Questions tagged [greens-functions]
A Green's function is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions, thereby restricting that equation's fundamental solution. In QFT, it is essentially the propagator.
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Born approximation for 1D scattering using Green's function
For $1D$ scattering, we can write a recursive equation for the wave function:
$$\psi(x) = Ae^{ikx} + \int dx'\frac{e^{ik|(x-x')|}}{2ik}\frac{2m}{\hbar^2}V(x')\psi(x')$$
I am trying to show that ...
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Explicit Form of Feynman Propagator for a Scalar Field in Position-space: Derivation Details
This is Problem (6.1) from Schwartz's QFT and the Standard Model. I am trying to directly calculate, by performing the integral over momenta, the explicit position-space expression of the Feynman ...
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Proving that Retarded K-G Propagator is Green function (Peskin & Schroeder 2.56) [closed]
I am trying to derive Peskin & Schroeders expression $2.56$:
$$(\partial^2 +m^2)D_R(x-y)=-i\delta^{(4)}(x-y)\tag{2.56}$$
with $$D_R(x-y)=\theta(x^0-y^0)\langle 0|[\phi(x),\phi(y)]|0\rangle.\tag{2....
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How to choose contour of integration prescription Klein- Gordon Propagator? [duplicate]
I am going through the complex integral in peskin & Schroeder's intro to QFT (equation 2.54, deriving the Free Klein-Gordon Propagator):
$$\langle0|[\phi(x),\phi(y)]|0\rangle=\int \frac{d^3p}{(2\...
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Computing inverse kinetic operator for in-in formalism
Consider the kinetic operator in in-in formalism.
\begin{align}
-\frac{1}{2} \int_{x, y} \phi^{A}_{x} K^{AB}_{x, y} \phi^{B}_{y}
\end{align}
where $K^{AB}$ is the kinetic operator with the form
\begin{...
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Calculation of $ \gamma(\lambda) $ in massless renormalizable scalar field theory
In Peskin & Schroeder p.413 and 414, the Callan-Symanzik equation for a 2-point Green's function is used to calculate $ \gamma(\lambda) $ for a massless renormalizable scalar field theory. The two-...
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Some calculation in Mahan book, p73 [closed]
On page 73 of Mahan, Many-particle physics, 3rd edition, one finds
$$
_0\langle|S(-\infty,0) = e^{-iL}_0\langle|S(\infty,-\infty)S(-\infty,0).
$$
I'm wondering why this is true, as in the previous ...
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Feynman propagator from Hadamard propagator
The Feynman propagator is defined as
$$i G_F = \theta(t-t')G^+ + \theta(t'-t)G^-. \tag{1}$$
Using
$$G^{(1)} = G^+ + G^-,$$
$$G_R = -\theta(t-t')G, $$
$$G_A = \theta(t'-t)G, $$
$$\bar{G} = \frac{1}{2}(...
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Fourier transform of Green function using residue theroem
I want to compute the Fourier transform of a Green function in $k$-space :
$$
G^R_{n,m}(\omega)=\int_0^{2\pi}\frac{dk}{2\pi}\frac{e^{ik(n-m)}}{\omega+i\eta-\epsilon_k}
$$
By substituting $\omega$ and ...
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Finding scalar propagators in QFT for specific spacetime dimension $d$ and mass $m$
I need to understand how in practice one finds propagators for given $d$ and $m$ in quantum field theory. I can write down the theory provided for it but I don't know how to use it.
We will compute ...
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What is the equation describing the boundary of a 2D charge density?
Consider a charge density $\rho(x,y)$ distributed on the 2D plane. The charge density follows the Poisson equation:
$$\nabla^2 \mathbf{\phi}=\mathbf{ \nabla\cdot E}=-4\pi\rho(x,y),$$
where $\phi(x,y)$ ...
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How to show a propagator of massive spin-1 field is the Green function to its equation of motion?
During my QFT course my teacher said the propagator of the massive spin-1 field is the green function of the equation of motion derived from the Proca Lagrangian, which is shown in the next two ...
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Three-Point correlation function in cosmology
I have been studying this review article on Non-Gaussianity from inflation. It was mentioned that $n$-point correlation function can be obtained by the expression
\begin{equation}\label{eq:1}
\langle\...
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Is Kirchhoff's scalar theory of diffraction mathematically inconsistent?
I've heard that Kirchhoff's scalar diffraction theory is mathematically inconsistent. Is this true? If so, where in the formulation does this inconsistency arise and are there ways to remedy it?
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Laplace Green function on $R \times S^3$
On flat Euclidean $R^4$ the Laplace operator has the Green function $G(x,y) = \frac{1}{4\pi^2(x-y)^2}$, i.e.
$$-\Delta G(x,y) = \delta^4(x-y).$$
What would be the corresponding Green function on $R \...
