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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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How to solve this particular problem of Wick's theorem?

So I know the basics of Wick's theorem, but unsure about how to solve this time ordered product of a term that involves normal ordering. Is it just simply the sum of all possible contractions, but no ...
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Gutzwiller renormalization factors

I am computing the expectation value of the kinetic term of a tight-binding model, respect to the Gutzwiller wavefunction, in the limit of infinite lattice-coordination, i.e using these constraints (...
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$\psi\psi \longrightarrow \psi\psi$ scattering in Scalar Yukawa model

In David Tong's lecture notes, Equation 3.48 In line 2, how is $|0\rangle \langle 0|$ introduced between $\psi^{\dagger}(x_1)\psi^{\dagger}(x_2)\psi(x_1)\psi(x_2)?$ Why is $\langle p_2',p_1'|\psi^{\...
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Expectation Value Of The Double Occupancy Operators' Product

I want to prove the relation \eqref{eq:Metz_relation} that i found in this article. \begin{equation} \left\langle\varPhi_0\right|\prod_{i} \hat{D}_i\left|\varPhi_0\right\rangle= \left\langle\varPhi_0\...
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Physical Meaning of the Gutzwiller Constraints

I have a doubt on the constraints for the expecation values obtained by Bünemann et all. First i want to introduce my notation To analytically solve a tight-binding model, \begin{equation} \hat{H}= ...
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What is radial ordering?

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
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Expectation value of a path-ordered exponential

Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$: $$ \overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
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Symmetry factor in $\phi^4$ theory

I'm having trouble while trying to understand what the symmetry factor of a Feynman diagram really is. From books I get that it is a geometrical factor that you get by the number of ways in which you ...
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Calculation of a 4-point function by path integrals

In Srednicki's book in chapter 8 a four-point function is computed as a sum of products of propagators: $$<0|T\phi(x_1)\phi(x_2) \phi(x_3)\phi(x_4)|0> = \frac{1}{i^2}[\Delta(x_1 -x_2)\Delta(...
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Time ordering, normal ordering and Wick's contraction

I'm reading chapter 4 of Peskin & Schröder, and I'm confused how they express the time ordering of two fields: let $T$ denote time ordering, $N$ the normal ordering and I use $C$ for the ...
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Kac-Moody algebra, proof of parameters calculation

I'm following the notes "Ginsparg - Applied Conformal Field Theory" (https://arxiv.org/abs/hep-th/9108028) and I'm stuck on a proof at page 140 about Kac-Moody algebras. I would like to prove that $\...
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Conformal transformation of a vertex operator before normal ordering

Let us consider a free scalar boson $\varphi(z,\bar{z})$ on the complex plane and assume the following two-point correlation function \begin{eqnarray} \langle\varphi(z,\bar{z})\varphi(w,\bar{w})\...
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Apparent problem in using Wick's theorem to calculate matrix elements of two body operators

In the second quantized notation, a two body operator $\hat{O}$ can be written as $$\hat{O} = \sum\limits_{x_1,x_2,x_3,x_4} O_{x_1,x_2,x_3,x_4} a^\dagger_{x_1}a^\dagger_{x_2}a_{x_4}a_{x_3} ,$$ where ...
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Operator product expansion of more than 2 operators in CFT

I’m confused about OPE in 2d CFT. I’ve found difficulties in taking OPE of the product of 3 operators. Consider the following operator product. \begin{align*} O_1(z) :O_2(w) O_3(w): = O_1(z)\frac{1}{...
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What exactly are we doing when we “invent” Feynman Diagrams?

So, I am trying to derive the Feynman rules for Yukawa theory (following the section in Peskin). Specifically, for the process 2 fermions $\rightarrow$ 2 fermions. To second order, I then have that ...
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Problem with Wick's theorem (normal ordering of a contraction)

Taking the example of two bosonic fields, Wick's theorem is \begin{equation} T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\} + N\{(\phi\phi^\dagger)_c\} \end{equation} where the subscript $c$ ...
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Radial ordered commutation relation

In the book Conformal Field Theory of Francesco, Mathieu and Sénéchal, in Sec. 6.1.2, the authors state that the integral $$ \oint_w \mathrm{d}z~ a(z)b(w) ~=~ \oint_{C_1} \mathrm{d}z~ a(z)b(w) - \...
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There are too many Wick's Theorems!

