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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Conflicting definitions of Bulk-to-Boundary propagators in AdS

This problem has to do with bulk reconstruction in AdS/CFT. It is given that the bulk-to-boundary propagator can be obtained from the bulk-to-bulk propagator by the following relation (c.f. https://...
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Temperature Dependence of the Kubo Formula

I'm trying to calculate the DC conductivity of a Renormalized Fermi Liquid with Green's function \begin{equation} G(i\omega,k)=\frac{Z}{i\omega-Z\tilde{\epsilon}_k-ig\omega^2} \end{equation} where $...
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Green function derived from the idea of position representation in quantum mechanics

I am trying to study green's function by using a lecture by NPTEL its link is https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s I don't get this step by him at the minute of ten. $$D_x G(x,x^{'...
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45 views

How to know how the self-energy changes the mass?

Suppose we have a Green's function of the typical form \begin{equation} G(k)=\frac{1}{k^2-m^2-\Sigma(k)} \end{equation} where $\Sigma(k)$ is the self energy of that particle. How exactly can we ...
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A question in imaginary time Green's function

I am learning many-body quantum field theory with Bruus and Flensberg's Introduction to Many-body Quantum Theory in Condensed Matter Physics, there is a derivation that confuses me a lot. To add ...
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60 views

Relationship between Dyson equations from different problems

Recently, I noticed that the Dyson equation $$G=G_0+G_0\Sigma G$$ is used not only in quantum field theory but in some other branches of physics. For instance: 1. Wave equation From the wave ...
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109 views

How to obtain the quasiparticle equation from Dyson equation?

The problem is formulated as follows: Dyson equation for zero temperature Green's function: \begin{equation} \left[ i\dfrac{\partial}{\partial t_1} - h(\vec{r}_1) \right] G(1,2)-\int d3 \Sigma(1,3)G(...
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73 views

How to derive the vertex function from mass operator in Hedin's equations?

I am stuck from the mass operator to vertex function in the derivation of Hedin's equations. The problem could be organized as follows: Mass operator: $$M(1,2)=i\hbar\int d(34)v(1^+,3)\dfrac{G_1(1,4)}...
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37 views

References regarding Green's function on a square domain in 2D

Premise: I know this question would be better suited to MathSE, but since I endeavour to solve a free CQFT on a bounded domain, I'm confident I'll find a more exhaustive answer here. I'm trying to ...
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How can I prove/understand the following functional derivative? [duplicate]

Assume that $F[h(\xi);x,y]$ be the inverse of $G[h(\xi);x,y]$ in the sense that the following identity is satisfied: \begin{equation} \int dz F[h(\xi);x,z]G[h(\xi);z,y] \equiv \delta(x-y) \end{...
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How to compute thermodynamic magnitudes with the Green's function?

I'm studying the SYK model and there seems two equivalent approaches for solving it. One is the diagrammatic expansion in the large $N$ limit, where we get self-consistent equations (in imaginary time)...
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86 views

Momentum distribution Fermi liquid and spectral representation

In a Fermi liquid the momentum distribution shows a jump at the Fermi surface, i.e. \begin{equation}\langle n_{k_F-\delta k} - n_{k_F+\delta k}\rangle = Z_{k_F}\end{equation} with $Z_k$ the strength ...
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Real and Imaginary time Green's Functions

In real time, one can calculate the two point function of a given theory using \begin{equation} G(\vec{x},t)=\langle \Omega | \phi(\vec{x},t)\phi^\dagger (0,0)|\Omega\rangle =\int_{\phi(0,0)}^{\phi(\...
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60 views

Regularising the Green's function in 2D

The Green's function for the 2D Helmholtz equation satisfies the following equation: $$(\nabla^2+k_0^2+\mathrm{i}\eta)\,{\mathsf{G}}_{2\mathrm{D}}(\mathbf{r}-\mathbf{r}',k_o)=\delta^{(2)}(\mathbf{r}-\...
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Integrating the dyadic Green function of a dipole source

I would like to ask if anyone know how to derive the equation (4) from this paper? I could not figure out how they derive the term $-\mathbf{P}_\mathrm{LO}(\mathbf{r})/3\varepsilon_0\varepsilon_B(\...
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How to derive the Galitski-Migdal formula from the definition of zero temperature Green's function?

