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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110 views

On the interpretation of Feynman diagrams

I am currently trying to find out about what exactly Feynman diagrams are, and up until now I have mainly used the lecture notes 'Mathematical ideas and notions of quantum field theory' by Etingof. In ...
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QFT: relation between Cutkosky's cutting rules and the optical theorem

I'm self-studying QFT and am trying to see the relation between Cutkosky rules and the optical theorem, which are presented together as consequences of unitarity in almost every elementary ...
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Two-body correlation function vs. two-body reduced density matrix

The notions of two-body correlation function and two-body reduced density matrix appear frequently in the literature. But what are the precise meanings of them? A notion more often heard of than the ...
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How to properly use integral identity for loop calculations

Setup For the following let us define $$I_{\alpha}(\sigma) :=\mu^{2 \epsilon} \int \frac{\mathrm{d}^{D} k}{(2 \pi)^{D}} \left(\frac{1}{k^{2}-\sigma^{2}}\right)^\alpha $$ and assume that the following ...
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Do correlations in local quantum spin systems always decay exponentially or algebraically?

Consider translation-invariant quantum spin systems, that is qu-d-its on a lattice with a geometrically local Hamiltonian. Usually, such models are either gapped (in an ordered/disordered phase) or ...
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How are propagator and two-point function related?

Assume that we have a QFT with one scalar field $\phi$ with mass $m$ and the Lagrangian $$\begin{aligned} \mathcal{L}_{\mathrm{EFT}, \mathrm{off}}=& \frac{1}{2}\left(\partial_{\mu} \phi\right)^{2}-...
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Bosonization and peculiarities of 1-D systems of interacting fermions

I'm studying bosonization and from what I've understood the main reasons why it's useful are that: For models such as the Hubbard model the Bethe Ansatz, though it allows to evaluate eigenvalues and ...
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Good books on Green's functions [duplicate]

Could anyone recommend introductory books on Green's functions with applications in the framework of classical electrodynamics?
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Normal ordering of operators

Just a quick question regarding normal ordering. To begin given a field $\phi(x)$ we can normal order it in respect to some state $|G\rangle$: $$:\phi(x):=\phi(x)-\langle \phi(x)\rangle $$ Typically ...
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How to calculate the “vortex” correlation function in 2D free system?

I want to calculate the following correlation function in 2D square lattice: $$G(i, j, \tau) \equiv\left\langle e^{-\frac{i}{2}\left[\hat{\Phi}_{i}(\tau)-\hat{\Phi}_{j}(0)\right]}\right\rangle_{0}$$ $\...
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Change of Mutual Information in Isolated Quantum Systems

I was reading some publications regarding correlation and mutual information for composite quantum systems. I noticed that most papers give the expression for the mutual information to be: $$\Delta I(...
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Partial Trace of correlated states

If given a state of the form $\rho_{AB} = \rho_A \otimes \rho_B$, I know that the partial trace with respect to one of the two subsystems will return me the reduced density matrix of the other: $$\...
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Classical and Quantum Correlations [closed]

I am just now trying to get started with the idea of classical and quantum correlations in physics. Could anyone provide me with the reference of some introductory material(books/publications/articles/...
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A short question on Ryder's proof of LSZ formula

I am reading Ryder's derivation of LSZ formula and I do not follow one intermediate step. It first solves the inhomogeneous Klein Gordon equation using Green's function. The result with retard Green's ...
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Can a mere change of variable make any physical difference?

As far as I am aware, a change of variable is intended to transforms a problem into an equivalent and usually simpler problem. But I came across an expression in a statistics class which seems to ...
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Interpretation of Feynman's propagator

What is the interpretation of the Feynman's propagator $$D(x-y) :=\langle 0 |\phi(t,x)\phi(t',y)|0\rangle~?$$ As far as I understand, it is the following. $|D(x-y)|^{2}$ is the probability density of ...
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How to solve this path integral over region?

