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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Arbitrarity of $i$ in the propagator

My question is simple: how arbitrary can the factor in front of the propagator be? What I mean by that is, if we call the wave operator $K$ and the propagator $G$, I've seen different books use ...
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Frequency Integration of Green's Function

If one has a Green's Function that has been projected into the Helcity basis (from spin) of the from \begin{equation} G(\mathbf{k},\omega)=\sum_s\frac{1}{\omega-\epsilon_{\mathbf{k},s}+\mu+isgn(\omega)...
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Deriving Solution for Wave Equation Using Retarded Green Function

In the book Advanced Classical Electromagnetism by Robert Wald, it was shown if $\psi$ satisfy the homogeneous wave equation $$\square\psi=0$$ then we have $$\psi\left(x^{\mu}\right)=-\frac{1}{c} \...
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Scalar theory with non-trivial boundary conditions. Green function

Let us consider the free scalar field theory $\varphi(x,t)$ in a space of dimension 1+1, with Minkowski metric $\eta_{\mu\nu}=diag(+,-)$. It is well known that it is described by the classical action ...
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Why does a pole in the Green function correspond to a bound state?

Consider the many-body (zero temperature) fermion Green function $$ G(a,b;t)=-i\theta(t)\langle\psi_a(t)\psi_b^\dagger\rangle $$ Where I'm restricting $t>0$ for causality and that the free ...
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How to derive the equations of motion of finite temperature Green function?

I'm having a trouble deriving the equation of motion of a Green function. My understanding of the derivation is the following. Given a set of fermionic creation annihilation operators $$\{a_\alpha (\...
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Struggling with Peskin and Schroeder equation (12.49) and the constraint of renormalizability

In peskin and schroeder it's written that any renormalizable massless scalar field theory has a 2-point greens function of the form: I don't get how we can know that the 1-loop diagrams have exactly ...
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Time-independent amplitude to go from one point to another in Feynman lectures (free particle)

In the third chapter of Feynman Lectures Volume III, I found this expression Suppose a particle with a definite energy is going in empty space from a location $\boldsymbol{r_1}$ to a location $\...
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Correlation function of 4-currents on a general QFT

Given $V^\mu(x)$ a 4-current of a general unitary, Poincaré invariant QFT, I need to show that the correlation function: $$iC_{\mu\nu}(x-y) = \langle {\tilde{0}}|T\left[V_\mu(x)V_{\nu}^\dagger(y)\...
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On tensor manipulation and algebra

I am reading Quantum Field Theory in a Nutshell by Anthony Zee. On page 33, I can't figure out how he got equation 3. The initial equation is $$[-(k^2 - m^2)g^{\mu\nu} + k^{\mu}k^{\nu}]D_{\nu\lambda}(...
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Issues with imposing periodic boundary conditions in the presence of a gauge field

The issue is if there is a phase winding (like a $U(1)$ vortex), then the PBC are broken due to different phase windings across the lattice. Open BC should be used and these can be built out of the ...
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Does the Hamiltonian act on a Heaviside theta function?

I am doing some revision on theoretical physics, specifically propagator theory. This is talking about how to work out the probability amplitude at some time $t_{f}$ and position $x){f}$, given that ...
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Correlation functions with different momenta

In Mahan's "Many-particle physics" p.653 the following correlation function is evaluated \begin{equation} \Phi(i\omega)=-\sum_{\textbf{k},\textbf{p},\sigma}\sum_{\textbf{k}',\textbf{p}',\...
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Green function dependence on temperature

Consider the retarded Green function for fermions $$ G_{ij}^r(t)=-\frac{i}{\hbar} \theta(t)\langle[c_i(t),c^\dagger_j(0)]_{+}\rangle $$ They can be understood as the $(i,j)$ entry of the matrix $G^r(t)...
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Deriving Lorentz-covariant expression for the retarded Green's function of wave equation in $n+1$ dimensions

Consider spacetime to be homogeneous and isotropic. Then, the Green's function for the wave equation satisfies \begin{equation} \square G(x^{\mu}) = \delta^{(n+1)}(x^{\mu}).\tag{1} \end{equation} In $...
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Analytical solution to Poisson's equation for gravity

I am studying Poisson's equation for gravity. $$\nabla^2 \varphi = 4\pi G\rho$$ I have read that it is solved analytically using some Green's function, to give the well known formula of potential $$ \...
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Stuck solving an Inhomogenious differential equation using Green's Function

