5
$\begingroup$

I'm on Polchinski's p39.

  1. Can someone please tell me the steps in the equivalence below? $$\exp\left[\frac{\alpha'}4\int d^2z_4 d^2z_5\ln|z_5-z_4|^2\frac{\delta}{\delta X^\mu(z_4,\bar z_4)}\frac{\delta}{\delta X_\mu(z_5,\bar z_5)}\right] $$ $$\times X^{\mu_1}(z_1,\bar z_1)X^{\mu_2}(z_2,\bar z_2)X^{\mu_3}(z_3,\bar z_3)$$ $$=X^{\mu_1}(z_1,\bar z_1)X^{\mu_2}(z_2,\bar z_2)X^{\mu_3}(z_3,\bar z_3)+\frac{\alpha'}{2}\eta^{\mu_1\mu_2}\ln|z_2-z_1|^2X^{\mu_3}(z_3,\bar z_3)+(2\text{ permutations})~?$$

  2. Why do the variations in the exponential act on $X^{\mu_1}(z_1,\bar z_1)X^{\mu_2}(z_2,\bar z_2)X^{\mu_3}(z_3,\bar z_3)$?

  3. Exactly how does the integration in the exponential go and how does the exponentiation give RHS?

  4. The permutations look reasonable but the first term on RHS comes from when $\exp[...]=1$?

$\endgroup$
  • 1
    $\begingroup$ Hint : in your case, the exponential is practically reduced to the two first terms $e^x=1+x$ $\endgroup$ – Trimok Sep 1 '14 at 8:16
  • $\begingroup$ Thanks - this is exactly what I needed. Very stupid of me! $\endgroup$ – LorentzNoether Sep 1 '14 at 14:29
3
$\begingroup$

First of all, note that the radial operator ordering ${\cal R}$ is implicitly implied in many textbooks of CFT (e.g. Ref. 1). For instance, eq. (2.2.7) on p. 39 in Ref. 1 is discussing Wick's theorem between two operator ordering prescriptions. In this case between normal ordering $:~:$ and radial ordering ${\cal R}$. See also e.g. this Phys.SE post. The basic 2-point relation of Wick's theorem is

$$ \tag{1} : \hat{X}_i \hat{X}_j : ~=~ {\cal R}(\hat{X}_i \hat{X}_j)+ C_{ij}, $$

where the so-called contraction $C_{ij}$ is assumed to be a $c$-number. [More precisely: $C_{ij}$ is assumed to be a central element.] Here the indices $i,j,k,\ldots$ are a shorthand for all possible discrete and continuous labels of the operators $\hat{X}_i ,\hat{X}_j,\hat{X}_k, \ldots $, cf. DeWitt condensed notation.

OP's sought-for 3-point relation of Wick's theorem is$^1$

$$ \tag{2} : \hat{X}_i \hat{X}_j\hat{X}_k : ~=~ {\cal R}(\hat{X}_i \hat{X}_j\hat{X}_k) +C_{ij} {\cal R}(\hat{X}_k) + C_{ik}{\cal R}(\hat{X}_j) + {\cal R}(\hat{X}_i)C_{jk}. $$

Eqs. (1) and (2) can be formally generalized to eq. (2.2.7) of Ref. 1

$$\tag{3} :{\cal F}: ~=~ \exp \left(\frac{1}{2} \sum_{i,j} C_{ij}\frac{\partial}{\partial \hat{X}_i} \frac{\partial}{\partial \hat{X}_j} \right) {\cal R}({\cal F}), $$

where the operator ${\cal F}$ is a function of the operators $\hat{X}_i$. Note that the operators are treated as commutative objects under the two ordering symbols $:~:$ and ${\cal R}$. Eq. (3) is a convenient formal shorthand/mnemonic of the various $n$-point relations of Wick's theorem.

References:

  1. J. Polchinski, String Theory, Vol. 1; p.39.

--

$^1$ In this answer we have assumed for simplicity that all operators $\hat{X}_i$ are Grassmann-even. If some of the operators $\hat{X}_i$ are Grassmann-odd, there will be additional sign factors in eqs. (2) and (3).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.