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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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Feynman rule for scalar QED vertex

A popular problem in QFT textbooks and courses is to derive the Feynman rules for scalar QED. Usually, this theory is presented via the following Lagrangian density: $$\mathcal{L} = (D_\mu\phi)^\...
Rafael Grossi's user avatar
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Number Operator "Ordering" for Higher Order Bosonic Operators

I'm considering the algebra of a single harmonic oscillator where $[\hat{a},\hat{a}^\dagger]=\hat{\mathbb{I}}$. Typically, one is interested in normal, antinormal or symmetric ordering. I am ...
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Correlation functions of exponentials of fields

I've been trying to solve for scattering amplitudes for 4 graviton scattering in string theory. However, while going through Schwarz, Witten and Green book for string theory, I come across the ...
Nakshatra Gangopadhay's user avatar
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Equivalent definitions of Wick ordering

Let $\phi$ denote a field consisting of creation and annihilation operators. In physics, the Wick ordering of $\phi$, denoted $:\phi:$, is defined so that all creation are to the left of all ...
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Hartree-Fock Hamiltonian and higher-order terms

I'm diving into Hartree-Fock methods, and I'm confused on why the Hartree-Fock Hamiltonian reduces into a single particle Hamiltonian. When applying Wick's theorem to the Fermi Sea vacuum, we use the ...
lukealk98's user avatar
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Radial ordering in CFT

Consider the following quantum two-point function (without assuming radial time ordering), $$\begin{align} \langle 0 | \hat{T}(y)\hat{T}(z) |0 \rangle & = \sum_{n,m}y^{-(m+2)}z^{-(n+2)}\...
phonon's user avatar
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Non-perturbative matrix element calculation

Following Peskin & Schroeder's Sec.7's notation, I would like to compute the matrix element $$ \left<\lambda_\vec{p}| \phi(x)^2 |\Omega\right>\tag{1} $$ where $\langle\lambda_{\vec{p}}|$ is ...
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Wick's theorem for an interacting theory in the $n=4$ case

I was working with the following expression related to the Wick's theorem for four fermionic operators. $$ \langle c^\dagger_i c_j c^\dagger_p c_q \rangle = \langle c^\dagger_i c_q \rangle \langle c_j ...
Bio's user avatar
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How does one go about calculating non-time ordered correlation functions in scalar field theory?

I am aware that expressions for the time-ordered expectation values of field operators can be derived using Wick’s theorem. My question is, how would one go about finding the corresponding non-time-...
Jack's user avatar
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Issue in Feynman propagator with derivative of a scalar field: which Feynman rule do I apply?

While solving a QFT exercise, I'm trying to calculate the Feynman propagator $$ \underbrace{\partial_\mu \phi(x) \phi(y)} = \langle 0 \vert T { \partial_\mu \phi(x)\phi(y) } \vert 0\rangle $$ where ...
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Confusion regarding simplifying normal ordered products in CFT

I am studying CFT on my own and have some confusion regarding applications of Wick's Theorem to simplify normal ordered products to time ordered products. Wick's theorem is fairly straightforward, ...
QFTheorist's user avatar
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Calculate first-order term of the $S$-matrix for the $\phi^{4}$ theory [closed]

Before I ask a question, I will start with a small introduction. I want to evaluate the $S$-matrix order-by-order in an expansion in small $\lambda$ for a $2 \rightarrow 2$ scattering in $\phi^{4}$ ...
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Wick contraction between two scalar fields

I have a short question about Wick contraction. It is given that $$\phi\left(x\right) = \phi^{+}\left(x\right) + \phi^{-}\left(x\right)\tag{1}$$ where: $$\phi^{+}\left(x\right) = \int \frac{d^3p}{\...
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Doubt regarding use of Wick contractions

I'm currently taking my first course in QFT and am learning about finding transition amplitudes using Wick's theorem. As far as I'm aware, Wick's theorem gives us a way to change from a time-ordered ...
Samuele Fossati's user avatar
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Doubt on scattering amplitude in scalar Yukawa theory

I'm currently following David Tong's notes on QFT. In the section on calculating transition amplitudes using Wick's theorem, he gives an example using a scalar Yukawa theory with real scalar field $\...
Samuele Fossati's user avatar
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Vacuum expecation value of normal ordered product, Peskin/Schroeder

I'm working through Peskin/Schroeder where they treat time ordered product of fields in the interactin picture. The Wick Theorem was introduced as $$T\{\phi(x_1)\dots\phi(x_n)\}=N\{\phi(x_1)\dots\phi(...
go_science's user avatar
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Path integral derivation of exact identity for bosonic field

Let $\eta(t)$ be a non-dynamical Euclidean Gaussian bosonic field with partition function $$ Z=\int D[\eta]\exp\left(-\frac{1}{2\sigma^2}\int_0^t \mathrm{d}\tau\,\eta(\tau)^2\right) $$ so that $\left\...
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How to apply multiple Klein-Gordon operators to products of propagators?

