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Questions tagged [correlation-functions]

A correlation function is a statistical correlation between random variables at two different points in space, time, or other parameter space, usually as a function of the variable distance between these points. In QFT, field autocorrelation functions are propagators, so use the "propagator" tag, instead.

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Two-point correlation function of a scalar field $\langle 0 | \phi(x) \phi(0)| 0 \rangle$

I'm trying to find the two point correlation function for a massless scalar field obeying $\square \phi = 0$. I can write $$\langle 0 | \phi(x) \phi(0)| 0 \rangle = \int \frac{d^dk}{(2\pi)^d} \delta(...
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Assumptions behind Ornstein-Zernike correlation function

Let $S(\mathbf q)$ be come correlation function in Fourier space ($\mathbf q$ = wavevector). In the study of condensed matter systems, I have often encountered the statements that a reasonable form ...
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Four point function with complex momenta?

It is well known that the four-point function $$\int_{\mathbb{R}^{3,1}}\frac {d^4 q}{((q+p_1)^2-i\epsilon)((q+p_2)^2-i\epsilon)((q+p_3)^2-i\epsilon)((q+p_4)^2-i\epsilon)}$$ can be computed using the ...
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What does Lorentz index structure say about a full-fledged correlator?

I have a probably dumb question. Consider the following position space correlation function in a YM-theory (with or without matter fields): $$f_{\mu_1\cdots \mu_n}^{a_1\cdots a_n}(x_1,\ldots,x_n)=\...
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Bond order correlation function

I am trying to compute the bond order correlation function, $g_6$. It is defined based on the bond order parameter: $$\psi_6(x_i) = \frac{1}{N_i}\sum_{i=1}^{N_i}{\exp(i6\theta_i^j)}$$ where $\theta_i^...
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Different Schwinger-Dyson Equations

In the literature on QFT there are a lot of different equations that are all called "Schwinger-Dyson equation" so I wanted to know how are they related and if they have proper names. The first ...
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What substitutions are allowed within time-ordered products?

I always thought of the time-ordering in QFTs as an explicit operation. Meaning the time-ordering "operator" just takes everything I write inside it and shuffles the operators around until they are in ...
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LSZ reduction formula for massive vector bosons

What is the precise form of the LSZ reduction formula for massive vector bosons? The LSZ formula for scalar bosons, fermions, and photons is given e.g. in the textbook "Quantum field theory" by ...
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Path integral and Out-of-time-ordered (OTOC) correlator

A simple observation that any insertions within the path integral are classical variables (Not operators) and hence, objects inside the path integral "commute" (is symmetric under exchange). Hence, ...
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What are three-point functions?

I came across this term while I was trying to read this paper related CFT correlators. Can some please take some time out to explain what does it mean in general?
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Contradicting definitions of time-correlation functions

The correlation of two quantities $A(t)$ and $B(t)$ is usually given as $$\left\langle A(t)B(t')\right\rangle,$$ not specifying what one is supposed to integrate over. My first guess would be to ...
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A confusing point in linear response theory on the ground state

Information about a quantum system could be drawn from its response to a small perturbation. This is formulated in what is known as linear response theory. In second quantization, consider a ...
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Calculation of time-ordered propagations and correlators

I am reading the following paper M. H. S. Amin and D. V. Averin, “Macroscopic Resonant Tunneling in the Presence of Low Frequency Noise,” Phys. Rev. Lett., vol. 100, no. 19, p. 197001, May 2008. I ...
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Bound on Quantum Chaos

I am currently reading the paper A Bound on Chaos. In this paper, they evaluate the quantity C(t), which is an out-of-time-order correlator (OTOC), and use very clever arguments to show that there ...
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How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate

If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean? When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, ...
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Tricky 3 particle distance correlation question for quantum mechanics

Three particles are detected/placed/focused at position $x=y=z=t=0$. (So that according to quantum mechanics their momentum/energy are completely unknown). They are non-interacting Fermions each with ...
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Faddeev-Popov determinant and topology of the worldline

I am studying the path integral quantization of relativistic particles, using the BRST quantization method. I have to compute the integral \begin{equation} Z\sim \int Dx \det(\partial_\tau)e^{-\int_0^...
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How to understand the two-point correlation function in momentum space?

Let's take the Ising model as an example and study the two point spin spin correlation function: $$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{...
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A puzzle about Green's function and S-matrix

From the discussion in some posts example1, example2, we know that the S-matrix is the residue of the corresponding Green's function. On the other hand, S-matix is a physical observable in QFT, but ...
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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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Closed set of operators under renormalization

While reading the article http://inspirehep.net/record/61135, I came across the concept of "closed set under renormalization". The definition they give is the following. In any renormalizable field ...
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Deriving Ward identity directly from a given formula for the conserved current only using the equal-time canonical commutation relation

I have a very technical question on deriving a Ward identity directly from a given explicit form of the "conserved current". Let me emphasize that I do not start with an apriori knowledge on the ...
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Constraints on correlation functions of Quasi Primary Fields

I have problems understanding constraints on correlation functions of quasi primary fields (QPF) following DiFrancesco's Conformal field theory book. In chapter 4, section 4.2.1, a QFP is defined as a ...
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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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Feynman Rules for Two Different Interacting Fields

I'm currently studying how to deduce Feynman rules for general theories, and I've managed to deduce them for $\phi^3$ and $\phi^4$ theories. Up to this point I've considered the same field for all ...
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Electron correlation - difference between correlation and dependence

When we talk about electron correlation in condensed matter physics or chemical physics, we usually refer to the fact that the pair-density $$ P(r,r') = N(N-1) \int |\psi(r,r',r_3,...,r_N)|^2 \; \...
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Obtaining real-space correlations from reciprocal space correlations

Consider a system of Ising variable $s = \pm 1$ on a rectangular lattice which has open boundary conditions on the top and bottom and periodic boundary conditions to the left and right. In other words,...
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How to calculate correlation functions for fermionic operators?

