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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Green function in 2D, unit disk and Poisson kernel

First, I know that the Green function of 2D Laplace operator is given by $$G(z,w)\propto \ln\frac{|z-w|}{|z-\bar{w}|}.$$ Second, I also understand how can I obtain the Green function on unit disk, $$...
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Quantum Field theory, solving delta / green function

I have read an equation as follow $$[-(k^2-m^2)g^{\mu\nu}+k^{\mu}k^{\nu}]D_{\nu\lambda}(k)=\delta^{\mu}_{\lambda}$$ The solution is given as: $$D_{\nu\lambda}(k)=\large{\frac{-g_{\nu\lambda}+k_{\nu}k_{...
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41 views

Getting Green's Function for Laplace's Equation in Cylindrical Coordinates

I am trying to understand a derivation for finding the Green's function of Laplace's eq in cylindrical coordinates. Let the Green's function be written as: $G(r,\theta,z,r',\theta',z') = G(\mathbf{r},\...
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Average of Two successive momenta $m\frac{x_{k+1}-x_k}{\epsilon}m\frac{x_k-x_{k-1}}{\epsilon}$ using rules of path integral

A Problem from Feynman's Path Integral Book Let $x_i$ be coordinates at different time instances, prove that $$ \langle\chi|m\frac{x_{k+1}-x_k}{\epsilon}m\frac{x_k-x_{k-1}}{\epsilon}|\psi\rangle=\int\...
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Fourier Transform of Green function

I have a question regarding the (discrete) Fourier Transform of the retarded Green function (I neglect hats on operators): $$G(i,j;t)=-i\theta(t)\langle \{c_i(t),c^\dagger_j \} \rangle $$ specifically ...
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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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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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Relation between many-body system's single-particle retarded Green' function and Propagator for single particle wavefunction

In the 8th Chapter of book Many-body Quantum Theory in Condensed Matter Physics by Henrik Bruus and Karsten Flensberg, they give an explicit form of retarded Green's function (GF) for the many-body ...
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D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone)

How one differentiates a delta-function of a space-time interval? Namely, $$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$ Somewhere I saw that the ...
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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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How to realize Heaviside $\theta(t-t')$ and Dirac $\delta(t-t')$ as matrices in numerics?

As is well known, single-particle Green's functions in the time domain might involve $$\theta(t-t')$$ for the retarded and advanced Green's functions. Sometimes, we also need $$\delta(t-t')$$ to ...
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Are propagators in QFT Wightman Functions?

I am studying relativistic quantum mechanics and I can't really understand how propagators arise from the theory. They are generally defined as the Wightman function $$ W_F (t',\vec{x}', t,\vec{x}) \...
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Free-space dyadic Green function

My professor derived the free-space dyadic Green function from the general Green function: $$\tag{1} \stackrel{\leftrightarrow}{\boldsymbol{G}}=\left[\stackrel{\leftrightarrow}{\mathbb{I}}+\frac{1}{k^{...
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The gradient of the d'Alembertian Green's function

So I have to prove that the d'alembertian of the associated green's function $G(t,t',\vec{r},\vec{r}')$ is equal to zero when given that $\vec{r}\neq\vec{r}'$ $$\left(\frac{1}{c^2}\partial^2 t-\Delta\...
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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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Question regarding a special identity for $2\pi\delta(E-\epsilon_\alpha)$

I am reading Datta's book about Quantum Transport at the moment and I stumbled over an identity for the Dirac delta distribution, which is correct since it fullfils all the requirements for the Dirac ...
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Mysterious factor 2 in the Schrödinger equation derived from Feynman's Kernel

Feynman's Quantum Mechanics and Path Integral has a vivid physical interpretation of the path integral formalism. But I was stumbled on some mathematical details while following his derivation of the ...
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How to take 'non-local' functional derivatives?

