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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Correlation functions of currents on torus at critical level

In a paper "CORRELATION FUNCTIONS OF CURRENT-ALGEBRA THEORIES ON THE TORUS" by Mathur and Mukhi the torus correlation function of currents are computed. Crucially, they use the relation $$ \sum_a (J^a ...
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Cross-correlation power spectra

I am trying to understand how cross-correlation power spectra (PS) are interpreted. If I have understood correctly, the following hold: A PS can either be an autocorrelation PS or a cross-correlation ...
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Monopole operator: correlation functions

Let's consider free Maxwell theory: $$ S_{Maxwell} = \int d^dx \; -\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu} $$ In such theory one can define monopole operator using path integral via correlation functions ...
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Klein-Gordon equation multiple Green's functions

I am trying to understand Green's functions for a Klein-Gordon equation: $ (\frac{\partial^2}{\partial t^2} - \nabla^2 +m^2) \phi(\vec{x},t) = 0$ and $ (\frac{\partial^2}{\partial t^2} - \nabla^2 +...
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Getting Feynman propagator using path integral

In QM using Feynman path integral(FPI) we derive the propagator of free particle which comes out to $$(f(t))e^{iS_{cl}/\hbar}$$ But in QFT the Feynman propagator is derived using the differential ...
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How to connect Green function to propagator?

I know that there has already been many questions related to this question, such as in Differentiating Propagator, Green's function, Correlation function, etc. However, that question mainly ...
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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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Does the degeneracy in dominant eigenvalues of MPS transfer matrix necessarily mean long-range correlation?

A short-range correlated MPS state generally has one non-degenerate dominant eigenvalue in its transfer matrix, and the degeneracy in transfer matrix generally leads to long-range correlation (I'm ...
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Two-point functions don't signal spontaneous symmetry breaking?

I am aware that when some operator $\phi_n(x)$ which transforms nontrivially with respect to some symmetry group acquires a VEV, it signals the spontaneous breakdown of a particular symmetry, since ...
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Renormalization conditions of the Callan-Symanzik equation

Assume we have a massive $\phi^4$ theory the exact two-point correlation function is given as $$G=\frac{iZ}{p^2-m_r^2}+\text{terms regular at } p^2=m_r^2 $$ and if I want to apply renormalized ...
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Relation between the Glauber correlation functions and statistical correlation

As stated in Wikipedia's page: Correlation or dependence is any statistical relationship, whether causal or not, between two random variables. Now in "The Quantum Theory of Optical Coherence" ...
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Causality, branch cut choice and analytic continuation of Euclidean 2-pt. correlator in 2D CFT

In 2D CFT, the Euclidean two point correlator of a primary operator $\mathcal{O}$ with conformal weights $h$, $\bar{h}$ is given by $$ \begin{align} \langle\mathcal{O}(z,\bar{z})\mathcal{O}(0,0)\...
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Non-zero Euclidean commutator in 2D CFT?

In a Euclidean QFT, commutators of operators vanish for any spacetime separation. This can be argued very simply by using the path integral representation of the correlator, wherein operators become ...
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Validity of Linear Response Theory

Suppose the perturbation of the hamiltonian is some multiple of the free hamiltonian, that is $$H=H_0+H_1=H_0+\lambda H_0=(1+\lambda)H_0.$$ Here, certain operators apparently have no response due to ...
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Trace class of the position autocorrelation function

Let us consider a quantum system of $N$ distinguishable particles, and let us tag the configuration $\hat q_j$ of one of those. I am interested in checking whether the position auto-correlation ...
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Why traditional turbulence theory concerns so much about statistics such as correlations?

I have been wondering why the traditional turbulence theory, e.g., Kolmogorov's 1941 theory, concerns so much about things like two-point correlations, structure functions, their scalings, and so ...
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Fourier Transform of Density Operator boosting momentum

I am currently reading Girvin and Yang's book on Condensed Matter Physics. I am reading up on the dynamical structure factor and in Section 4.3 where they talk about how $\rho_{+q}$, which is defined ...
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Change sign for response function

There is a argument about response function: according to the Kramers-Kronig relation$$G(\omega)=\int_{-\infty}^{+\infty}d\omega' \frac{A(\omega')}{\omega+i0_+-\omega'}$$ response function will change ...
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Propagators and Green functions for general fields

In my QFT class we have defined the Feynman propagator of a field $\phi^r$ (where $r$ could be a vector or spinor index, or even a multiindex if $\phi$ is a tensor field etc.) as $$ \Delta^{rs}_F(x - ...
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Are the momenta of the $1/2$-BPS operators in this $2$-point function on-shell?