In the paper by Peschel (2003) https://arxiv.org/pdf/cond-mat/0212631.pdf How does one derive the following relation: $$ \langle c_{n}^\dagger c_{m}^\dagger c_{k}c_{l}\rangle = \langle c_{n}^\...
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Kosterlitz-Thouless transition and correlation function

I’m studying Kosterlitz transition on this book: https://tinymachines.weebly.com/uploads/5/1/8/8/51885267/kardar._statistical_physics_of_fields__2007_.pdf#page173 . At page 165 it says:” The gradient ...
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What is the link between statistical and QFT correlation functions?

I'm studying statistical mechanics in particular correlation function: https://en.wikipedia.org/wiki/Correlation_function_(statistical_mechanics) and I have understood it. Now searching on internet ...
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Simple reason why the correlator of a vector with two identical scalars vanishes?

In scalar QED, the photon interacts with a charged scalar and the three point function of a vector, scalar and scalar bar is nonzero. I remember an argument that very simply proved that if you try ...
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Brownian motion from two gaussian noise processes

Consider some brownian motion for which we obtained the following solution for the langevin equations $$ u\left(t\right)=e^{-\alpha t}\int_{0}^{t}e^{\alpha s}\left(\xi\left(s\right)-\xi'\left(s\right)...
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Truncated $N$-Point Functions

In Quantum Field Theory, truncated N-Point functions (or truncated Green's functions) are the N-Point functions of diagrams with their external legs chopped off. I was told that the truncated N-Point ...
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How can a Dirac delta function that does not occur under an integral be used to describe a transition rate?

In his excellent notes (found here), Mark Tuckerman shows that the transition rate of absorption between quantum states i and f, coupled by operator B, can be expressed as the fourier transform of the ...
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Density density correlations of a simple Brownian particle [closed]

Suppose, I have a particle satisfying the equation \begin{equation} \frac{dX}{dt}=\eta(t) \end{equation} Where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$. I can now define a density like $\rho(x,...
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One-point correlation function of any quasi-primary operator is zero

How do I show the following: In a general CFT (with no boundary), the one-point correlation function of any primary operator $A_Δ$ is $0$ (unless we are talking about the identity operator): $$<...
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Equipartition and correlations

This question references C.L.Henley's paper on arxiv. Page 3, section B: Effective free energy and correlations. There is an ice polarization field $\mathbf{P}(\mathbf{r})$, that has been coarse ...
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Two Time Correlation function calculated from Born rule

Update Below I'm having a hard time reconciling two different calculations of the quantum two time correlation function. Consider quantum operator $A$ with eigenvectors $\{|\phi_i\rangle\}$ and ...
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Is there any difference between radial distribution function and pair correlation function?

I have understood the part that for solids pair correlation function is the measure of the probability of finding the center of another particle in the neighborhood of a given particle. On the other ...
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Current-current correlator in superconductivity

I was reading a paper on superconductivity. There is a claim that if the momentum space current-current correlation function $\langle\mathcal{T} j_{\mu}(k)j_{\nu}(-k) \rangle$ has a pole at zero ...
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What's The 2pt Correlation Function For The Spin Fields For The 3D Critical Ising Model?

The title it self explanatory. What's The 2pt Correlation Function For The Spin Fields For The 3D Ising Model? I know the form of the four point function and have worked out how to express it in terms ...
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statistical errors associated to Monte Carlo sampling

I have $n$ successive observation $A_\mu $ of a quantity $A$ and I need to understand how the expectation values of the square of the statistical error depends from the autocorrelation time but a ...
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correlation function in Fourier space

I'm reading this paper and want to prove eq (8): The field $\psi(\mathbf{x}) \in \mathbb{C}$ exists in a finite periodic 2D square box (of side length $L$), and has a Fourier series expansion, and ...
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“Contraction Property of thermal density matrix” in the Maldacena's paper of A Bound of Chaos

In the paper, https://arxiv.org/abs/1503.01409 (Maldacena, et al. “A Bound on Chaos.”) in equation (24), the authors write an inequality, $$ Tr( y^{1+\eta} V y^{3-\eta} V ) \leq Tr(y V y^2 V) $$Where $...
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How to understand that correlators measure physical correlation?

Background In physics, we always come across $n$-point correlators (e.g. 2-point correlators). For instance, in phase transitions, one is interested in finding the correlation function between order ...
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Help with Correlation/Green's Function of Rotated Variables (Keldysh Rotation)

I'm working through this paper, and have encountered "a little algebra shows that...", yet I'm not familiar enough with the topic at hand to figure this out. Here is the paper: https://arxiv.org/abs/...
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Shear strain correlations from displacement Fourier transform

Currently working with molecular dynamics simulations, I would like to compute shear strain correlations in my 2-dimensional system. How I used to do things Accumulated shear strain at position $\...
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Are correlation functions in CFTs always single-valued functions?

I am wondering whether correlation functions e.g. in 2D CFTs such as $$\langle \phi\phi\phi\rangle=C\frac{1}{z_{12}^{h_1+h_2-h_3}z_{23}^{h_2+h_3-h_1}z_{31}^{h_3+h_1-h_2}}\frac{1}{\bar z_{12}^{\bar ...
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Simplifying CMB correlation function with spherical harmonics

I originally asked this on the physics Stack Exchange site, but perhaps it could be more easily answered here. Given the definition of the correlation function for CMB temperature fluctuations as $$ ...
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How are correlation length and cluster size related in the 2D Ising model?

What is the relationship between correlation length and cluster size? Does the correlation length give the average cluster size, or is the cluster size something different?