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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Correlations in the Hellings-Downs curve for the NANOGrav 15-year dataset
I'm trying to reproduce the Figure 1c from the paper "The NANOGrav 15-year Data Set: Evidence for a Gravitational-Wave Background". This graph corresponds to pulsar pair correlations of the ...
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How are point-dependent concepts such as correlation functions, renormalization, cluster decomposition...etc interpreted in axiomatic QFT?
In traditional quantum field theory we often speak about things happening at a point. For example, the correlation function
$$\langle 0 | \phi(x) \phi(y)\ | 0\rangle \tag{1}$$
can be thought of the ...
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How was the critical exponents related to the scaling dimensions of the local operators?
On "The Conformal Bootstrap: Theory, Numerical Techniques, and Applications"(arXiv:1805.04405
) by David Poland, Slava Rychkov, Alessandro Vichi page 5
Consider for example the critical ...
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Partial gauge fixing a point to infinity in conformal field theory [duplicate]
While deriving the structure of a 4 pt. function in CFT, we write the conformal block with respect to the cross ratios
$$ u = \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \; v = \frac{x_{14}^2x_{23}^2}{...
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Evaluation of Hadamard function for a free particle
I'm currently using Fulling's book (Aspects of Quantum Field Theory in Curved Spacetime) to study the Hadamard state and Hadamard's functions. In chapter 4, he shows that for a free particle one ...
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Autocorrelation functions - haven ratio
I'm studying Haven coefficients (I created similar question but with MSD). Right now I have question referring to calculating haven ratio using velocity-velocity autocorrelation functions. What I have ...
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Gauge Fixing in Derivation of Lorentzian OPE Inversion Formula in 2D CFT
I have been looking through the following article: https://arxiv.org/abs/1711.03816 and wish to understand the derivation from scratch. The definition of conformal partial waves and the object of ...
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Calculating charge diffusion coefficient from Green Kubo relation
I'm trying to calculate Haven coefficient in some crystal that I have assembled coordinates of N ions of oxygen in k time steps. I looked up equations in this article: Pressure-Dependent Diffusion ...
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What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?
For background, I'm primarily a mathematics student, studying geometric Langlands and related areas. I've recently been trying to catch up on the vast amount of physics knowledge I'm lacking, but I've ...
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Velocity correlations of different particles
In section 13.3.2 of Statistical Mechanics: Theory and Molecular Simulation by Mark E. Tuckerman, the author derives the Green-Kubo relations for the diffusion constant. In the derivation, he makes ...
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Renormalization group equation and method of characteristics
All of this question refers to ref. 1. The equation are numbered alike.
The author claims to solve a renormalization group (RG) equation using the Method of characteristics, but there is a passage ...
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How do we interpret disconnected diagrams in scattering theory?
It is apparent that disconnected diagram contributes additional delta functions to the corresponding matrix element. For example, we consider the scalar $\phi^3$ theory and the following $2\...
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Propagator for a massless scalar field in $d$-dimensional spacetime [closed]
I'm trying to show that for a free massless scalar field, the 2-point correlation function in $d$-dimensional spacetime has the following form:
$$<\phi(x)\phi(y)> = \int \frac{d^d{p}}{(2\pi)^d}\...
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Description about cancellation of bubble diagrams while computing correlation function by M. Schwartz
I'm trying to understand M.Schwartz's description on his own QFT & SM book, which is about cancellation of disconnected diagrams so called bubbles when we compute two point correlation function ...
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Graphs about four-point Feynman diagrams for $N$ scalar fields
Consider following Lagrangian for $N$ scalar fields $\phi^a, a=1, \ldots, N$ :
$$
L=\frac{1}{2} \partial_\mu \phi^a \partial^\mu \phi^a-\frac{1}{2} \mu_0^2 \phi^a \phi^a-\frac{1}{8} \lambda_0\left(\...
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Peskin and Schroeder QFT Eq.(12.66), the Renormalization Group equation
I am troubled for the derivation of Eq.$(12.66)$ on Peskin and Schroeder's QFT book.
$$
\left[p \frac{\partial}{\partial p}-\beta(\lambda) \frac{\partial}{\partial \lambda}+2-2 \gamma(\lambda)\right] ...
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Coherence, Correlation and causality in Quantum Field Theory
I've started to study in details quantum optics and I find difficulties in linking the concepts of coherence and correlation among fields, especially because I'm building right now a background on ...
