Questions tagged [wick-theorem]

A combinatoric procedure in QFT of reducing arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. A string of such operators is rewritten as the normal-ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.

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It there a Generalised Wick theorem for rectangular matrices?

Say we have several rectangular matrices (J,K) of different shapes, and we want to evaluate the Expected value, or Trace, of their Product $$ \mathbb{E}[J_{1}^{T}J_{2}^{T}J_{3}^{T}K_{3}K_{2}K_{1}] $$ ...
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Relationship between normal-ordered vacuum state and parity operator

In the paper "Operator ordering in quantum optics theory and the development of Dirac’s symbolic method" by Hong-yi Fan, as referenced in this question, the authors mention the property $$:A:...
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Help with Wick's theorem in a $\phi^4$ QFT

QFT noob here. I am currently working out the momentum space two-point function for a $\phi^4$ qft in Euclidean space time, and considering the $\lambda^1$ order contribution, I am encountering a ...
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Demonstration of the variance on a $C_{\ell}$ : can't make appear into demonstration a term "$-1$"

Regardings the definition of $C_{\ell}$ on a survey, we measure all the $2 \ell+1$ coefficients. We are thus led to define an estimator of the observed power spectrum $$ \hat{C}_{\ell}=\frac{1}{2 \ell+...
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How do Wick contractions and OPEs relate?

I am trying to understand how to recover the factorisation of the four-point function (assume $\langle \phi\rangle = 0$) of some free Gaussian field $$\langle \phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4) \...
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Higher-order Langevin noise correlation

Supposing Langevin noises are white noise, we know that the noises F are Gaussian and higher-order noise correlations, $\langle F_{t1}F_{t2}...F_{tn}\rangle$ can be decomposed by the second-order ...
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Linked Cluster Theorem in Conformal Field Theories

I am trying to compute an effective action for the source fields $J(x)$ in some theory $$S=S_\mathrm{CFT} + S_J= S_\mathrm{CFT} + \int \phi^\ast J + h.c. $$ where $\phi$ is a primary of my (...
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How do I show that the $n$-point correlator $\left\langle\phi(x_1)\phi(x_2)...\phi(x_n)\right\rangle$ is equal to this expression?

Given the Euclidean action \begin{equation} S_E(\phi) = \int d^d x \frac{1}{2}\big(\nabla\phi\cdot\nabla\phi + m^2\phi^2\big)\end{equation} and the partition function \begin{equation}\mathcal{Z} = \...
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Polchinski OPE of spacetime translation current

I am trying to derive $$ j^\mu(z):e^{ik\cdot X(0,0)}: \;\sim \frac{k^\mu}{2z}:e^{ik\cdot X(0,0)} \tag{2.3.14a} $$ from Polchinski's String Theory vol.1 equation (2.3.14a). using $j^{\mu}=\frac{i}{\...
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Feynman propagator for interacting field

For scalar field, Feynman propagator is commonly defined as $$ \Delta_F(x-y) = \langle 0 | T\phi(x)\phi(y)|0 \rangle . $$ For free theory, field satisfy equation of motion is $$\phi(x) = \int\frac{dp^...
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Operator expectation value for system of non-interacting particles (Fermions)

I am reading the book "Electronic Structure" by Richard Martin which poses the following problem: Show that the expectation value of an operator $\hat O$ in a system of identical, non-...
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OPE's from Spacetime Lorentz Invariance of the Polyakov action

How to explicitly determine the other singular terms Polchinski (2.4.14) using wick's theorem $$ T(z)A(0,0) = ...+\frac{h}{z^2} A(0,0)+ \frac{1}{z} \partial A(0,0)+... $$ if $$z \rightarrow z' = az+b $...
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Meaning of a strange Feynman diagram for the $\phi^3$ scalar Field theory

Background I am considering a scalar field theory with $\sim\phi^3$ interaction term, with Lagrangian \begin{equation} \mathcal{L} = \frac{1}{2}\left( \partial_\mu\phi\right)^2 - \frac{m^2}{2}\phi^2 - ...
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Doubt in using Wick thoerem in using OPE

While calculating OPE of $T(z)\partial_w\phi(w)$ in Francesco CFT book, I can't understand how Wick theorem is used. The calculation is like following: $$T(z)\partial_w\phi(w)=-2\pi g:\partial_z\phi(z)...
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Complex Fields, Propagators, Wick's Theorem and Feynman Diagrams

I'm having quite the problem connecting all these concepts, so apologies for the lengthy post. Complex Fields and Feynman propagators Let $\psi$ denote a complex field. Then, after quantization, we'll ...
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Particle Creation by a Source

I am currently self-studying Quantum Field Theory and am using the textbook Introduction to Quantum Field Theory by Peskin and Schroeder. Currently I am in chapter 4, and am doing the first problem in ...
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Contractions of probability amplitude in quantum field theory

