Mar 2004, Rene Meyer wrote:

    > The first one is GSW's statement just before 2.2.54, p. 88, that the
    > L_m's of the Virasoro algebra generate transformations like
    >
    > tau -> tau -ie^{im tau}


    To see this just go the other way round. Check that a field of conformal
    weight J has the commutator given in (2.2.54).

    > The second one is the statement on p. 92 that for the two conditions
    > k_mu zeta^munu = 0 and tr zeta = 0 the tensor zeta should be a
    > symmetric traceless tensor. Tracelessness is clear, but how to show
    > that under this condition the tensor should be symmetric?


    Symmetry follows implicitly from the form of formula (2.2.66), where
    antisymmetric components of zeta do not contribute. k_m zeta^mn = 0
    is responsible for the statement "... to be the polarization tensor..."
    because this removes the unphysical timelike polarization.
     
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