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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Why do we "normal-order" instead of just subtracting off vacuum energy?

The Hamiltonian is arbitrary upto a constant anyway. Why don't we just subtract off the vacuum energy? The Hamiltonian was always observable only upto a constant. Instead, we do normal-ordering, in ...
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Normal Ordered Product in Operator Product Expansions

In an example of operator product expansion applied to $\phi^4$ theory of the book QFT an integrated approach, where the Eulerian Lagrangian is $$\mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi\right)...
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Is the Chronological product defined in Zavialov's book coherent?

In Zavialov's book "Renormalized Quantum Field Theory" he defines the Chronological Product for arbitrary operators as follows (I-47): Given two operators in the form $$A(\phi) = \sum_{n=0}^\...
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Wick Theorem: number of contractions [closed]

I have to prove that the number of contractions in Wick's Theorem is equal to: $$\frac{n!}{(n/2)! \ 2^{n/2}} \ \ \ where \ \ n \ \ is \ even$$ I don't know how to start, if someone can help.
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Proof of Wick's theorem for general Gaussian states

For boson modes $a^\dagger_i, a_i$, consider the density matrix which is an exponential of quadratic operators in $a^\dagger_i$ and $a_i$: $$\rho = e^{-H_{ij} a^\dagger_i a_j + (K_{ij} a^\dagger_i a^\...
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Defining normal-ordering in an OPE calculation

Suppose $\phi_1, \phi_2, \psi$ and $\bar{\psi}$ are free fields of a two-dimensional CFT with propagators on the plane given by $$\phi_1(y)\phi_1(z) \sim -\log(y-z),\quad\phi_2(y)\phi_2(z) \sim -\log(...
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Normal ordering of number operators $n$th power

In resources I keep seeing the normally ordered form of the number operator to the $n$th power, $${(a^\dagger a)}^n=\sum_{k=1}^n S(n,k){(a^\dagger)}^ka^k.$$ Why are we interested in the number ...
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Wick's theorem: From operators to fields

I understand Wick's Theorem when operators are involved to be, $$\mathcal{N}(f(a,a^\dagger) = :\!\sum\textbf{All contractions}\!:$$ But I'm getting slightly confused when this is expanded to fields, I'...
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Wick's Theorem and Functional Derivative

In the Quantum Field Theory An Integrated Approach, Fradkin, the author derived the partition functional for a free scalar field (after analytic continuation to imaginary time ) as $$Z_{E}[J]=Z_{E}[0] ...
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Different values for the Normal ordering

I've come across 2 examples approaching the ordering of $a^2({a^\dagger})^2$, each reach different results: $a^2({a^\dagger})^2=\;:\!\sum\text{all contractions}\!:\;=\;:\!aaa^\dagger a^\dagger\!:+\;4:...
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Using Wick's Theorem in an example with the harmonic oscillator

I understand Wick's theorem to be, $$T(x)=\mathcal{N}(x)=\sum:\textbf{all contractions}:$$ And I'm researching combinatorics and quantum theory in general. How would one connect Wicks theorem to the ...
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Wick theorem and explicit vanishing of $\text{vacuum}\to e\bar{e}$

I want to see explicitly how the transition amplitude for $\text{vacuum}\to e\bar{e}$ vanishes in QED but I'm struggling with Wick's theorem . I take $|i\rangle=|0\rangle$ and $|f\rangle=b^\dagger(\...
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How to calculate the OPE of the $X_L(z_1)X_L(z_2)$ in the free boson theory from the mode expansion?

From the polchinski page 238, given \begin{equation} [x_L,p_L] =[x_R,p_R]=i\tag{8.2.14} \end{equation} and the mode expansion $$\begin{equation} \begin{split} X_L(z) = x_L -i\frac{\alpha'}{2}p_L \ln z ...
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Time ordering nucleon-nucleon scattering

I am trying to compute the nucleon scattering $\psi \psi \rightarrow \psi \psi$ described by: \begin{equation} \begin{array}{l} |i\rangle=\sqrt{2 E_{p_{1}}} \sqrt{2 E_{p_{2}}} a^{\dagger}\left(\mathbf{...
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Relation of Wick theorems

In the context of quantum stat mech it is common to use Wick's theorem to refer to the factorisation $$ \langle f_1 f_2 f_3 \cdots f_N\rangle = \sum_{\text{pairings}\, \pi} (\pm 1)^{|\pi|} \langle f_{\...
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Intuitive Approach to Wick's Theorem

Context I'm currently reading Many-Particle Physics by Gerald D. Mahan. In section 2.4 it explains Wick's theorem and he gives the example $$ _0\langle|T \hat{C}_\alpha(t) \hat{C}_\beta^\dagger(t_1) \...
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Peskin and Schroder Equation 4.37 invalid

In chapter 4 of Peskin & Schroeder, \begin{align} T\{\phi(x)\phi(y)\} &= N\{ \phi(x)\phi(y)+ \text{Contraction}({\phi(x),\phi(y)}) \}, \tag{4.37}\\ & = N\{ \phi(x)\phi(y)\}+ N\{\text{...
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Wick's theorem on $C_\ell$ : where does factor $\dfrac{1}{2\ell+1}$ come from?

Just a question that bothers me. This concerns Wick's theorem, which my book gives as: Wick's theorem: If $G = (G_1, \dots, G_n)$ is a centered gaussian multivariate random variable ($\langle G_1 \...
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Simplification of nested time-ordered products

I'm trying to progress towards understanding, and perhaps finding a proof for, the "nested" Wick's theorem for time-ordered products $T\{ \ldots \}$ alluded to in part (II) of this answer. ...
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QFT: Normal Ordering Interaction Hamiltonian Before Using Wick's Theorem

It has recently come to my attention, though reading the notes of a course on QFT that I've started, that there seems to be an "ambiguity" in, or at least two distinct ways of, calculating ...
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How can we relate the Feynman rules obtained from the Lagrangian with the usual definition from canonical quantization?

I am following a course on QFT that is based upon canonical quantization and not path integrals. When calculating scattering amplitudes we compute, at the relevant order, the corresponding matrix ...
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Scalar Yukawa theory, contraction

In the scalar Yukawa theory ($\Phi$ is real scalar field and $\phi$ is a complex scalar field): \begin{equation} \mathcal{L}_{S Y}=\left(|\partial \phi|^{2}-m^{2}|\phi|^{2}\right)+\frac{1}{2}\left((\...
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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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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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