Given a Lagrangian $$\mathcal{L}=\frac12(\partial_\mu \phi(x))^2+\frac12m^2 \phi(x)^2-\frac{\lambda}{4!} \phi(x)^4$$ The probability amplitude in each order is given by: $$A=(2\pi)^3 (2E_q)^{1/2} (...
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Applying Wick's theorem in second-order products

Consider the effective current$\times$current interaction Hamiltonian of weak interaction at the quark level, \begin{equation} \mathcal{H}=\frac{G_F}{\sqrt{2}}[\overline{e}\gamma^\mu(1-\gamma_5)\nu_e][...
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Wick's theorem, contracting field operators at the same point

I want to calculate the amplitude for nucleon meson scattering $\psi \varphi \to \psi \phi$ in scalar Yukawa theory, with interaction term: $$H_{I} = g \int d^{3}x \psi^{\dagger} \psi \varphi.\tag{3....
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What is my misunderstanding in Wick's theorem?

Trying to understand Wick's theorem, I took most of my knowledge from the corresponding Wikipedia article. The statement is that given the definition of normal ordering of operators $A,B,C,\ldots$ any ...
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2-point correlator of the isospin current

given the isospin current $$J_{\mu}(x) \propto \bar{u}(x)\gamma_{\mu}u(x)-\bar{d}(x)\gamma_{\mu}d(x)\,,$$ I want to evaluate the 2-point correlator $$\langle \bar{\psi}(x)\gamma_{\mu}\psi(x)\bar{\psi}(...
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On contractions and S-Matrix in $\phi^4$ scalar theory

If you have a self-interacting Lagrangian for a scalar field theory: $$L= L_0 + L_I = \frac{1}{2} (\partial_\mu\phi)^2 - \frac{1}{2} m^2\phi^2- \frac{g}{4!}\phi^4$$ where $g$ is the coupling constant, ...
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How to find all possible Wick contractions of 5 fields?

I need to find all possible contractions (in the sense of Wick contractions) for 5 fields. One can of course start drawing randomly, but I'm sure there is some kind of algorithm to do this ...
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If $|\psi\rangle$ is a free fermionic state, why does its reduced density matrix $\text{Tr}_C(|\psi\rangle \langle \psi|)$ also obey Wick's theorem?

I have recently been trying to understand this paper. So far I understand why, given a free fermionic state $|\psi\rangle$, it is fully characterised by its 2-point correlation matrix (i.e. obeys ...
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Proof of this version of Wick’s theorem?

I can’t prove/find the following version of Wick’s theorem: Say we have a system of free fermions with Hamiltonian $$ H = \sum_{ij} t_{ij}c^{\dagger}_ic_j\quad \longrightarrow \quad H = \sum_k E_k d^{\...
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White noise approach to Feynman integrals: why do we use this renormalization?

I start by saying that I know very little about Feynman integrals so please bear with me. In Kuo's book "White noise distribution analysis" or in Hida, et al. "White noise analysis"...
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How did Wick's theorem work for Feynman propagator in Dirac Equation?

In David Tong's Quantum Field Theory Lecture Notes, Page 115 Eq. 5.34, the Feynman propagator was defined to be $$ S_F(x-y)=\langle 0|T\psi(x)\bar\psi (y)|0\rangle \newcommand{\normord}[1]{:\mathrel{#...
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Feynman rules for scalar electrodynamics

The issue I have been learning perturbation theory in QFT, but due to the weird nature of the course I was attending I still haven't learned how to properly do it by Feynman diagrams. I think the best ...
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Number of Wick contractions for $\left< x ( t^\prime )^5 x ( t^{\prime \prime} )^5 \right>$

I am considering the possible Wick contractions for the following expression: \begin{align*} \left< x ( t' )^5 x ( t'' )^5 \right> = \left< x( t' ) x( t' ) x( t' ) x( t' ) x( t' ) x( t'') x(...
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Normal ordering of operators

Just a quick question regarding normal ordering. To begin given a field $\phi(x)$ we can normal order it in respect to some state $|G\rangle$: $$:\phi(x):=\phi(x)-\langle \phi(x)\rangle $$ Typically ...
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BCS and Feynman Diagrams

I am currently studying the strong coupling theory (by Eliashberg) of superconductivity, which is derived by BCS by considering self-energy diagrams and the following interaction potential $$ V(q, i\...
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Time ordering on Kelydsh countour

I want to compute a time-ordering product but I have a question concerning this time ordering product. First, we consider A,E,I,L,N,O and V some second quantized operators without specifing what ...
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Is there no mode expansion for $:e^{i k_\mu X^\mu(z, \bar z)}:$ in the free boson CFT?

We are told that in a 2D CFT, all primary operators can be written in the form of \begin{equation} \tag{1} \mathcal{O}(z, \bar{z} ) = \sum_{m,n \in \mathbb{Z} } \frac{\mathcal{O}_{m,n} }{z^{m + h} ...
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Are time and normal ordering idempotent (meta-)operators?

