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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Dyson series expression for the two-point Green function
On Chapter 7 of Fetter & Walecka, the authors prove Dyson formula for the (imaginary time) propagator $U(t,t_{0}) = e^{H_{0}t_{0}}e^{-H(t_{0}-t)}e^{-H_{0}t}$, where I am ommitting the $\hbar$'s. ...
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Understanding how second quantization recovers full $N$-body correlations from classical Schrödinger fields
I'm trying to understand a conceptual point in many-body quantum mechanics and quantum field theory.
Starting point: Consider a classical Schrödinger field $\psi(\mathbf r, t)$ with interactions, ...
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How can we read Feynman rules from the equations of motion?
Reading Feynman rules from a Lagrangian is a quite standard procedure. However I have seen papers (for example Appendix A of arXiv:2412.14858) where this is done from the Equations of Motion instead ...
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What is the meaning of "knowing all the Green functions implies knowledge of the full theory"?
I have encountered many times the sentence in the title. Either written in books or told by more experienced friends, there seems to be a consensus that "If we know all Green functions of a ...
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Two-loop diagrams for the 4-point function in $\phi^4$ theory
My book Peskin & Schroeder says on p. 326 that 0-loop & 1-loop order diagrams are:
Also 2-loop order diagrams are listed in eq. (10.51) on p. 338:
I have no idea what $s$ means in the last ...
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When is Wick Rotation Justified?
Suppose I have a QFT defined by a Lagrangian in Minkowski space and one in Euclidean space related by a Wick Rotation. What sort of objects/properties in general stay the same between either theory; ...
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What is the average formula for interacting picture at positive temperature?
Although this looks a very simple question, it has been difficult to find an answer on textbooks since many of them develop the theory of interacting many particles for zero temperature and just ...
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Understanding how certain loop integrals correspond to certain renormalized parameters of a theory
I am learning about Renormalization and I am reading various scripts/ books and online texts. In the wikipedia article with the same name:
https://en.wikipedia.org/wiki/Renormalization#...
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Defining quantum field theory via path integrals
For simplicity, consider a quantum field theory with a single quantum field $\phi$. It is well known that if we know all correlation functions of the field $\phi$, that is we know all functions
\begin{...
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Kallen-Lehmann-like Representation for Arbitrary Operators
Is it true that
$$ G_{AB}(p^2) = \int d^4x e^{ip\cdot x} \langle 0| T A(x) B(0)| 0 \rangle = \int_0^\infty dM^2 \frac{i}{p^2 - M^2 + i\epsilon}\rho_{AB}(M^2) $$
where
$$\rho_{AB}(p^2) = \sum_n \...
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Green's function/propagator has 1 or 2 arguments?
I am having trouble to understand how do we define green's function just like that because previously it was just an inverse d'Alembertian and d'Alembertian only depends on 1 variable, however in this ...
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Measurement correlation between position and momentum
Consider a collision of a Gaussian with a sharp potential barrier from which it will be both reflected and transmitted with equal probability.
Please note: I am not interested in the potential barrier ...
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Application of LSZ formula to propagator and 1PI diagrams
I've been trying for a while to prove how Peskin and Schroeder come up with formula 7.57 in their book (reported in the image below). But I cannot seem to understand how to derive the result. Moreover ...
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How to show that correlation functions with an odd number of fermionic fields vanish, given only the Wightman axioms?
I'm reading Streater-Wightman's "PCT, Spin and Statistics, and All That," and I've come across a computation that I don't know to justify from the Wightman axioms.
Specifically, immediately ...
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Derivation of LSZ reduction formula in Peskin & Schroeder QFT
I am currently trying to understand the derivation of the LSZ reduction formula showed in the Book "Introduction to Quantum Field Theory" by Peskin and Schroeder. Here eq. (7.36) states:
$$
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Derivation of <$\sigma \sigma \sigma \sigma$ > in Di Francesco's CFT book (Equation 11.39)
I am reading section 11.2.3 from the conformal field theory textbook by Di Francesco, et. al. In equation (11.39), I don't see why the second equal sign holds. The equation looks like:
$$
\langle \...
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Am I right about the Green’s function in QFT?
Fact 1
In standard quantum mechanics, expectation values are invariant across different pictures (Schrödinger, Heisenberg, and interaction):
\begin{equation}
\langle A\rangle=\langle\psi(t)| A|\psi(t)\...
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Density matrix operator in Green's function
When computing the Green's function, in the Heisenberg picture, the Green's function can be written as:
\begin{equation}
G\left(t_1,t_2\right)=-i \operatorname{Tr}\left(\rho_H(t_0) \left(\psi_{H}(t_1) ...
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Dispersion relation for heavy quark correlator
In particle physics, we often encounter correlators $\Pi(q^2)$ which are functions of the squared momentum transfer $q^2$. These functions are real-valued for some $q^2$ below a threshold $M^2$, and ...
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Does the order still matter in $\phi_1(z_1) \phi_2(z_2) \phi_3(z_3)$ in 2D CFT?
Consider the associativity of the $\phi_1(z_1) \phi_2(z_2) \phi_3(z_3)$. The operation of the OPE only make sense if two operator $\phi_i(z_i)\phi_j(z_j)$ are within the radius of the convergence, ...
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2D CFT decoupling of holomorphic and anti-holomorphic parts
I saw this post but it didn't really help me Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT
I am trying to fully understand as to why the holomorpic and anti-holomorphic part decouple. ...
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2-point function in 2D CFT
I was reading https://arxiv.org/abs/2208.05180, the part about 2-point functions in 2D CFT and im confussed as to why they can just work out the holomorphic part and still get the anti-holomorphic ...
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Contractions for Dirac Field
In QFT, one can define a contraction between bosonic fields rather simply:
\begin{equation}\Delta(x-y)=T(\phi(x)\phi(y))-N(\phi(x)\phi(y))\end{equation}
I do not know how to insert the bar notation of ...
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Can Keldysh Green’s functions be used to compute the expectation value or correlation function at any given time?
In many condensed matter and non-equilibrium contexts, Keldysh Green’s functions are widely used to compute correlation functions when the system is out of equilibrium.
However, my motivation for ...
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What does the spectral density have to do with representations of the Lorentz group?
The Källén–Lehmann representation says that the two point function of any quantum field theory can be written as an integral over all possible masses of free theory two point functions with fixed mass,...
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