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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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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Questions about the equivalent forms of Wick's theorem?

NOTE: The problems have been editing with more details. I have met Wick's theorem first in this book fundamentals of many-body physics when talking about the perturbation expansion of zero ...
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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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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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Stratonovich integral in quantum field theory

I'm reading a paper on Wick renormalization and there are a couple of things that are not that clear to me. The paper ends with the following sentence: In Euclidean quantum field theory, the ...
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Mode operators in the Virasoro algebra

This questions concerns Exercise 2.11 in Polchinski. We are asked to compute the commutator $$L_{m}(L_{-m}|0;0\rangle) - L_{-m}(L_{m} |0;0\rangle)$$ By plugging the mode expansions, we use the ...
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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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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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How to prove Wick's Theorem (Zee's eq. I.2 (16)) via Gaussian integration?

I'm working through Zee's QFT in a Nutshell but there's an integral [I.2 (16)] I couldn't quite derive. The problem is to find $$\langle x_i x_j ... x_k x_l\rangle=\frac{\int ... \int dx_1 ... dx_n ...
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Wicks contractions of stress-energy tensor and plane partitions

I am working out the number of wick contraction of a number $n$ of stress-energy tensor in 4D CFT. The strategy is as follows: For 1 stress energy tensor $T_{\alpha\beta}$, you have only one ...
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OPE Kac-Moody Currents

We have the following operators: \begin{align} J^a(z) = \frac{1}{2}\psi_s^{\dagger}(z)\sigma^a_{s s'}\psi_{s'}(z), \hspace{10 mm} \bar{J}^a(z) = \frac{1}{2}\psi_s^{\dagger}(\bar{z})\sigma^a_{s s'}\...
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Error in proof of Wick's theorem

I am having a little trouble proving wick's theorem. I'll start from the last step that I know is correct. We define $$\left\langle \prod_{j=1}^{2m}x_{i_j}\right\rangle:=\left.\frac{\partial^{2m}}{\...
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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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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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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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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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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 contractions with respect to an arbitrary state

I've been working through the following set of slides related to Wick's theorem: http://www.euroschoolonexoticbeams.be/site/files/2008_JDobaczewski_lecture.pdf From slide 19 onwards the following is ...
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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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Calculating OPE of Graviton Vertex Operator

Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
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Explicit evaluation of a radially ordered product

I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in $...
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Factor of $1/2$ in the Sugawara construction

I'm trying to reproduce the Sugawara construction calculation using this reference (page 14). The normal-ordering of two local operators is defined as $$ N(XY)(w)=\frac{1}{2\pi i} \oint_w \frac{dx}{...
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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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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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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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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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654 views

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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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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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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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 ...
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Is this time ordering correct?

Suppose we have a current $J_\mu(0)$ and some (say scalar) fields $\phi(x)\psi(x)\chi(x)$. Suppose also that we don't know the commutation relations between the current and the other fields. Isn't it ...
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Wick renormalization

I'm trying to understand the Wick renormalization in the framework of the Ito integral. I saw the Wick theorem as presented on Wikipedia in a QFT course and I would like to understand how that is ...
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Help with two dimensional polar axis Fourier transform

This is a problem that I met in real-life physics research. This question is related to Wick's theorem. The question is: 1. In two dimensional plane with polar axis, why do we have the following ...
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Wick contraction in proton-pion production

Proton-pion production $\gamma + p \rightarrow \pi^0 + p$ occurs through the interaction hamiltonian $$\mathcal H_{int} = ig \bar \psi^{(p)} \gamma_5 \psi^{(p)} \phi + e \bar \psi^{(p)} \gamma_{\mu} \...
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Obtaining the $s,t,u$ Feynman diagrams by Wick contraction

Consider a real scalar field described through the following lagrangian $$\mathcal L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}m^2 \phi^2 - \frac{g}{3!}\phi^3$$ The second ...
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Linear Dilaton CFT

I´m doing the exercises on the Tong lectures of String Theory, in particular Problem Sheet 2: Consider the tensor: $T(z) = \frac{-1}{\alpha '} :\partial X(z) \partial X(z): - Q \partial^2 X$. By ...
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How to apply Wick's theorem in 2nd quantization for Spin Density Operators?

I am trying to work out a correlation function consisting of two spin density operators. Once I rewrite everything in 2nd quantized form, I am unsure of how to apply wicks theorem because the paul ...
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Which orderings are commonly used in modern physics?

I am curious as to which mathematical orderings are used in contemporary theoretical physics, and in what context/situations. So far, I have encountered the following: Time ordering: Commonly used ...
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Euclidean Feynman Rules derivation using Wick Theorem

When studying perturbation theory and Feynman rules, the standard derivation seems to start from the S-Matrix element in the interaction picture, and expands it into some series, after which Wick's ...
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How does one calculate Fourier transform of Feynman propagator?

I am struggling with calculating the following integral on Sredinicki: How did he get the second line of (10.6)? That is, how did he calculate the Fourier transform of Feynman propagator?
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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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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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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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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))...