I recently asked this question about the Fourier transform of a $2$-point function, and it lead me to another question. I am looking at $2$-pt functions of $1/2$-BPS operators in $\mathcal{N}=4$ SYM ...
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Three point correlation function 2D Ising model

What is the expected behaviour of the three point function $<\sigma_i \sigma_j \sigma_k>$ of the Ising 2D model at the critical point where conformal symmetry is valid? Do they have a power-law ...
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Three-point function in CFT

Reason for deriving the 3-point function Searching for a derivation of the three-point function constraints in CFT online I have realised that there is no derivation of the 3-point function. Most ...
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Meaning of momentum after Fourier transforming a propagator

Let the (free, massless) scalar propagator in $4$d be defined in position space as: $$\left\langle \phi(x_1) \phi(x_2) \right\rangle \sim \frac{1}{x_{12}^2} \tag{1}$$ with $x_{12}:=x_1 - x_2$ and ...
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Constraining the 2-point correlation function

Consider the two-point function $$ \langle\mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\rangle=f(x_1,x_2) $$ If the operators are in a CFT, we can constrain this function using the symmetries of the theory. ...
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Unsubtracted correlation function to show hyperscaling?

I am looking at the second problem of this old statistical physics course: http://www.lassp.cornell.edu/clh/p653/hw04.pdf. However, I am stuck at 4.2. How would I go about showing, that there exists a ...
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Measurement of $n$-point functions

The $n$-point correlation functions appear frequently in physics. I think I can understand how the concepts can be exploited to understand physical processes. What I struggle to understand is how they ...
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Argument for exponential decay of correlation function

Suppose we have some lattice classical spin system. The correlation function is defined as $$\Gamma_{ij}=\langle S_iS_j\rangle -\langle S_i\rangle \langle S_j\rangle $$ It is often said that $$ \...
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Will an entangled system give different measurement then just correlation? [duplicate]

I have read this question: Correlation vs. entanglement for composite quantum system Entangled states can produce nonclassical correlations, but this is not necessarily the case. So far so good. ...
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Analytical Continuation of Correlated function

The real time linear response function is given by Kubo's formula $$\chi_{AB}(t-t')=-i\Theta(t-t')\langle [A,B]\rangle.$$ This can also be obtained by analytically continuing the imaginary time ...
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How many-body density $n(\vec{r},t)$ can be viewed as a kind of correlation function?

I am reading Martin's book: interacting electrons. In chapter five about the definition of the correlation function, some points about density as correlation function confused me. The author adopted ...
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Maximally uncorrelated observables as a consequence of unbiasedness relation

In the case of a fully correlated scenario of $A$ and $B$, which are descirbed in mutually unbiased bases, why do the other observables have to be maximally uncorrelated. How does this follow from the ...
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Question about the Dyson-Schwinger equation for 4-point function of the SYK model

In studying the SYK model, I have no idea how to get the kernel $K(t_a,t_b,t_3,t_4)$ and $\Gamma_0(t_1,t_2,t_3,t_4)$ and also ladder diagrams in the Dyson-Schwinger equation for 4 point function of ...
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Doubt about the derivation of the Callan-Symanzik equation

I was reading about the Callan Symanzik equation from Peskin and Schroeder. On page 411, they assume that since $G^{(n)}$, the connected Green's function is renormalized, the $\beta$ and $\gamma$ ...
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Calculation of pafaffian matrix form from spin-spin correlation function in free fermions theory

I want to calculate the following correlation functions \begin{equation} \langle\sigma^{x}_{\ell}\sigma^{x}_{m} \rangle=\langle B_{\ell}A_{\ell+1}B_{\ell+1}\ldots A_{m-1}B_{m-1}A_{m}\rangle \end{...
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Vanishisng vaccum Feynman diagrams in $A-B-\phi$ scattering model

