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Normal Ordering in String Theory: Polchinsky vs. all others

Polchinsky defines normal ordering in string theory as: $$:X^\mu(z,\bar z)X^\nu(w,\bar w): = X^\mu(z,\bar z) X^\nu(w, \bar w) + \frac{\alpha'}{2} \eta^{\mu\nu} \log |z-w|^2$$ and for more ...
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Central charge in energy-momentum tensor OPE

I think that general point of view about central charge in books is considering OPE $T(z) T(w)$ for different field theories and finding that general expression for the most singular term is about to ...
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Matching symmetry factor when a heavy vector field is integrated out

Let us consider the lagrangian $$\mathcal{L} = \alpha \bar{u}\gamma^\mu u V_\mu + \frac{\beta^2}{2}V_\mu V^\mu$$ there $V_\mu$ is a heavy vector field and $u$ is a massless SU(3)-colored quark. If ...
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Time ordering of normal ordered product

I would like to calculate $$<0|T(:x^4::y^4:)|0>$$ for scalar fields $x$, $y$ "by hand", but I don't understand yet how. With Wicks theorem I'd say this is strictly 0. Is this correct? By hand I ...
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Why is normal ordering a valid operation?

Why is normal ordering even a valid operation in the first place? I mean it can give us some nice results, but why can we do the ordering for the operators like that? Is its definition motivated by ...
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Time-ordering vs normal-ordering and the two-point function/propagator

I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between $::$) is ...
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How to find the number of distinct contraction cases in Wick's Theorem?

Let $\mathcal{G}^8_{un}:=(t_1,t_2,t_1'^3,t_2'^3)=\langle 0 \mid T[Q_{un}(t_1)Q_{un}(t_2)Q(t_1')^3Q(t_2')^3] \mid 0 \rangle_{un}$ We want to use Wicks theorem to write this function as the sum of 2-...
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Perturbation theory : quadratic external field

I'm trying to derive the explicit form of S-matrix of an interaction Hamiltonian $$H' = \frac{1}{2} \lambda \left[ \int d^3 x \rho({\vec x}) \phi({\vec x}, t)\right]^2\tag{1}$$ Even though the ...
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Why is the Wick contraction in HFB or BCS equal to a single-particle density?

I'm trying to understand how in Hartree-Fock-Bogoliubov (HFB) or BCS theory we can write a product of creation/annihilation operators as single-particle densities under the guise of "Wick's theorem". ...
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Wick theorem applying to partly ordered operator

I symbolize $T$ as the time-ordered operator and $::$ as normal order symbol. I know that in quantum field theory generally we have: $$T\phi_1(x_1)\dots\phi_n(x_n)=:\phi_1(x_1)\dots\phi_n(x_n):+A$$ ...
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Wick's Theorem examples

Does anyone know of websites or texts that have an abundance of examples of computing time-ordered products of fields using Wick's Theorem for both bosons and fermions? I'm not just talking about the ...
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Functional integral aproach for Feynman rules [duplicate]

I am familiar with the basic ideas of quantum field theory but I feel uncomfortable when I have to derive Feynman rules by myself for a given action (for example in non-linear sigma models or ...
Consider the mode expansion of a (chiral) scalar field confined to a disc with circumference L: $$\phi(x) = \phi_{0} + p_{\phi} \frac{2\pi}{L} x + \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} e^{-(k_{n}a)... 2answers 336 views Chronological and normal ordering I've realized I'm little bit confused when I want to treat elements like this$$\left<\phi_0|T\{a_p(t)a_p^+(t')V(t_1)V(t_2)\}|\phi_0\right>$$with$$V(t)=\dfrac12 \dfrac{1}{(2\pi \hbar)^9}\...
I've been struggling with a detail in Second Quantization which I really need to clear out of my head. If I expand the S-matrix of a theory with an interaction Hamiltonian $H_I(x)$ then I have  S ...