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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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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How to calculate the invariant amplitude for a decay process using wick's theorem?

I have difficulties to apply Wick's theorem to the following problem-set: We have three free scalar fields $\phi_1, \phi_2, \phi_3$. While the field $\phi_3$ has the mass $M$ while $\phi_1$ and $\...
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Ordering ambiguity in the Feynman propagators obtained using Wick's theorem

Applying Wick's theorem to a string of four field operators, $\phi_a\equiv\phi(x_a)$: $$T(\phi_1\phi_2\phi_3\phi_4)=\{...\}, \tag{1}$$ we obtain several terms, three of which are fully contracted ...
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Correlation functions - Polchinski equation 6.2.18

At some point of Polchinski book, we are interested in calculate the following correlation function: $$\left\langle \prod_{j=1}^n[e^{ik_i\cdot X(z_i,\bar{z}_i)}]_r\prod_{j=1}^p\partial X^{\mu_j}(z_j')...
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OPEs of exponentials

Given a holomorphic field $H(z)$ with OPE: $$H(z)H(0)\sim -\ln z$$ What is the most smart way to calculate the OPE's of the exponential operators $e^{\pm iH(z)}$, given as follows? $$e^{iH(z)}e^{-iH(0)...
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How do disentangling and reordering of exponential operators work?

I have seen in several sources that by invoking Lie groups, $$e^{\alpha_1 g_1+\alpha_2 g_2 + \dots} = e^{\beta_1 g_1}e^{\beta_2 g_2}\dots $$ where $g_i$ are elements of a Lie algbera. For example, ...
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Where does this formula for normal ordering in QFT come from? [duplicate]

On this Wikipedia page you can find the following equation for free fields $$:\phi(x)\chi(y):=\phi(x)\chi(y)-\langle0| \phi(x)\chi(y)|0\rangle\tag{1}$$ But I don't understand where this comes from ...
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Summing over disconnected diagrams - Peskin and Schroeder

In Peskin & Schroeder, page 97, the following expression is given as part of the demonstration of how the $n$-point correlation function is calculated using connected diagrams: $$\sum_{\text{...
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Disconnected 2nd order terms in 2-point correlation function in $\phi^3$ theory

I'm trying to figure out a detail on the calculation of correlation functions in the $\phi^3$ theory. So, I know we can calculate a 2-point correlation function as: $$G(x_1, x_2)=\frac{\langle0|\...
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Rearrangement lemma in normal ordering

I try to calculate energy-momentum tensor from sugawara construction of wakimoto representation for SU(2)K current in 2d CFT. but at first I have to understand rearrangement lemma. so can anybody ...
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Which Wick contractions are allowed?

TLDR: Given a Lagragian $\mathcal{L}$ depending on some fields $\{\phi_a\}$, which contractions between the fields are permissible? Example to illustrate my problem Consider the following Lagragian $...
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Wick theorem exercise

I'm a newbie in QFT and I have some doubts with this simple exercise: Using the Wick Theorem evaluate $$\langle0|T(\phi^4(x)\phi^4(y)|0\rangle$$ Use a diagrammatic approach to represent the possible ...
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Wick's theorem contributions to nucleon scattering

I am following David Tong's notes on QFT. On page 58, he applies Wick's theorem to $\psi\psi\rightarrow\psi\psi$ scattering for a scalar field with a Yukwara interaction term. The claim is, that by ...
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OPEs of vertex operators

Suppose we have two chiral bosons $a(z)$ and $b(z)$ with operator product expansions (OPEs), $$a(z)a(w) = \log(z-w), \quad b(z)b(w) = -\log(z-w)$$ as well as $a(z)b(w)=0$, including only singular ...
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Wick contractions and antisymmetric differential operator

The problem I'm trying to rederive Equation S6 in the supplementary of the article PRL 114, 126602 (2015) - ''Spin pumping with spin-orbit coupling''. The starting point is the imaginary time retarded ...
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Evaluating expectation value in second quantization using Wick's theorem

I am working in the context of quantum chemistry. Given this expectation value in second quantization $$ \langle 0 \vert [p^\dagger q, \kappa] \vert 0 \rangle $$ with $$ \kappa = \sum_{ai} k_{ai}(a^...
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Why does Peskin and Schroeder move normal ordering move outside a commutator?

The equation trying to prove that Wick's theorem by induction in P&S on page 90 implies that normal ordering can be moved outside a commutator (at least with a positive frequency field), which I ...
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Is the symmetry factor of a Majorana fermion doubled compared to Dirac fermions?

I think the title says it all. If I have to contract e.g. an expression of the following form: $$\left\langle (...)\ \bar{\psi}(x_1)\ \psi(x_1)\ (...)\ \bar{\psi}(x_2)\ \psi(x_2)\ (...) \right\rangle,...
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Am I reading this Feynman diagram in $\mathcal{N}=4$ SYM right?