I have the 4-point correlation function for a scalar free field $$ \langle{0} | T \phi_1 \phi_2 \phi_3 \phi_4 | 0 \rangle = -\left[ \Delta_F(x_1-x_2) \Delta_F(x_3-x_4) + \Delta_F(x_1-x_3) \Delta_F(x_2-...
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Contractions in current-current correlator for Kubo formula

To calculate the conductance $\sigma_{ij}(\mathbf{q}, \omega)$ of e.g. a disordered electron gas using the Kubo formula, one must compute the (imaginary-time-ordered) current-current correlation ...
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How to recognize Feynman diagrams from the $S$-matrix expansion?

I'm studying scattering processes in QED and one usually have to compute first of all the Scattering matrix $$\hat{S}=T\biggl (\exp\{-i\int d^{4}x:\bar{\psi}(x)\gamma_{\mu}\hat{A}^{\mu}(x)\hat{\psi}(x)...
Filippo's user avatar
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Symmetry Factor and Wicks Theorem

I have a problem with a particular kind of exercise. The question is: Consider $\phi^4$-theory with $\mathcal{L}_\text{int}=-\frac{\lambda}{4!}\phi^4$. Give the symmetry factors of the diagram and ...
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Product of normally ordered exponentials as a normal ordering of product of exponentials

I want to simplify a product of normally ordered exponentials that are in the following form $$:e^{x(\hat{a}^\dagger+\alpha_x^*)(\hat{a}+\alpha_x)}:\times :e^{y(\hat{a}^\dagger+\alpha_y^*)(\hat{a}+\...
Saurabh Shringarpure's user avatar
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Applying Wick's theorem to mass-terms in a free field theory

I'm studying a free field theory in $d$ dimension which consists of two sets of $O(N)$-scalars \begin{equation} S = \int_{\mathbb{R}^d}d^dx \left( \frac{(\partial_\mu\phi^i)^2}{2} + \frac{(\partial_\...
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Wick contraction with exponential operator in 2D CFT

Consider a 2d CFT in radial quantization. Let $A(z)$ be some primary field and $Q$ a charge that can be written as $$Q = \oint dz \, z^{(h-1)}J(z)\tag{1}$$ for some holomorphic current $J(z)$. I will ...
Oblonski's user avatar
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The vanishing of vacuum expectation value

I have some difficulty understanding why the vacuum expectation value vanishes. As illustrated in my notes, we can split the field into two parts: $$ \phi(x) = \phi^+(x) + \phi^-(x), $$ where $\phi^+(...
user174967's user avatar
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Greiner´s Field Quantization question [closed]

I upload a screenshot of Greiner´s book on QFT. I don´t understand one step. I need help understanding equation (3), what are the mathematical steps in between? Greiner, Field Quantization, page 245 (...
cmc's user avatar
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Relationship with double summing of $a_{\ell m}$

I would like to convince myself of the following relationship in an astrophysical context: \begin{aligned} & \sum_{m}\sum_{m^{\prime}}\left\langle a_{\ell m} a_{\ell m}^* a_{\ell m^{\prime}} a_{\...
guizmo133's user avatar
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Free fermion OPE

In Di Francesco's Conformal Field Theory, the propagator for the free Majorana fermion theory is given by $$ \langle{\psi(z) \psi(w)}\rangle = \dfrac{1}{2\pi g} \dfrac{1}{z-w}$$ and the energy-...
phenolphthalein's user avatar
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Kardar "Statistical Physics of Fields" expectation value identity

In Eq. (5.7) of his book "Statistical Physics of Fields", M. Kardar proposes the identity $$ \langle e^{\sum_i a_ix_i} \rangle =\exp{\left[\sum_{ij}\frac{a_ia_j}{2}\langle x_ix_j \rangle\...
CW279's user avatar
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Defining Wick/normal ordering beyond rearranging the order of annihilation and creation operators [duplicate]

Most introductory quantum field theory books define Wick ordering as rearranging a product of creation and annihilation operators such that all the creation operators appear to the left of any ...
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What justifies the interpretation of Feynman Diagrams of physical processes? [duplicate]

Feynman diagrams arise mathematically essentially as neat graphical ways of organising the terms in Wick's theorem for time-ordering. But at the same time we're supposed to interpret them as some sort ...
Jarah Fluxman's user avatar
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Wick's contractions

According to Wick's theorem, we have $${\left\langle {T\left[ {\hat A\hat B\hat C\hat D} \right]} \right\rangle _0} = {\left\langle {T\left[ {\hat A\hat B} \right]} \right\rangle _0}{\left\langle {T\...
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Normal ordering in Sine-Gordon model [duplicate]

I am studying Bosonization from Giamarchi's book (Quantum Physics in 1D), in Appendix E while doing RG analysis at second order he says (Eq. E.18) that we can NOT expand cosine directly because field $...
Barry's user avatar
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Query in the proof of Wick's theorem

I am looking at the proof of Wick's theorem in the notes here: https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qft/handouts/...
user3678252's user avatar
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Why are all the possible permutationss in the perturbative $S$-matrix calculations added together?