Any idea how to solve this functional integral?: $$\Delta_\Sigma(x,y) \propto \int \exp\left(i \int\limits_\Sigma \phi(x)(\eta^{\mu\nu}\partial_\mu \partial_\nu -m^2)\phi(x) dx^4 \right) \phi(x)\phi(...
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Correlation function of the displacement operator

I am interested in the correlation function $\langle D(\tau) D(0)\rangle$ of the displacement operator $D = \exp(\sum_n \alpha_n(b_n^\dagger - b_n))$ with real $\alpha$. The Hamiltonian consists of $n$...
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Why is there an extra $Z_3$ in Ward-Takahashi identity? [duplicate]

I'm trying to derive Ward-Takahashi identity $$k_\mu V^\mu(p,q,k)=Z_1 Z_2^{-1}e(S^{-1}(q)-S^{-1}(p))\tag{68.12}$$ using Schwinger-Dyson equation and Ward identity. In renormalized spinor QED, the ...
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Correlation function of time derivative of fields

Suppose you have calculated a two point function for a field $\phi$, and the result is some function of the positions (it can be a generic function, not necessary a function of the distance $x_1-x_2$):...
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Estimate correlation function by information in the subsystems

Motivation $\newcommand{\expect}[1]{\langle \hat{#1}\rangle}$ Let me start with thoughts I had so far; there is a TLDR with my question at the bottom. Imagine a pure system $\rho$ and with a bipartite ...
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Why are $n$-point correlation functions called Greens functions even for $n>2$?

So in QFT the main way we get results are from objects of the form: $$ \langle \phi...\rangle. $$ Why are these sometimes referred to as greens functions? Do they solve a differential equation like: ...
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What does the correlation function look like in first-order transition?

I know the correlation function for critical phenomena $$G(r)\sim \frac{1}{|r|^{d-2+\eta}}$$ for $r\ll \xi$ and $$G(r)\sim e^{-|r|/\xi}$$ for $r\gg\xi$.
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Classical field theory correlation function

I'm studying QFT from Schwartz's "Quantum Field Theory and the Standard Model", and in chapter 7 he derives the Schwinger-Dyson equations for the correlation functions in a scalar field ...
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1answer
66 views

LSZ reduction formula for free fields

I'm reading the section on the LSZ reduction formula in Schwartz's QFT book and he talks about the action of free fields in the formula. Specifically he says (sec. 6.1.1, p. 73): The LSZ reduction ...
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Perturbative expansion and path integrals

We’ve started studying path integrals and perturbative expansion. We wrote the action as $S[x]= S_0[x] +S_{int}[x]$ where the first term is the action for the model which we can solve exactly, while ...
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Third-order Feynman diagrams of 2-point function in $\phi^4$-theory [closed]

$\newcommand{\Braket}[1]{\left<\Omega|#1|\Omega\right>}$ Hello, I am currently studying QFT and have a problem concerning the 2-point correlation function in $\phi^4$-theory. When I draw all the ...
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Computing a four-point function by contraction of fields

I have some problems in understanding the following method that an author used to compute a four point function. I am referring to https://arxiv.org/abs/1711.08482, by G. Sarosi pages 21-22, and https:...
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1answer
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Cancellation of vacuum bubbles in the correlation function

$$ G_K = \langle\Omega|T\prod_{i=1}^{2K}\phi(x_i)|\Omega\rangle. $$ A problem sheet question asks you to 'Show that the contribution to $G_K$ of $O(λ_L)$, $L$ integer, in which all external points are ...
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Is X-ray scattering just measuring a correlation function?

In my textbook it says that X rays measures the density-density correlation function. If the scattering is elastic, we get the equal time correlation function. They mention that intensity of the ...
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Derivation of Correlation function in Open Quantum Systems and Master Equation

I have a question and I have searched a long time about it without any success. It is about how to include convolution broadening incorrelation functions of bath operators, when using Open Quantum ...
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74 views

Correlation Functions of the one dimensional spin-1/2 XY Model

I am currently working on studying how to diagonalize the spin-1/2 XY model using the method included in " Annals of Physics 16.3 (1961): 407-466" by Lieb et al. In fact, I'd like someone to ...
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1answer
64 views

Correlation Functions: How can I prove this simple equation [closed]

The correlation functions of the Transverse Ising Model is beautifully explained in "Quantum Ising Phases and Transitions in Transverse Ising Models" Quantum Ising Phases and Transitions in ...
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Confusion on different propagators of QFT

I have a very naive confusion regarding the propagators of QFT. I have come across the terms: (i) Retarded propagator, (ii) Advanced propagator, (iii) Feynman propagator ... I can comprehend the ...
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Is there a Bell inequality for $2\times 2$ and $1\times1\times1\times1$ configuration?