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LSZ Reduction formula: Peskin and Schroeder
On page 224 of P&S, they have the following expression (7.36),
The integral over $d^3q$ gives us all the $q \to p$, then the integral over $dx^0$ is computed. The RHS given matches when only the $...
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The Callan-Symanzik equation on Peskin & Schroeder's QFT
On Peskin & Schroeder's QFT page 411, the Callan-Symanzik (CS) equation reads
$$
\left[M \frac{\partial}{\partial M}+\beta(\lambda) \frac{\partial}{\partial \lambda}+n \gamma(\lambda)\right] G^{(n)...
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confusing when Hermitian conjugation involve with time ordering
I encounter a problem when I try to calculate the Hermitian conjugation about a correlation function. For example the propagator of free real scale field
$$\tag{1}\langle0|T\{\phi(x_1)\phi(x_2)\}|0\...
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Relationship between anti-commutators and correlation
Ballentine (in his solution at the back of the book to his Problem 8.10) writes that
$$[Tr(\rho \{A,B\}/2)]^2$$
is related to the correlation between the observables represented by $A,B$, but gives no ...
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Massless limit of Dirac fermion correlation functions
In the 2D massless Dirac fermion CFT we have correlation functions like
$$\langle J(z,\bar{z})J(0)\rangle \sim \frac{1}{z^2},$$
where in terms of real Euclidean coordinates $x^0,x^1$, we have $z=x^0+...
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What are the conformal Ward identities associated with the correlation functions of the stress-energy tensor?
I found two papers on this matter, but am having trouble parsing the answer from either of them.
https://arxiv.org/abs/1911.05359
https://arxiv.org/abs/2108.06767
For that matter, what even are the ...
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Determination of Intermediate Scattering Function in numerical simulation
The Fourier transform of single particle density :
$$\rho(\textbf{k},t)=\int\rho(\textbf r,t)\exp(-i\textbf k.\textbf r)d\textbf r=\sum_j \exp(-i\textbf k.\textbf r(t))$$
The Intermediate Scattering ...
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Understanding Classical Contributions to the Quantum Field Theory Path Integral
I'm a data scientist with no physics background who sometimes reads about physics in my spare time, so please take it easy on me, I know these are really obvious questions to physicists but feel free ...
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Mutual information between energy levels
I have the following partition function
$$Z(\beta_1,\beta_2) = Z(\beta_1)Z(\beta_2) +Z(\beta_1,\beta_2)_c$$
Thus, I have a non-zero mutual information $I(Z(\beta_1);Z(\beta_2)) = S(Z(\beta_1))+S(Z(\...
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Functional derivative of the generating functional with respect to the source term
To be specific, let us use the notations in W. Metzner et al., Rev. Mod. Phys. 84,299 (2012).
The generating functional $G[\eta, \bar\eta]$ is given by [Eq. 4]
\begin{align}
G[\eta, \bar\eta] = - \ln \...
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Exact form of two-time correlators in out of equilibrium evolution from factorized distributions
Suppose that the have an initial product state at some time $t=0$ written in a computational basis $|0\rangle,|1\rangle$, for instance the state $|1011010\rangle$. The associated density matrix $\...
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Correlation function of tachyon vertex operators
In Polchinski's first volume on String Theory, it is claimed in exercise 2.3 (or analogously in eq.(6.2.17)) that
$$ \Bigl\langle \prod_{i=1}^N : e^{i k_i \cdot X_i(z_i,\bar{z}_i)} : \Bigr \rangle = i ...
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Relation between power dissipation and imag part of susceptibility [duplicate]
I am trying to understand the following relation between power dissipation and the imaginary part of the susceptibility, from Sethna's Statistical Mechanics textbook. Why does the integral equal the ...
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How an autocorrelation function is calculated experimentally in a Dynamic Light Scattering experiment
As said in the title, I would like to know how an autocorrelation function of scattered light intensities defined as $G(\tau)=\langle I(t)I(t+\tau)\rangle$ is computed experimentally. Let's say we ...
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Symmetrical Delayed Choice Quantum Eraser
This is a modification to a Kim et al.-type quantum eraser experiment. According to the research paper which proposed this modified experiment,
This modification makes the similarity to Bell-type ...
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What does the two-point correlator of the energy-momentum tensor mean (clarification)?
I had a final project in a QFT 2 course on c-theorems in QFT where quantities such as $\langle T_{zz}(x)T_{zz}(0)\rangle$ where $T_{zz}$ is the energy-momentum tensor in holomorphic coordinates, was ...
