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6

As Lubos Motl mentions in a comment, for all practical purposes, the sought-for equation (v1) is proved via Wick's Theorem. It is interesting to try to generalize Wick's Theorem and to try to minimize the number of assumptions that go into it. Here we will outline one possible approach. I) Assume that a family $(\hat{A}_i)_{i\in I}$ of operators ...

3

We can use $$\begin{split} : F : : G: = \exp \left( - \frac{\alpha'}{2} \int d^2 z_1 d^2 z_2 \log|z_{12}|^2\frac{\delta }{\delta X_F^\mu(z_1, {\bar z}_1)} \frac{\delta }{\delta X_{G\mu}(z_2, {\bar z}_2)} \right) :F G: \end{split}$$ This gives \begin{split} : \frac{i}{\alpha'} \partial X^\mu(z) : : e^{i k \cdot ...

3

I think there is a typo in the first formula. Let me propose this (partial) answer for the $3$ first formulae: Because $H(z)H(0) \sim -ln(z)$, we may write the OPE for any pair of operators $F(H), G(H)$ functions of $H$ (in analogy with formula $2.2.10$ p.$39$ vol $1$) $$:F::G: = e^{- \large \int dz_1 dz_2 ln z_{12} \frac{\partial}{\partial ... 3 You don't need to use that. You can simply do the cross-contractions by hand. Let's do that. Note that I only care about the the \frac{1}{z^4} term to evaluate the central charge. We have \begin{split} T(z) T(w) & =\left( : \partial_z b c(z): - \lambda \partial_z : b c (z): \right) \left( : \partial_w b c(w): - \lambda \partial_w : ... 3 Here we will outline a strategy to prove the sought-for operator identity (4) from the following definitions of what the commutator and the normal order of two mode operators \alpha_m and \alpha_n mean:$$ [\alpha_m, \alpha_n]~=~ \hbar m~\delta_{m+n}^0, \qquad\qquad(1) :\alpha_m \alpha_n:~=~\Theta(n-m) \alpha_m \alpha_n ~+~ \Theta(m-n) \alpha_n ...

2

The first term in your second equation does not contain any singularities and is hence part of the "non-singular terms" at the end of the last expression. To find the final form you just need to perform the derivatives of the logarithm terms and Taylor expand the term $:\!\partial X^\mu(z)\partial'X_\mu(z')\!:$ around $z=z'$. The singular contributions from ...

2

In the operator$^1$ formulation, the vital assumption, which makes the standard Wick's theorem hold, is the assumption that the contractions are in the center of the pertinent operator algebra. This is often stated casually as the contractions should be $c$-numbers, meaning that the contractions should (super)commute with all pertinent operators. $^1$ A ...

2

In the particular case where $f(\vec x)$ is a sum of expressions, where each expression has a total even power of the $x^i$ like : $f(x) = x_1x_2 + x_1^2 x_2 x_3 + ...$, . we may present general expressions. we have : $$I(\Sigma)=\int d^{n} x\, e^{-\frac{1}{2} \vec x \cdot \Sigma^{-1}\cdot \vec x}\tag{1} = \sqrt{(2 \pi)^ndet \Sigma}$$ We have : ...

1

OP wrote (v1): In the beginning of proof of this theorem one says that permutation of fields in the left and right side of (1) doesn't change it. Answer: This is essentially due to the fact that operator ordering prescriptions (such as e.g. time ordering $T$ or normal ordering $::$) are (graded) symmetric $$... 1 An other way is to begin with (2.2.1.4)$$:e^{i k_1.X(z,\bar z)}:~ :e^{i k_2.X(0,0)}: ~\sim ~|z|^{\alpha' k_1.k_2} ~:e^{i (k_1 + k_2).X(0,0)}:$$Now, derive this expression relatively to k_1^\mu, then doing k_1=0, we get :$$:i X_\mu(z, \bar z):~:e^{i k_2.X(0,0)}: ~\sim(\alpha' (k_2)_{\mu} \ln|z| + i :X_\mu(0,0):):e^{i k_2.X(0,0)}:$$Now, we derive ... 1 I was able to confirm both of your formulas, assuming that the operators are fermionic and normal ordered. Here is the general algorithm for generating a cumulant average of n variables (taken from page page 34 of G.W.Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, attributed to van Kampen):$$G_c(X_1,X_2,\ldots, ...

1

You can always use Wick's theorem when you're describing expectation values with respect to free field states (that is, non-interacting, like a single Slater determinant state). For example, in Hubbard-Stratonovich or (generally) Variational Mean-Field Theory (like Bogliubov-deGennes) you take the interaction terms and decouple them into a model that is ...

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