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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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Vanishing correlation function

Mirror Symmetry p. 206, Eq. 10.192. I have an operator $\mathcal{O}$ that commutes with my supercharge $\overline{Q}_+ $, $\left[\overline{Q}_+, \mathcal{O} \right]=0$. Why does the correlation ...
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Is there correlation in these pictures? [closed]

Is correlation in these pictures please? Thank you.
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How to derive equation (N.15) in Ashcroft and Mermin's Solid State Physics?

They state in their book on page 792 the following: It can be proved, however, that if $A$ and $B$ are operators linear in the $u(R)$ and $P(R)$ of a harmonic crystal, then: $$\langle e^A e^B \...
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Correlation function at zero distance

I'm confused about the definition of the correlation function (at equal time). I know it is defined from the thermal average of the scalar product of two random variables (for example the spins of a ...
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Does somebody know where to find original paper of Lehmann, Symanzik and Zimmerman translated in English?

does anybody know where to find original paper about LSZ reduction translated in english? unfortunantely, Ive found only original German article. H. Lehmann, K. Symanzik, and W. Zimmerman, "Zur ...
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Spinwaves, Mermin-Wagner theorem, Two-point correlation function and Heisenberg model

I was looking at the Mermin-Wagner theorem (as following the previous question) and the Heisenberg model seems to be presented, and they split the Hamiltonian $H$ in the matrix or vector n-components ...
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Expectation value of descendant fields

I'm trying to calculate the following quantity: $ \left<(L_{-1}\phi)(w_1)(L_{-1}\phi)(w_2) \ldots (L_{-1}\phi)(w_N) \right>$ where $\phi(w_i)$ is a primary operator and $L_{-1}$ is the ...
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Quantum field theory: corrections to excited state correlation functions

I want to know how to calculate the lowest-order-in-the-coupling-constant correction to $$M(x, y,k,p)=\langle k|\phi(x)\phi(y)|p\rangle$$ in $\phi^4$ scalar field theory in a relatively general ...
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Is the two-point correlaction function equivalent to a sum of two-point correlaction function?

I'm doing a research and I came across an expression that to move forward, since $<\varphi(x)^2> = \left.\frac{\delta^2 W(j)}{\delta j(x)^2}\right|_{j=0}$, I have to assume that $<\varphi(x)...
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Asymptotic LSZ reduction formula (Peskin & Schroeder)

Peskin & Schroeder, An Introduction to Quantum Field Theory, write at page 224 $$\int d^{4} x e^{i p \cdot x}\left\langle\Omega\left|T\left\{\phi(x) \phi\left(z_{1}\right) \cdots\right\}\right| ...
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Relation between standard and Kubo-transformed quantum correlations

Via path integral molecular dynamics it is possible to measure the Kubo transformed correlation function between two operators $\hat A$ and $\hat B$ \begin{equation*} K_{\hat A\hat B} = \frac 1 {Z_\...
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How do I fit a resonance curve?

In an experiment, I collected data points $ (ω,υ(ω))$ that are modeled by the equation: $$ υ(ω)=\frac{ωC}{\sqrt{(ω^2-ω_0^2)^2+γ^2ω^2}} \,.$$ How can do I fit the data to the above correlation? And ...
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Linear response treatment of the magnetization of a system of noninteracting fermions

While trying to solve an exercise, I ran into what looks like a contradiction. I'm sure I'm making some kind of mistake, but I couldn't spot it. I'm not asking for help in solving the exercise, which ...
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Error in histogram measurements

I ran into the following statement here and here but I believe it's more general. Let's suppose we're running a simulation of a system and we are interested in the distribution of a quantity (say $M$...
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Calculation of current from path integral

I would like to calculate $\langle\bar{\psi}\psi\rangle$ in free theory. I start from the following generating functional: $$Z[J]=\int\mathcal{D}[\bar{\psi},\,\psi]\exp\left(i\int d^dx\,[\bar{\psi}(i\...
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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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369 views

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,...