In my Quantum Mechanics homework, I had to solve the following differential equation $$ \left(\frac{d^2}{dx^2} + k\right) \psi = \lambda (\delta(x-a) + \delta(x+a)) $$ Which comes from the potential $...
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Obtaining the Green's function for a 2D Poisson equation ( in polar coordinates)

I am trying to solve the following BVP within an annular region of radii $r_1$, and $r_2$ : $$ \begin{cases} \nabla^2u=f\\ u(r_1) = p\\ u(r_2) = q \end{cases} $$ If we define an auxiliary problem in ...
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Schwartz's derivation of the Feynman rules for scalar fields

In his book "Quantum field theory and the standard model", Schwartz derives the position-space Feynman rules starting from the Schwinger-Dyson formula (section 7.1.1). I have two questions ...
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Generating functional in $\phi^4$ theory calculation upto 1st order

This question is based on section $1.2$ of Gauge Theory of Elementary Particle Physics by Ta-Pei Cheng and Ling-Fong Li. In $\phi^4$ or $-\frac{\lambda}{4!}\phi^4$ theory let $W[J]$ be the vacuum-to-...
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What is the meaning of a propagator of a Dirac field and how to get a probability of a process from it?

Let me first present what is my understanding of a propagator. What we measure in the experiment is a probability of scattering. We try to construct a theory predicting these measurements. What we are ...
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Interpretation of the surface integral in the general solution for poisson‘s equation with dirichlet BC

If we are given $\Delta \Phi(x)=-\frac{\rho(x)}{\epsilon_0}, x \in V$ and $\Phi(x)=\omega(x), x \in \partial V$, we can find a solution to $\Phi(x)$ on $V$ using Green‘s function: $\Phi(x)=\frac{1}{\...
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Calculating some functional derivative

I am reading Mark Srednicki's quantum field theory, p.50~p.52 (Part I section 7). In the section, he derives a the formula for the ground state to ground state transition amplitude of harmonic ...
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Intuitive Approach to Wick's Theorem

Context I'm currently reading Many-Particle Physics by Gerald D. Mahan. In section 2.4 it explains Wick's theorem and he gives the example $$ _0\langle|T \hat{C}_\alpha(t) \hat{C}_\beta^\dagger(t_1) \...
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Why do Hankel functions represent travelling waves?

I was trying to find the fundamental solutions for the Helmholtz equation in $\mathbb{R}^d$ when I found this answer. Here, and in some other places, it is stated that Hankel functions represent ...
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Schrodinger operator with matrix potential and Green's function

Clearly Schrodinger operators with matrix potentials appear very naturally in molecular dynamics/quantum chemistry, particularly when considering a crude adiabatic basis or diabatic basis for an ...
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Calculate the energy gap using Green's function

Can I calculate the energy gap of the given Hamiltonian by Green's function? Is there any basic code in MATLAB to do that?
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Why is the Propagator given by the Green's Function for a General Field in Canonical Quantisation?

In canonical quantisation, it is taught that the propagator for the Klein-Gordon field is defined as $$\Delta_F(\vec x - \vec y) \equiv \left < 0 \right | \overleftarrow{\mathcal T} \phi(\vec x) \...
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Equivalence between the Schrödinger and Green's function frames for a free particle

I want to show the mathematical equivalence between the Schrödinger: $$\Psi(x,t)=e^{-it \hat{H}/\hbar} \psi'(x,0), \tag{1}$$ and Green's function (propagator method) : $$\Psi(x,t)=\int_{-\infty}^{+\...
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Matrix elements of the self-energy in the GW approximation

I have been teaching myself the $GW$ approximation (or the $G_0W_0$ version of it). I am able to follow the entire derivation of Hedin's equations, etc. and the $GW$ approximation itself. However, I ...
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Harmonic oscillator propagator in Euclidean time

I'm following Nastase's book on Quantum Field Theory but this question is just about quantum mechanics in the path integral formalism. In chapter 8 he considers the propagator equation for a harmonic ...
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Boundary condition of dyadic Green’s function for planarly layered media

The dyadic Green’s function of free space in terms of orthonormal system for TE/TM polarized waves is: where, now by applying boundary conditions to the first equation, the dyadic Green’s function ...
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Shift in renormalized Green's function

In chapter 12.2 p. 410 of Peskin and Schroeder the Callan-Symanzik equation is derived. I understand the relation between (connected) renormalized and non-renormalized Green's functions given by $$ G^{...
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Is the retarded propagator exactly the Green's function?