I already know that $T\,:\,\,:\,=T$ and $\,:\,\,:\,T=\,:\,\,:\,$ simply because when one operator acts after the other, play a bit with what the previous has done and do what he wants to actually do, ...
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Demonstration Wick's theorem

I read a lot about Wick's theorem, on the internet and on this site, but I'm unable to find the answer to my question. I have a problem trying to demonstrate Wick's theorem: I know it is done by ...
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Why does the 3-point function of a real scalar field vanish?

$$\langle0\lvert T\hat\phi(x_1)\hat\phi(x_2)\hat\phi(x_3)\rvert0\rangle$$ I'm looking for an intuition to it, if not an actual interpretation. Otherwise, I know how to get the result 0 using Wick's ...
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Example of truncated expectation which gives a disconnected diagram. Where am I wrong?

My notes introduce the truncated expectation in the following way: given $S_0$ a quadratic form, consider a generating function $$e^{W(J)} = \int \prod_x d\varphi_x e^{-S_0+ JO} = \int P(d\varphi) e^{...
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Wick's theorem for fermions

I'm currently studying Wick's theorem for fermions with Peskin's and Schroeder's Introduction to QFT (p.115 & p.116). Here, Wick contractions are defined as $$ \psi^\bullet (x)\bar{\psi}^\bullet(y)...
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On which propagator does the field self-contraction loop go on this Feynman diagram?

This question relates to page 111 in Peskin and Schroeder. I am trying to do the derivation of the 2-particle to 2-particle Feynman diagrams in $\phi^4$ theory by hand, following Peskin and Schroeder. ...
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$\phi^4$-theory, S-matrix Feynman diagram to first order from Peskin and Schroeder

This relates to page 111 in Peskin and Schroeder. We have the $\phi^4$ S-matrix for a 2-particle to 2-particle scattering reaction: $$-i\frac{\lambda}{4!}\int d^4x \langle p_1p_2|\mathcal T\left(\phi(...
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How to compute $\langle n'| (a^++a)^k | n \rangle$ for arbitray $k$? [duplicate]

I'm trying to compute the 2nd order correction to the energy spectrum of a 1D quantum harmonic oscillator when a perturbation of the form $\gamma\,\hat{x}^k$ (with $\gamma\ll1$) is added to the ...
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Commutation relations of derivatives of fermionic fields from the commutation relations of the original fields

I have a general question regarding such type of calculations, but let me start with a concrete question. Consider the $bc$- free fermion CFT so that $b(z)$ and $c(z)$ are free fermions with OPE, $$b(...
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Proof for Wick's theorem (non-interacting case)

I follow the proof of Wick's theorem for a non-interacting case in "Many-Body Quantum Theory in Condensed Matter Physics (Bruus, Flensberg)", Ch 11.6. In that chapter, above (11.75), the ...
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Schwinger-Dyson equation for connected correlation functions

Could someone tell me what's the Schwinger-Dyson equation for connected correlation functions? I'm looking for a formula that relates a connected $n+1$-point function to connected lower point ...
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Non-polynomial correlators for stochastic path integrals

What is the right way to evaluate a stochastic path-integral of the form: $$\int \mathcal{D}x \mathcal{D}\tilde{x} \left( \int_0^T \sin(x(t_2)) x(t_2) dt_2 \int_0^T \tilde{x}(t_1) dt_1 \right) e^{-\...
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What is going on with this generating functional? (QFT)

I am reading Peskin and Schroeder's chapter on functional methods and they compute the following correlation function: \begin{equation*} \begin{split} \langle 0| T\phi_1\phi_2\phi_1\phi_3 |0\rangle ...
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Normal ordering of hamiltonian

I came across this in the lecture notes of quantum field theory by David Tong. Inside time ordering interactions aren’t taken to be normal ordered. Interaction hamiltonian should be normal ordered ...
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Why commutation relations is applied in normal order for antiparticle relation ($t_1<t_2$) in wick theorem

I am reading Wick theorem from "Student Friendly Quantum Field Theory" By Robert D. Kaluber. I understand how normal orderd vanishes and only contraction remains. But in page no. 205, it is ...
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Clarification of derivation in Tong's lectures on String theory

I'm reading Tong's Lectures on String Theory chapter 4 on conformal field theory. The PDF can be found here. I'm trying to understand his proof of claim 2 in section 4.3.3, but I can't seem to grasp ...
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Wick's theorem for non-equilibrium steady state

I am working on a grand canonical Hamiltonian which has the form: $$ \hat{K}=\hat{H}_{SC}+\hat{H}_{tip}+\hat{H}_{T}-\mu\hat{N}_{SC}-(\mu+eV)\hat{N}_{tip} $$ where $\hat{H}_{T}=-t_0\sum_{\sigma}(c^{\...

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