This is a follow-up question to QMechanic's great answer in this question. They give a formulation of Wick's theorem as a purely combinatoric statement relating two total orders $\mathcal T$ and $\...
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Connection between Fermi sphere and Wick's theorem

For Fermi sphere we can write down the Hamiltonian with the second quantization language: $$H_0 =\sum_k E_n(k) c_k^\dagger c_k$$ For this simple noninteracting Hamiltonian, we can define the ...
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OPE double-contractions between $T$ and $e^{ikX}$

I am reading David Tong's lecture notes chapter 4 http://www.damtp.cam.ac.uk/user/tong/string.html On the top of page 82 in the eq. before eq. (4.27), we are computing the OPE between $T$ and $e^{ikX}...
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What is the calculation rule of the normal ordering operator?

Here $\phi_I$ is just the free Klein-Gordon field. So, this field is decomposed of two components shown above. Now let $N$ be the normal ordering operator. Then, I think that $N(\phi_I^+(x)\phi_I^-(y))...
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What is a contraction in QFT?

I have been reading QFT and I am stumbling upon the idea of Wick's theorem. The correlation functions have something to do with "contractions". I want to understand what the physical meaning of a ...
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Wick contraction and propagator confusion

I am having trouble understanding the how the Wick contraction leads to the Feynman propagator for scalar fields. The Feynman propagator can be written as $$ D_F(x-y)=\langle 0 | T(\phi(x) \phi(y)) | ...
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Ising Model — Higher order moments

We assume that $p(\boldsymbol{x})$ is the probability mass function of the Ising Model with zero external field, then $$p(\boldsymbol{x})=\frac{1}{Z(\boldsymbol{\theta})}e^{x^T \boldsymbol{\theta} x}$$...
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Computing the OPE of $T : \mathrm{e}^{ikX} : $ [closed]

I've hit a stumbling block where I'm just not seeing how to get from line to line in the following calculation from David Tong's strings notes. Can someone spell out how line 1 becomes line 2 in the $\...
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Why do we have $[\phi_1^+,:\phi_2\phi_3:]=:[\phi_1^+,\phi_2^-]\phi_3:+:\phi_2[\phi_1^+,\phi_3^-]:$?

How $$[\phi_1^+,:\phi_2\phi_3:]=:[\phi_1^+,\phi_2^-]\phi_3:+:\phi_2[\phi_1^+,\phi_3^-]:$$ with $\phi_i=\phi(x_i)$ field operators ($\phi_i^+$ is the annihilation part while $\phi_i^-$ is the creation ...
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Time ordered product of bilinear functions for Dirac-field

If we have two Observables (bilinears of Diracfield $\psi(x)$) $O_1(x)=\bar{\psi}(x)\Gamma_1\psi(x)$ and $O_2(y)=\bar{\psi}(y)\Gamma_2\psi(y)$ and if we calculate their time ordered product $T(O_1(x) ...
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Wick Theorem: Performing contractions in the right order

The first line is one of four terms that one gets after applying Wick theorem to the time-ordered product of these field operators and as far as i understand it is just a short-hand notation for which ...
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Mean field approximation in BCS theory

Bardeen, Cooper and Schrieffer's (BCS) theory describes spinful Fermions that mutually interact via an attractive contact interaction. The general Hamiltonian reads in second quantization $$H = \sum_{...
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Fermion density with Wick's theorem

I want to calculate the expectation value \begin{equation} \langle\textrm{F}\rvert\Psi^\dagger_{m_1}(x_1)\Psi_{m_1}(x_1)\Psi^\dagger_{m_2}(x_2)\Psi_{m_2}(x_2)\lvert\textrm{F}\rangle \end{equation} ...
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Ambiguity in free field operators

I am interested in the ambiguities which exist in defining the composite free field operators--i.e., operators corresponding to monomials of the fundamental field operator (and their derivatives). In ...
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Question on Wick's theorem for fermions

I have a guilty suspicion this should be obvious. What is the difference between these two expectations taken over the same measure ($\int \mathrm{d}\mu(\bar\psi,\psi)\exp{\sum \bar\psi A\psi}$ for ...
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Wick contraction corresponding to a connected diagram in $\phi^4$-theory to second order

I am trying to understand the diagrams that comes from a two-point correlation function, $$\langle \Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle$$, in $\phi^4$-theory. The zeroth order contribution, i.e. $\...
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Questions about the equivalent forms of Wick's theorem?

