Friday, 17 November 2017

homework and exercises - Computing the OPE of T:mathrmeikX:



I've hit a stumbling block where I'm just not seeing how to get from line to line in the following calculation from David Tong's strings notes. Can someone spell out how line 1 becomes line 2 in the X:eikX: calculation and how that result is actually being employed in the calculation for T:eikX:?


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Answer



Contraction with exponential is actually more easy than polynomials. This is so because the exponential is the eigenfunctional of δδXμ. The cross contractions of (...)A and (...)B are given by


(...)A(...)B=exp(α2ημνd2z1d2z2ln|z12|2δδXμA(z1,ˉz1)δδXνB(z2,ˉz2))(...)AB


and we have the eigenfunctional equation



δδXμ(z1,ˉz1)exp(ik.X(z,ˉz))=ikμδ2(zz1;ˉzˉz1)exp(ik.X(z,ˉz))


then if (...)B=exp(ik.X(z2,ˉz2)) we have


(...)A(...)B=exp(iαkμ2d2z1ln|z2z1|2δδXμA(z1,ˉz1))(...)AB


just contractions on (...)A with:


iαkμ2ln|z2z1|2


for each Xμ in (...)A. So, if (...)A=XμXμ, we have one 0-contraction (0) plus two identical 1-contraction (1) plus one 2-contractions (2), and only this:


(0) :XμXμ(z1)exp(ik.X(z2,ˉz2)):


(1) 2:1(iαkμ2ln|z2z1|2)Xμ(z1)exp(ik.X(z2,ˉz2)):


(2) :1(iαkμ2ln|z2z1|2)1(iαkμ2ln|z2z1|2)exp(ik.X(z2,ˉz2)):


Adding all together you get the answer.



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