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Calculating gauge propagator in minimally coupled, non-relativistic fermion system
For context, I am trying to derive Eq. 4.1 of $T_c$ superconductors">this paper. Consider the action
$$S[\psi^\dagger, \psi, a] = -\int d\tau \int d^2r \sum_\sigma \psi^\dagger (D_0-\mu_F-\frac{1}{...
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How to calculate the Green function of 1D Kitaev chain?
After performing Jordan-Wigner transformation, a uniform transverse Ising model becomes a 1D Kitaev chain as
$\hat{H}_{p=0,1} = -J\sum_{j=1}^{L}{(\hat{c}_{j}^{\dagger}\hat{c}_{j+1}+\hat{c}_{j}^{\...
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Finite temperature Greens function simplification
I am currently studying topological insulators using Topological insulators and superconductors by Bernevig. In chapter 3, section 3.2.2, he has derived the finite temperature Green's function using ...
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Feynman's rule for Green's function of particle in the external field and with interaction
I'm learning the quantum field theory for condensed matter systems, and I've just learnt about the single particle Green's function with interaction and its Feynman rule. But I met some problems when ...
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Two-point Wightman function in $\phi^4$ scalar theory
There is a known perturbative expansion of the time-ordered 2-point correlation function $$⟨\Omega|T\phi(x_1)\phi(x_2)|\Omega⟩,$$ where $|\Omega⟩$ is the vacuum of the interacting theory. What is the ...
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Why does the transmission oscillate when using a numerical dispersion?
I understand that the transmission probability oscillates when the electron is considered to be a plane wave - this is because of the interference effects of these waves.
However, when a different ...
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Proof that $-\partial^2 G(x, y) = \delta(x-y)$ for free field propagator
I recently realized that there is a slightly pedantic issue when one normally proves that the equations of motion acting on the free field propagator gives a delta function which I have become ...
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General interpretation of the poles of the propagator
I am somewhat familiar with the fact that the poles of the Feynman propagator in QFT give the momentum of particle states. I'm also familiar with the KL spectral representation in that context (See ...
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Inverse of an operator [closed]
I want to understand how to find the Inverse of an operator.
I know it involves the use of Green's function but I can't seem to figure out how.
Here is the actual problem:
On page 302 of Peskin&...
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Klein-Gordon and Green's Function
I want to prove the following relation:
$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)|0\rangle = -i\delta^{(4)}(x - x_1).$$
My Approach: Consider LHS
$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)...
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Derivation of Feynman rules in many-body theory
In textbooks on many-body quantum physics (e.g. Fetter and Walecka), Feynman diagrams are typically introduced after formulating the Dyson perturbative expansion of the Green's function using Wicks ...
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How to interpret a Green's function containing a branch cut from $-\infty$ to $+\infty$ at negative imaginary frequency?
From some loop calculation in an EFT, I have found the Green's function containing this square root:
$$G(\omega, k)\sim \sqrt{\frac{D k^2 \tau- (i+\tau \omega)^2}{(i+\tau \omega)^2(-D k^2 \tau + \...
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A problem in solving a PDE of the time-ordered Green function
In the article written by Jauho in 2006 Introduction to the Keldysh nonequilibrium Green function technique, I've tried to operating with whether $g_{k\alpha}^t$ or $g_{k\alpha}^{-1}$ from right to ...
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Curved spacetime generalization of Bethe-Salpeter equation
I am interested in the problem of bound states in QFT in curved spacetime. I was wondering if the generalization of the Bethe-Salpeter equation is as simple as replacing the Green’s functions in the ...
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Confusion Regarding the Propagator [duplicate]
To my understanding, the expression $$G^+=\theta(t_f-t_i)\langle x_f|\mathcal{\hat U}(t_f,t_i)|x_i\rangle$$ represents the probability amplitude that a particle starting at position $x_i$ at time $t_i$...
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Doublet impulse force harmonic oscillator
I initially asked this on a math forum, but I see now that physics was a better choice. I'm considering an at-rest simple harmonic oscillator (m,k) and want to model the force by a doublet (derivative ...
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Momentum Space Propagator from Path Integral Formulation of “Polyakov-style” action for a massive relativistic point particle
I have the derived the following expression for the propagator of a “Polyakov-style” action for a massive relativistic point particle:
\begin{equation*}
G(X_2 - X_1) = \mathcal{N}'\int_{0}^{\infty}...
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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:
$$<\phi(x)\phi(y)> = \int \frac{d^d{p}}{(2\pi)^d}\...