I am currently in the process of getting into linear response theory in general, and I have often met functional derivatives of the type $$\frac{\partial J[f(x)]}{\partial f(y)} = \chi(x,y).$$ I've ...
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Classical harmonic oscillator Green's functions

A recent paper (Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach) derived expressions for thermal conductivity in a system of harmonic oscillators that decay ...
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Derivation of Born's Approximation in 1D

I'm studying Quantum Mechanics and I'm curious about something related to Scattering Theory: Griffiths has a derivation from the Green's function to the Born's approximation in 3D but I was wondering ...
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Quantum Monte Carlo: arbitrary Observables in Worm Algorithm

Consider a generic Hamiltonian of interacting particles on a lattice, where $a^{\dagger}_i$ creates a particle at site $i$, and $n_i=a^{\dagger}_i a_i$ and $\langle i, j\rangle$ are nearest neighbors. ...
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How to prove zero classical classical Green's function contribution

Much of the Keldysh formalism is based on the following identity $$G^{T}(t,t') + G^{\tilde{T}}(t,t') = G^{<}(t,t') + G^{>}(t,t'),$$ which using their defintions is equivalent to $$G^{\text{cl,cl}...
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Why do Green functions show up in treating dielectric media microscopically?

I'm reading the treatment of Thomas-Fermi screening in Ashcroft and Mermin (ch. 17). They write some strange equations which I've never seen before, anywhere. First of all, they write the usual local ...
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Problem with infinitesimal factor in the expression for $G^{+}(x,y,E)$

In my book on QFT (Lancaster & Blundell) they give the following expression for the Green's function: $$G^{+}(x,y,E) = \sum \frac{i\phi _{n}(x)\phi ^{*}_{n}(y)}{E-E_{n}}.$$ However, they then ...
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Green's function for free scalar field theory as an inverse of $-(\partial^2+m^2)$

In his book [QFT in a Nutshell], Zee argues that the functional integral $$Z= \int D\varphi \; e^{i\int d^4x[ -\frac{1}{2}\varphi(\partial^2+m^2)\varphi + J\varphi]}$$ can be evaluated by discretizing ...
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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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Where did the boundary conditions go in the frequency-space solution to the Green's function for a damped harmonic oscillator?

In differential equations, Green's functions are only defined given boundary conditions. In fact, you need two of them for a second order differential equation. In a lot of physics lecture notes, a ...
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Why does photocurrent need $G^<$ among other Green functions?

In this highly-cited paper (or its pdf) on photoemission, Eq.(3) gives the current density in terms of $G^<$ $$ {\bf j}({\bf r},t) = 2\hbar\left( \frac{e\hbar}{2m} (\partial_{\bf r'} - \partial_{\...
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Noise in density of states plot using decimation method

I am using this(https://arxiv.org/abs/1604.02499) article to learn how to use the decimation method to calculate the density of states of a system. Included in the article is Julia code to calculate ...
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Numerical evaluation of Green function to evolve wavefunction for harmonic oscillator: Part II

This is part 2 of my original question. Briefly, I am numerically evaluating $$ \psi (x,t_2) = \int dy \, G(x,y) \psi(y,t_1),$$ with $t_2 > t_1$. $t_2-t_1=\Delta t = \epsilon $. My $G$ is given by $...
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55 views

Numerical evaluation of Green function to evolve wavefunction for harmonic oscillator [closed]

Inspired by the paper "Feynman's derivation of the Schrodinger equation", I'm trying to do a simple numerical evaluation of the following equation (4.1) from the paper: $$ \psi (x,t_2) = \...
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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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How to dress free Green functions with constant broadening?

I want to find a way to dress free Keldysh Green functions with the simplest level broadening. But there seems to be some quite unexpected result. Let's consider free Keldysh Green functions in ...
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Do sum rules for spectral function always hold?