Question data given below: Let us consider the 𝐴 βˆ’ 𝐡 βˆ’ πœ™ model that describes 3 real scalar fields 𝐴, 𝐡 and πœ™ governed by the following action $𝑆[𝐴, 𝐡,πœ™] = ∫ 𝑑^ 4 π‘₯ (β„’_𝐴 + β„’_𝐡 + β„’_πœ™ + ...
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Expression of Dirac Delta Correlation

spatio-temporal white noise $\xi(x,t)$ is often expressed as $$\langle\xi(x,t)\rangle=0,$$ $$\langle\xi(x_1,t_1)\xi(x_2,t_2)\rangle=\delta(t_2-t_1)\delta(x_2-x_1).$$ Now I understand that the first ...
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Mixed current correlators in QFT

The following is the definition for current correlator: $$\Pi_{B}^{\mu\nu}(q)=-\int d^4x \hspace{1mm}e^{iq.x}\left<0|\textbf{T}j_B^\mu(x)j_B^\nu(0)|0\right>$$where B=V, A stands for vector and ...
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Trace in correlations to compute Wigner transform

In the derivation of Wigner-transformed quantum time correlation functions, the following identity is used (in the case of a one-dimensional particle, for simplicity): \begin{align} C(t) &\equiv \...
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4-point function in CFT

Im currently trying to understand the form of the 4 point function in CFT, i.e. how to derive equation 4.62 in Di Francesco et al. In particular, the coefficients of the $x_{ij}=|x_i-x_j|$. For four ...
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Bond Orientational Correlation Function - how exactly to calculate

I have a question about this article. I am trying to compute the bond order correlation function, $g_6$. It is defined based on the bond order parameter: $$\psi_6(r_i) = \frac{1}{N}\sum_{j}^{}{\exp[...
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Equivalence between OPE associativity and four-point crossing symmetry

I'm reading Simmons-Duffin CFT Lecture Notes, where it's stated that one can recover the OPE associativity from the four-point correlator crossing symmetry. It seems supposed to be a very trivial ...
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Why is the Feynman propagator $\langle 0|T \phi(x) \phi^\dagger(y) |0 \rangle$ instead of $\langle 0 |T \phi^\dagger(x) \phi(y) |0 \rangle$?

If we expand a complex scalar field in terms of Fourier components, we can see that $\langle 0 |T \phi^\dagger(x) \phi(y) |0 \rangle$ contains terms which are not contained in $\langle 0|T \phi(x) \...
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Why can the retarded propagator be defined in terms of a commutator or without it?

In many textbooks [e.g. Peskin & SchrΓΆder p. 30 eq. (2.55), or Tong's notes p. 41 eq. (2.101)], the retarded propagator is defined as $$G_R = \Theta(x^0-y^0) \left< [\phi(x), \phi(y)] \right>...
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Relations between correlation function and commutators of two observables

In quantum theory (QFT or Quantum statistical mechanics), are correlation of two observable related to their commutator? Is there any explicit bound of the correlation by the norm of the commutator? ...
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Are propagators in QFT really Green functions?

In many textbooks the notions Green function and propagator are used interchangeably. But are they really the same thing? This popular answer argues that a retarded propagator function $D_R(x,t,x',t')...
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Eigenfunctionals and their application in physics

Is there any sensible meaning of the term eigenfunctionals? The object I want to describe is a solution to the following equation $$ {\mathscr D}_x F[g] = f(x) F[g] $$ where $ {\mathscr D}_x$ is an ...
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Structure factor and pair distribution function relation in 2D

I am trying to drive the integral form relation between pair distribution function and structure factor in 2 dimensions. In 3D we get: Where What would be the answer for $g(r)$ in 2D, my answer is:...
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Hartree-Fock factorization

I am studying c-field methods applied to Bose-Einstein condensates to understand how one gets to e.g. the dissipative GPE. To do so, one splits the field operator for the Bose gas into a low- and a ...
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Time dependence of Drude-like correlation function obtained from Matsubara formalism

I'm trying to calculate the real time dependence of the correlation function that I've obtained in my effective model (it is closely related to the electron density correlation function), given in ...
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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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