I have met Wick's theorem first in this book fundamentals of many-body physics (by Wolfgang Nolting) when talking about the perturbation expansion of zero temperature Green's function. Later in the ...
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Feynman diagrams - swapped only by vertex label

Consider as an example $\phi^3$ theory, which contains at second order both the contractions: $$\newcommand{\mean}[1]{\langle #1 \rangle} \mean{\hat a_q \phi(\color{red}{x})}\mean{T\phi(x)\phi(x)}\...
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Operator product expansion involving derivatives

I have questions regarding the equation (2.2.4) in Polchinski Vol 1: $$ X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2) = -\frac{\alpha'}{2}\eta^{\mu\nu} \ln|z_{12}|^2 + \sum_{k=1}^\infty \frac{1}{k!}\...
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Different consequences of Wick's theorem in fermionic and bosonic condensed matter systems

Based on Wick's theorem, the time-ordered product of operators can be written as a sum of normal-ordered product and products involving all types of contractions. Upon taking the ground state ...
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Proof of Wicks Theorem, 3 Fields

The problem statement, all variables and given/known data Question attached: Relevant equations Using the result from two fields that $ T(\phi(x) \phi(y))= : \phi(x) \phi(y) : + G(x-y)$ Where $G(...
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Normal order vs Time order for fermions

For a conformal field $X$, Polchinski gives a relation between the time ordering $T$ (or equivalently the radial ordering ${\cal R}$) of a functional of identical fields and the normal ordering, which ...
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Calculating vertex factor for scalar field theory

I am practising basic QFT and am having some trouble with calculating the vertex factor of an interacting theory involving two real scalar fields, $\phi_{1}$ and $\phi_{2}$. If I create a generic ...
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How to use Wick's theorem to compute this matrix element?

I wanted to see how to use Wick's theorem in practice (I know with Feynman diagrams it is better, but here I want to do this with Wick's theorem only), so I considered computing the matrix element for ...
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Appending a Noether current to a Feynman rule

Background Typically in QFT one derives the Feynman rules by differentiating certain terms in the Lagrangian w.r.t the relevant fields. So for instance if our term is $\mathscr{L} =\phi_1\phi_2\phi_{\...
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Actually calculating something using Wick's Theorem

I am still struggling to get my head around QFT and whilst I think I understand the method of generating functionals to compute correlation functions (as in my question here), my course notes often ...
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Normal ordering of the commutator between annihilation and creation operator

According to the commutation relation of annihilation and creation operators, $$[a,a^{\dagger}]=1. \tag{1}$$ I would like to calculate the vacuum expectation value of the normal order of this ...
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Wick's Theorem for Calculating the Vacuum Functional

The vacuum functional of a theory of free fermions is the overlap between the bare vacuum and the interacting vacuum (i.e. the true groundstate of the Hamiltonian). If the theory preserves particle ...
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Symmetry factor for 1PI Feynman diagrams in $\phi^4$ theory

I am trying to understand the various factors that the Feynman amplitude will carry corresponding to the Feynman diagrams of Fig. 1 of this reference. I understand that the $n^{th}$ diagram containing ...
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Wick theorem and OPE

I'm trying to work out in detail how the Wick theorem is used for constructing OPEs in CFT. One of the first things which bothers me is the difference in definitions of normal ordered product and ...
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OPE of normal ordered operators

In what follows I use $\mathcal{N}\{\ldots\}$ for normal ordering, $\langle\ldots\rangle$ for contraction and $\operatorname{Reg}\{\ldots\}$ for the complete sequence of regular terms which is ...
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OPE of Lorentz current with tachyon vertex

This is a question related to chapter 2 in Polchinski's string theory book. On page 43 Polchinski calculates the Noether current from spacetime translations and then calculates its OPE with the ...
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Does Wick's theorem still work for derivative fields

I am wondering if Wick's theorem still is useful for something like $$\langle0|T\ \partial\psi(x)\partial\psi(y)...\partial\psi(w)|0\rangle$$ can I say this things equals to all possible ...