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How do we interpret the second-order differential operator in the QFT path integral?
For the free scalar field theory, the path integral has a differential operator term in the exponent,
$$
Z[J] = \int \mathcal{D}\phi \, \exp\left( i \left[ -\frac{1}{2} \int d^d x \, \phi(x) A \phi(x) ...
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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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Problematic Factor of 2 in Klein-Gordon Propagator Derivation
I want to derive the Klein-Gordon Green function equation
$$(\Box_b + m^2) D_F(x_b - x_a) = - i \delta^4(x_b - x_a)$$
by using the same steps taken when fixing the 'exact' Green function of the non-...
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What's the quantum operator for the diffusion Green's function?
I am trying to understand the following equation about the diffusion Green's function ("Principles of condensed matter physics" by Paul Chaikin, chapter 7.4 Diffusion)
$$n(x,t)=\int G(x-x',t-...
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Inhomogeneous Solution to 2+1D wave-equation
The inhomogeneous wave equation
$$\left(\frac{1}{c^2} \partial_t^2 - \Delta\right)G(\vec{r},t)=\delta^3(\vec{r})\delta(t) \tag{0}$$
for a point-source in three spatial dimensions can be tackled by ...
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Peskin and Schroeder QFT Eq.(12.66), the Renormalization Group equation
I am troubled for the derivation of Eq.$(12.66)$ on Peskin and Schroeder's QFT book.
$$
\left[p \frac{\partial}{\partial p}-\beta(\lambda) \frac{\partial}{\partial \lambda}+2-2 \gamma(\lambda)\right] ...
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Fourier transformation of the inverse Klein-Gordon propagator
On Peskin & Schroeder's QFT, page 30, the scalar field propagator as the retarded Green function is defined as
$$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y) \tag{2.56}$$
The Fourier transformation is ...
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Notation Clarification of $G_{\uparrow\uparrow}$
I am a bit confused as to what the notation
$$G_{\uparrow\uparrow}$$
on the Green fucntions means, on a paper I have come across. Here $\hat{G}(k) = (i \omega_{l} - H_{I} - H_{P})^{-1}$.
The ...
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Can the Keldysh occupation function have a zero for bosons or a pole for fermions?
In the Keldysh framework for nonequilibrium dynamics of quantum systems we learn that there are essentially two Green's functions that characterize a system: the retarded Green's function $G^R(\omega)$...
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Is there a Hadamard form for 3-point functions?
Let's say we are trying to solve a wave equation for a scalar field in curved spacetime
\begin{equation}
\Big[\nabla^\mu \nabla_\mu +k(x)\Big]\phi(x)=-4\pi \rho (x)
\end{equation}
We are interested in ...
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Seeking Resources for Implementing Self-Consistent Numerical Calculation of Retarded Green's Function
I'm seeking resources or references to learn how to implement a self-consistent numerical calculation for the retarded Green's function $G^R(\vec{p},E)$, which is defined as:
$$
G^R(\vec{p},E) = \frac{...
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The Callan-Symanzik equation on Peskin & Schroeder's QFT
On Peskin & Schroeder's QFT page 411, the Callan-Symanzik (CS) equation reads
$$
\left[M \frac{\partial}{\partial M}+\beta(\lambda) \frac{\partial}{\partial \lambda}+n \gamma(\lambda)\right] G^{(n)...
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Why poles of two-point function corresponds to bound states?
In this article Two-time Green function method in quantum electrodynamics of high-Z few-electron atoms the author has:
Let $\mathcal{G}$ be fourier transform of the green function
$$
\begin{array}{r}
\...
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Greens Theorem for periodic functions
Ashcroft and Mermin supply the following proof of their equations (I.1/2), which get used often in computing integrals over the first Brillouin zone (in computing current densities etc.).
I find ...
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Retarded Green's function for Fermions vs Bosons
Normally, retarded non-interacting Green's function for Fermions is:
$$
G_0^R(k,E) = \frac{1}{E-\epsilon_k+i\eta}
$$
But recently, I read a few research articles (for example, equation (46) of [1]), ...
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Electric potential of uniformly charged wire with Green function
I want to calculate the electric potential of a uniformly charged wire with infinite length $\rho(\vec{r}') = \lambda \delta(x') \delta(y')$ with Green function $G(\vec{r}, \vec{r}') = \frac{1}{4\pi \...
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Fourier Transform of temperature Green Function
I am doing a calculation involving a temperature Green function for some operator $\hat{A}$:
$$G_{\hat{A}}(\tau)=-\Big\langle{T_\tau\big(\hat{A}(\tau)\hat{A}^\dagger \big)} \Big\rangle=-\theta(\tau)&...
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