I have a question regarding the calculation of the $S$-matrix. During the calculation of second order term of the $S$-matrix for e.g. the Møller scattering $(e^ − + e^ − → e^ − + e^ −)$ $|i\rangle=|e^...
Ozzy's user avatar
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Wick theorem for $N$-particle Matsubara green function: equal time contraction [duplicate]

I am wondering why, e.g., the book by Mahan, "Many-particle physics", mentions that contractions in Wick theorem for the $N$-particle Matsubara Green function between a pair of operators at ...
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Wick's theorem and Feynman propagator

(this is the image from book 'No nonsense QFT' by Jakob Schwichtenberg, page no, 426) The quantity $[\phi_-(x),\phi_+(y)]$ is like an operator inside the bra-kets $\langle 0|$ and $|0\rangle$. I'm not ...
Abhinav's user avatar
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Two-point correlation function of two complex scalar fields

For a lagrangian: $$ \mathcal{L}=\partial^\mu\phi_i^*\partial_\mu\phi_i-m_i^2|\phi_i|^2+\lambda(\phi_2^3\phi_1+\text{h.c.}). $$ where summation over $i=1,2$ is understood. I am trying to find the two ...
Nitzan R's user avatar
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Spin and scale dimension of canonical spin-1/2 fields in (1+1)d

I am reading the book "Non-perturbative methods in 2 dimensional quantum field theory" by Abdalla, Abdalla and Rothe and have some questions about the Chapter 2.4 "Bosonization of ...
alibengali's user avatar
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Multi-channel mean field theory

I have always been confused about the theoretical foundation of the mean field approximation. Below I follow the book Many-body Quantum Theory in Condensed Matter Physics by Bruus and Flensberg, ...
Zhengyuan Yue's user avatar
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2 answers
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Wick contraction in quantum field theory

I am reading Anthony Zee's "Quantum Field Theory in a Nutshell" (1st edition). On page 47, when evaluating the 4-point Green's function $G_{ijkl}^{(4)}$ to order $\lambda$ using Wick ...
rioiong's user avatar
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Time-ordering operators: Can we simplify expressions inside? [duplicate]

Clearly if we have two operators $\phi(t_1)$ and $\psi(t_2)$ and define a time ordering operator $T$ acting on operators such that $$T(\phi(t_1)\psi(t_2)):=\phi(t_1)\psi(t_2),~\text{if $t_1>t_2$ ...
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Using Wick's theorem on already normal-ordered functions

The book Quantum Field Theory of Many Body systems (X. G. Wen) defines Wick's theorem as follows. I would like to employ this definition to simplify the 4-operator product $$\hat{O}\;=\;a_p^\dagger ...
Amazon Forrest's user avatar
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How one can use Wick's theorem for the product $A:\mathrel{B^{n}}:$?

I try to use Wick's theorem in the case that some products we deal with are already normal ordered. My guess is that it could be something like \begin{equation} A:\mathrel{B^{n}}:~=~:\mathrel{AB^{n}}:+...
Paweł Korzeb's user avatar
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Does normal ordering (not) conflict with canonical commutators?

Normal ordering is pretty useful to stop expressions from diverging in quantum field theory and works out perfectly fine regarding this, but there is this little problem: Consider for example an ...
Samuel Adrian Antz's user avatar
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Wick's theorem on a four-point QED Green function in 2nd-order perturbation theory

Problem 13.1 in Mandl & Shaw's QFT. I need to calculate the second order contributions to the four point Green function $$ \langle A^{\mu}(x_1)A^{\nu}(x_2)\psi(x_3)\bar{\psi}(x_4) \rangle, $$ ...
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Calculating a four-point Green function using Wick's theorem (problem 12.1 in Mandl & Shaw)

In problem 12.1 in Quantum Field Theory, Mandl & Shaw the aim is to calculate the four point green function $$ G^{\mu\nu}(x,y,z,w) = \frac{\langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S\big)...
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Wick's Theorem Example from Stoof & Gubbels [duplicate]

I've been learning about Wick's theorem from a variety of sources when I came across this example from "Ultracold Quantum Fields" by Stoof et al. on page 148 (Here I have simplified the ...
FeldsparJustChilling's user avatar
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1 answer
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Computing $\langle 0|S |0\rangle$ in $\phi^4$ theory [closed]

$\newcommand{\bra}[1]{\langle #1|}$ $\newcommand{\ket}[1]{|#1\rangle}$ I have been reading David Tong's QFT notes. As part of an exercise, I am asked to examine $\bra{0} S \ket{0}$ to order $\lambda^2$...
Ando Bando's user avatar
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Normal ordered product of $4$ scalar fields $X^\mu$

I'm trying to get more familiarity with the conformal normal ordering used in Polchinski's String Theory vol. 1 and I'm currently trying to solve problem $2.2$ which asks to prove that the normal ...
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