Bell inequality is violated in entangle experiments is a known phenomenon. The standard experiment is conducted with two parties and is explained in https://en.wikipedia.org/wiki/Bell%27s_theorem#...
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Rescaling/renormalisation of the $n$-point function in $\phi^4$-theory by an unique $Z$?

In the chapter 12.2 of Peskin & Schroeder they introduce the rescaled renormalised $n$-point function respectively Green's function: $$\langle \Omega|T\phi(x_1)\phi(x_2)\ldots \phi(x_n)|\Omega\...
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Question about the Lehmann-Kallen representation in Srednicki [duplicate]

So when deriving the LK form of the exact propagator he says (chapter 13): $$ \langle 0| \phi(x)\phi(y)|0\rangle=\int d\tilde k e^{ik(x-y)}+\int_{4m^2}^\infty ds\rho\int d\tilde ke^{ik(x-y)}. \tag{13....
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Why commutator of positive and negative parts of scalar field is equal to the Feynman propagator?

Peskin & Schroeder state that the contraction of two fields, defined as the commutator: $$ [\phi^+(x),\phi^-(y)]\qquad \text{assuming}\ x^0>y^0$$ is equal to the Feynman propagator $D_F(x-y)$. ...
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Feynman diagram for the 2-point one-loop 1PI diagram

If I have the $\phi^3$-interactive theory. How can I draw the Feynman diagram for the 2-point one-loop 1PI diagram?
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Path integral formulation for Green's functions

In the first place, I am struggling when trying to derive the path integral formulation of the Green function for non-interacting particles $$G_{ij}(\tau)=-\frac{1}{Z}\int D(\bar{\psi},\psi) \psi_i(\...
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Positivity of correlation functions in the ferromagnetic Ising model

Is it true that all correlation functions of any even number of spins in the ferromagnetic Ising model with nearest neighbors interaction are nonnegative in any spatial dimension? In the one-...
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1answer
118 views

Spectral density of fluctuations (white noise/delta-correlated process)

Let I be the current flowing across some junction as a result of N charge carriers of charge q. And let $\langle I (t) \rangle$ be its average. Assume a particle number distribution such that its ...
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72 views

Super-ohmic bosonic bath correlation function

In quantum Brownian motion, bosonic/harmonic oscillator bath and interaction described by Hamiltonian $$ H_B = \sum_{n}\hbar\omega_n(b_n^\dagger b_n) \\ H_I = -\sigma_x \otimes B $$ and $$ B = \sum_n \...
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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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1answer
72 views

Renormalization group: Why is the correlation length only given by 'slow' modes

I am trying to understand the renormalization group approach to phase transitions. A central quantity is the correlation length $\xi$ given by $G(r) = \langle\phi(r)\phi(0)\rangle \sim e^{-r/\xi}$ for ...
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What is the physical interpretation of self-contraction being divergent?

If we take vacuum expectation value of two scalar field at the same point (2-point correlation function when 2 points coincide) it diverges. what is the physical reason(interpretation)behind this? ...
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Correlation function and spontaneous symmetry breaking

Let's look at ferromagnetic system as an example. The correlation function is defined: $$G(r,r_{0})=\langle(m_{r_{0}}-\langle m_{r_{0}}\rangle)(m_{r}-\langle m_{r}\rangle)\rangle=\frac{1}{r^{p}}e^{-\...
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1answer
147 views

Argument on why spin correlation functions in Ising model decay exponentially with a correlation length?

I'm reading Quantum Field Theory in Strongly Correlated Electronic Systems, Nagaosa. Consider 1D Ising model, $$H=J_z\sum_i S^z_iS^z_{i+1}.$$ on page 3, it says The groud stae is 2-fold degenerate ...
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32 views

When is white noise appropriate in physical systems?

Almost every textbook and a journal article that deals with thermal noise in the context of optical fields (such as signal processing, micro-ring resonators, laser sources, etc.) assumes that the ...
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What are $n$-body potentials, and $n$-body correlation functions?

In studying statistical mechanics, we ran into the concept of 3-body potentials, and in general $n$-body potentials. For $N$ interacting spherical particles in a closed volume $V$ in $d$-dimensional ...

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