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Finding initial conditions from the temperature autocorrelation function
So I was reading Mukhanov and in section 9.4 titled Correlation functions and multipoles, he talks about obtaining the auto-correlation function
\begin{equation}
C(\theta) = \bigg\langle \frac{\delta ...
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General validity of the Linked Cluster Theorem in QFT for arbitrary correlation functions
In QFT, either at zero or finite temperature, the Linked Cluster Theorem (LCT) ensures that all disconnected diagrams appearing in the numerator of the interacting Green's function exactly cancel with ...
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Most general form of the velocity correlation tensor
The most general form of velocity correlation tensor for homogeneous and isotropic turbulence is given by:
$R_{ij}(r)=A(r)r_{i}r_{j}+B(r)\delta_{ij}$
How can this depend on $r_{i}r_{j}$ if it is ...
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Why does the one-point closed string tachyon amplitude on the disk vanish at tree-level?
I would like to understand the amplitude of the coupling between $N$ open strings and one closed string at tree-level.
I know this coupling takes place in the disk topology.
Why does the amplitude ...
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Green function power-law behavior in real and momentum space
Given a Green's function with pawer-law behavior in $k$-space $g(k)\sim\frac{1}{k^a}$ (at least for small $k$), what is the asymptotic form for $g(x)$ in the real space?
In the paragraph above Eq. (56)...
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Propagator from free scalar path integral
Let $\langle{0|}T\phi(x)\phi(y)|0\rangle$ be the vacuum expectation value of the 2 point correlator for the free scalar field. Page six in these notes say that we can calculate this correlator by ...
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Question in Linear response theory
I am currently trying to do a calculation using linear response, but I am confused on one thing I never thought of. To start here is the relevant equation (I assumed the operators are self evident to ...
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How to check Virasoro symmetry at the level of four-point functions?
In a Conformal Field Theory in arbitrary dimensions, four-point functions are constrained to take the form
$$\langle O_1(x_1)O_2(x_2)O_3(x_3)O_4(x_4)\rangle = K(x_i)F(z,\bar z)$$
where $K(x_i)$ is a ...
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Holographic derivation of correlations functions in $\text{AdS}_3$/$\text{CFT}_2$
Intro
I am interested in finding the correlations functions $\langle A^{a_1}_z(z_1) \cdots A^{a_k}_z(z_k) \rangle$ in the frame of $\text{AdS}_3$/$\text{CFT}_2$ correspondance for 3D gravity and two ...
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Correlators $\langle \psi(z_1) \cdots \psi(z_N) \sigma(w_1) \sigma(w_2) \rangle$ in Ising conformal field theory
Question: How one obtains
$$\langle \psi(z_1) \cdots \psi(z_N) \sigma(w_1) \sigma(w_2) \rangle \sim \mathrm{Pf}\left( \frac{f(z_i,z_j; w_1, w_2)}{z_i-z_j}\right) \prod_{i=1}^N (z_i-w_1)^{-1/2} (z_i-...
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On the renormalization condition of the QED vertex function
In any renormalizable theory, we exploit the so-called renormalization conditions to determine the finite part of the counter-term couplings (namely $Z_1, Z_2$, etc in QED). In the on-shell scheme, we ...
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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_{\...
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Transition amplitude between field configurations from the path integral
In the path integral formulation of QFT, we should in principle be able to calculate the transition amplitude from a classical field configuration $\phi_{in}(x,t=0)$ to $\phi_{out}(x,t=T)$ using the ...
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How do I numerically compute the interquark potential from the correlation function of Polyakov Loops?
I know that the potential can be calculated in the following way:
$$
aV(r) =-\ln(<\sum_{\textbf{x}} (P(\textbf{x}+R)P^{\dagger}(\textbf{x}))>)/N_T.
$$
Now, suppone I have some procudure to ...
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Numerator of Electron Vertex Function in Peskin
In Peskin Chapter $6.3$ equation $6.38$ the integral numerator exhibit follwing expression:
$$
-ig_{\nu \rho}\bar{u}(p')(-ie\gamma^{\nu})i(\displaystyle{\not}k'+m)\gamma^{\\\mu}i(\displaystyle{\not}k+...
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OPE coefficients and three-point function [duplicate]
I am trying to prove that the OPE coefficients $C_{ij}^k$ are the same as the normalization of the three-point fuction $\lambda_{ijk}$.
$$\frac{\lambda_{ijk}}{|x_{12}|^{\Delta_i+\Delta_j-\Delta_k}|x_{...
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