In condensed matter physics, we can define the spectral function as $$ A_{\alpha}(\omega) = -\frac{1}{\pi}\mathrm{Im}G_{\alpha}^R (\omega) $$ It can be shown that this quantity satisfies the sum rule: ...
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Spectral function and bound states in condensed matter

In condensed matter physics (like in QFT) we can use Feynman's diagrams to compute the self-energy. From here we can obtain the spectral function as: $$ A_{\mathbf k} (\omega) = \frac{-\frac{1}{\pi}\...
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Dyson equation in terms of retarded Green function

Dyson equation, schematically written as $ G_0^{-1}-G^{-1}=\Sigma,$ holds for the causal Green function, which is the object used to formulate perturbation theory. However, working in a lattice model, ...
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Propagator as a Green's function in non-relativistic QM

I have a propagator $$ K=\sum_k\langle x|a_k\rangle\langle a_k| y\rangle \exp\left\{ \frac{-iE_k(t-t_0)}{\hbar} \right\} ~~,\tag{1} $$ which I know satisfies the time-dependent Schrodinger equation \...
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How to Fourier transform the massive propagator of a higher-dimensional Gaussian scalar field from momentum-space to position-space

Can anyone please help me to complete/check the explicit calculation of the following Fourier transform \begin{align} &\int \frac{d^Dk}{(2\pi)^D} \frac{e^{-i k.x}}{k^2+m^2} \\&\quad= \int_0^\...
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Green's functions for linearised gravitational waves?

Green's functions are ubiquitous in physics, or any situation where one would like to solve some set of partial differential equations with boundary conditions. It is therefore not so surprising that ...
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Scattering Greens function exactly at energy of bound state

I have a small bit of confusion about the expansion I am seeing in literature for the Greens function in time independent scattering theory. For example here is an excerpt from Scattering Theory of ...
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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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What is the importance of a proper, irreducible polarizability as opposed to reduced polarizability?

I am following Richard Martin et al. (2016) on interacting electrons. On page 150 he states (surprisingly perhaps) that there is a Dyson equation for screening. I cite the authors: As stressed before,...
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Are these Transformations of the Green's function equivalent?

The Green's function $G(E)$ can be constructed from the Hamiltonian $H$ $G(E) = [(E+i\epsilon)I - H]^{-1}$ where $I$ is the identity matrix. Say we want to perform a transformation into another basis ...
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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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Finding the solution of time-dependent Schrodinger equation

I want to solve the following Equation: $$\frac{d G(\tau,\tau')}{d\tau}=H(\tau)G(\tau,\tau')+\delta(\tau-\tau').$$ When $\tau \neq \tau'$, then, if $\tau>\tau'$, $$G(\tau,\tau')=T e^{\int_{\tau'}^{\...
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Dipole source directivity function? (Acoustics)

I'm trying to model the propagation of a dipole source and I need to know what is the directivity function (i.e. the polar pattern function) which describes it. I've read in an article that a dipole ...
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When could I find potential using Green's function? [closed]

In electrostatic, I have two equations to find the potential, when could I use the first or use the second? the first one:: $$ \displaystyle \Phi \ =\ -\frac{1}{4\pi } \ \int \phi ( x') \ \frac{\...
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Matching Two Point Function in momentum space using spherical coordinate

Background of the problem: The problem I am currently struggling is related to the momentum representation of Fourier transform. Briefly speaking, the integral in Minkowski under Cartesian coordinate ...
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1answer
25 views

Broadening of spectral function: interaction and temperature effect

Consider a non-interacting fermion system with Hamiltonian \begin{equation} H = \sum_{\nu}\epsilon_{\nu}c^{\dagger}_{\nu}c_{\nu}, \end{equation} where $\nu$ is some single-particle quantum number. It ...
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34 views

Imaginary-time Green's Function and Nonequilibrium Green's Function

I've been learning Green's Function approach recently, following Radi Jishi's book. It says that the imaginary-time Green's function (Matsubara function) cannot be used in non-equilibrium situations, ...

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