I am trying to prove that, for the real scalar field $\phi(x)$, the retarded propagator, which is defined as $$ D_{R}(x-y)=\theta(x^0-y^0)\langle 0 |[\phi(x),\phi(y)]|0\rangle $$ is the Green's ...
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How to obtain Green function for the Helmholtz equation?

all. I am following Jackson's Classical Electrodynamics. At Chapter 6.4, the book introduces how to obtain Green functions for the wave equation and the Helmholtz equation. I have a problem in fully ...
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Point charge above a ground plane without images

Suppose one wanted to find the electric field around a point charge located above an infinite ground plane as in the classic method of images example, but without using the method of images itself. (I....
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Intuition about non-equilibrium Green functions

In the following paper by Fotso and Freericks, the definitions of the Green functions are given as \begin{align} G^<(t,t') &= i \langle c^\dagger(t') c(t) \rangle \\ G^R(t,t') &= -i \theta(...
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Green's function for the inhomogenous free particle time-dependent Schrodinger equation in 3D

I need to solve the equation $$ \mathrm{i}\hbar \frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\left(\frac{\partial^2\Psi}{\partial x^2}+\frac{\partial^2\Psi}{\partial y^2}+\frac{\partial^2\Psi}{\...
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Does the photon propagator see the $Z$ boson propagator pole?

As I understand, the $Z$ boson (as it decays) has a pole in its propagator that is somewhere in the complex plane, shifted off the real line. Now, suppose that I look at the full (interacting) photon ...
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Is self-energy $\mathrm{Im}\Sigma^r<0$ always true?

Consider a one-particle retarded Green's function $$G^r(\alpha)=[\omega+i\eta-\varepsilon(\alpha)-\Sigma^r(\alpha)]^{-1}$$ with self-energy $\Sigma^r(\alpha)$ for some quantum number $\alpha$. It is ...
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Scalar field Bulk propagator

For a massless scalar field in $AdS_{d+1}$ the bulk propagator is \begin{align*} \Box_{\vec x, z} K_B(\vec x, z;\vec x') = \delta^{d} (\vec x - \vec x') \end{align*} if the solution to $K_B$ is \begin{...
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Help with Wick's theorem in a $\phi^4$ QFT

QFT noob here. I am currently working out the momentum space two-point function for a $\phi^4$ qft in Euclidean space time, and considering the $\lambda^1$ order contribution, I am encountering a ...
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Imaginary poles of the Green's function

I am dealing with a theory with the Green's function that has infinitely many poles on the imaginary axis: $$G(\omega)= \sum_{n=1}^{\infty} \frac{1}{\omega-i\mu_n}$$ where $\mu_n=\cosh(n\beta/2)$, and ...
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QFT generating functional and Green function and propagator

I am confused about why does the generating functional gives the propagator by differentiation, and why that propagator is the Green function. I understand how to take the functional derivative like ...
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Classical Green function

What is the physical reason why the classical Green's function is not defined as a principle value integral? In a recent discussion (Classical Green's function) it was said that the classical ...
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Relationship between KG equation and Yukawa potential

If we start from Klein Gordon Lagrangian density and work through canonical quantization, we could arrive at field operators for scalar fields. Now, if we solve for the free propagator we arrive at (...
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Euclidean-signature Klein-Gordon propagator in curved space don't match with the one in flat space

The Klein-Gordon propagator in euclidean signature and flat space can be written as: \begin{equation} G(x,y)\propto m \frac{K_1(m\sqrt{s})}{\sqrt{s}} \tag{1} \end{equation} With $s=(x-y)^2$ and $m$ a ...
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Meaning of correlators $\langle A(t)B \rangle$, $\langle [A(t),B] \rangle$, $\langle \{A(t),B\} \rangle$, etc

In quantum mechanics and many-body theory, one often encounters correlators like $$\langle A(t)B \rangle, \quad \langle [A(t),B] \rangle, \quad \langle \{A(t),B\} \rangle,$$ where $A$ and $B$ are two ...
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Do these Green's functions satisfy the Lorenz gauge condition?

I have read a little in "The wave equation on a curved space-time" by Friedlander. The book gives a construction of the retarded and advanced green's functions of a hyperbolic wave operator ...
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Understanding Green's function in the mirror charge method

Initially I want to point out that there are many way as to how we can define the green's function in vacuum or when other objects are taken into consideration, and the same goes for the Potential in ...
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