I would like to find the expression corresponding to the scalar-gluon sunset diagram in position space, for Euclidean $\mathcal{N}=4$ Super Yang-Mills theory. The diagram is the following: where the ...
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Wick contraction for unperturbative fields

Say I want to calculate the transition amplitude of a meson to 2 photons using QED only: $$\langle \gamma(q1) \gamma(q2)|meson\rangle$$ I can start with normal perturbation theory: $$\langle \gamma(...
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Scalar self-energy in $\mathcal{N}=4$ SYM in position space

In this paper (BMN Correlators and Operator Mixing in $\mathcal{N}=4$ Super Yang-Mills Theory, 2002), the authors give the following expression for the self-energy of the scalar propagator in $\...
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Correlation function of single annihilation/creation operator vanishes

I could not find anything on that on google, or here on physics stack exchange, which surprises me. My problem is, that I do not see, why exactly $<a> = <a^{\dagger}> = 0$ where <...> ...
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Unwarranted assumption of linearity of Normal Order

In the book "Advanced Quantum Mechanics" by Franz Schwabl on page 280 you can find a formula (13.1.18c) which assumes linearity of time ordering operator $::$ $ :\phi\left(x\right)\phi\left(y\right):=...
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Two questions to Wick's theorem: particle scattering vs. free field & evaluating a contraction

During a read in a self-study book about many-body physics, I came across Wick's theorem. There were two questions arising before I could start to grasp the idea of Wick's theorem. In the hope of ...
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Time ordering of Normal ordered products in Wick's theorem

I have a small doubt regarding wick's theorem. Is it normal ordered products are time ordered? Actually in wick's theorem we usually don't write the symbol of time ordering in front of normal ...
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Why don't we lose generality when we assume $z^0 > x^0 > y^0$ in the proof of Wick's theorem?

In proof's of Wick's theorem it's typically stated (see e.g. page 87 in Coleman's notes) that it's sufficient to consider just one possible time-ordering. For example for the product $T(\phi(z)\phi(x)...
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Rigorous formulation of Wick theorem at finite temperature

Consider a quadratic Hamiltonian in second quantization. For simplicity, let $H = \sum_{n =0}^\infty \epsilon_n c_n^* c_n$ where $\epsilon_n$ may satisfy some conditions. Let the average be $$ \langle ...
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Computing correlation function $\langle e^{i\beta \phi(x)}e^{-i\beta\phi(0)}\rangle$ for massless scalar field $\phi$

I am currently reading Shankar's "Bosonization: How to make it work for you in condensed matter" (http://inspirehep.net/record/408901/). In page 9, I am stuck with computing the correlation function ...
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How one goes to derive equation (2.4.19) from Polchinski's String Theory Volume 1?

Equation (2.4.19) states that an exponential times a general product of derivatives $$:(\Pi_i \partial^{m_i}X^{\mu_i})(\Pi_j \bar{\partial}^{n_j}X^{\nu_j})\exp(i k \cdot X ):\tag{2.4.18}$$ has weight ...
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Why in Wick theorem the expectation values of products of field operators is equal to the sum of expectation values of pairs of operators?

As stated in the title, I understand the expectation value of uncontracted operators are zero, and there are only certain type of pairs of operators remaining in the time ordered product of operators. ...
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The overlap of two Gaussian states

According to e.g. Serafini (Quantum Continuous Variables), the Hilbert-Schmidt product ('overlap') of two multimode Gaussian states $\rho_1,\rho_2$ is $$\text{Tr}[\rho_1\rho_2]=|\langle\psi_1|\psi_2\...
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Generalization of Wick's Theorem

Wick's theorem allow us to write a time-ordering of creation and annihilation operators as a normal-ordering of contractions of these operators. I am studying a system that consists of two kinds of ...
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How to obtain time-ordered density correlation function of free Bosonic system via Wick's theorem?

Consider a free Bosonic system. The Hamiltonian is given by $$ H=\sum_k \frac{k^2}{2m}a_k^\dagger a_k. $$ Since the spectrum is gapless, the ground state can be of any particle number (or even ...
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What happens if I Wick contract a trace operator internally?

In theories such as $\cal{N}=4$ supersymmetric Yang-Mills, we often consider operators such as $\cal{O}(x_1)=$Tr$(\phi(x_1)\phi(x_1))$, with $\phi$ the scalar field(s) of the theory. Then we go on ...
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Is there a program or a website able to perform all Wick contractions for a given expression?

Imagine I have an expression of the type: $$\langle \phi_{x_1} \phi_{x_1} \phi_{x_2} \phi_{x_2} \phi_{z_1} \phi_{z_1} A_{z_1} \phi_{z_2} \phi_{z_2} A_{z_2} \phi_{z_3} \phi_{z_3} A_{z_3} \phi_{z_4} \...
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Perturbation expansion with path integrals

This is from Hugh Osborn's 'Advanced Quantum Field Theory' notes, Lent 2013, page 15. I want to evaluate the expression $$ Z = \exp\Big(\frac{1}{2} \frac{\partial}{\partial \underline{x}} . A^{-1} \...
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Factor of $1/2$ in $TT$-OPE [closed]

I'm trying to calculate the TT OPE in a bosonic theory. I'm missing a factor of 1/2 in the least-singular term. We have (following Di Francesco) $$\langle \partial \phi(z) \partial \phi(0) \rangle = \...