Usually, in condensed matter physics the zero temperature Green's function is defined as: $$G(x,t,x',t')=-i \langle 0| \psi(x,t) \psi^\dagger (x',t')|0\rangle \qquad x\equiv(\vec{r},s)$$ in which $| 0 ...
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47 views

How to apply Wick's theorem in Anderson model

I'm trying to solve the non-interacting single impurity Anderson model where we consider free electrons in a conduction band: $$H_{cond} =\sum_k \varepsilon_k c_k^\dagger c_k$$ and an impurity with ...
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1answer
111 views

Complex integration in Peskin and Schroeder

In Peskin and Schroeder, I have a problem with a claim in equation (2.54), which I will rewrite more concisely here. He claims that we have the following equality : $$ \frac{1}{2E_p}e^{-iE_p(x_0)}-\...
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What is the Quantum Mechanical analogue of the Bethe-Salpeter equation?

For studying the bound states of quantum fields theories (e.g. studying excitons or mesons), the Bethe-Salpeter equation is often used as the starting point. Quoting Wikipedia the equation is: $$\...
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Combinatorics geometric series two-point function

In this answer Proof of geometric series two-point function it is said: Now what about the coefficients in front of each Feynman diagram? Due to the combinatorics/factorization involved it ...
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How to derive the analytical expression for the retarded Green's function with quadratic Hamiltonian?

For two operators, $A(t)$ and $B(t)$ the retarded Green’s function is defined as \begin{equation} G^R(t,t') \equiv \langle \langle A(t)|B(t) \rangle \rangle^R = -i\theta(t-t')\langle \{A(t),B(t')\} \...
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Finding the Green function of an operator in QFT

I'm working on some quantum field theory and have to operate on a field with the following operator: $$ (x^\mu \partial_\mu + 1)^{-1} $$ I've been trying to find an explicit form of this operator, ...
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About electrode self-energy and the relation between transmission functions and Green’s functions

I am reading Electronic Transport in Mesoscopic Systems by Supriyo Datta. I got stucked when deriving some formulas. On page 147, the book says "see exercise E.3.3" when it gives the formula (3.5.18), ...
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Two-dimensional bosonic field theory

I'm struggeling with the following question: Consider a two-dimensional bosonic field theory defined by the following action $$S =\frac{k}{2} \int dx_{1}dx_2 [(∂x_1 φ(x_1, x_2))^2 + (∂x_2 φ(x_1, ...
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Why does it matter that the propagator is related to the Green's function for the Schrodinger equation?

If $L = i \hbar \hat{H} - \dfrac{d}{dt}$, then $ L \psi(x,t) = 0$ is the Schrodinger equation. It is well known that we can solve the Schrodinger equation with initial condition $\psi(x,0) = f(x)$ ...
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64 views

Derivation of the QFT Propagator

I don't understand how we get from the RHS to the last line. \begin{eqnarray} \left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\...
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What is the physical meaning of Green's theorem and Green's identities?

During the derivation of Kirchhoff and Fresnel Diffraction integral, many lectures and websites I found online pretty much follows the exact same steps from Goodman(Introduction to Fourier optics) in ...
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Using Green's function to solve wave equation driven by oscillatory line source

I am asked to show that $$U(x, y, t)=\int dt'\cos(\omega t')\frac{c}{2\pi}\frac{\theta\left( c(t-t')-\lvert \mathbf r\rvert \right)}{\sqrt{c^2(t-t')^2-\lvert \mathbf r\rvert^2}},$$ where $\lvert \...
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How does the green function for the wave equation in three dimensions preserve the ordering of noises between a speaker and a listener

I was provided with the following equation in class for the Green's function of a three dimensional wave equation: However, I am confused as to how this form of the Greens function preserves the ...
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Determining the position and spectral weight of Andreev states from a retarded Green Function

I'm trying to understand a result from the paper Josephson current through a correlated quantum level: Andreev states and $\pi$ junction behavior by Vecino, Martin-Rodero and Yetati (2003). ArXiv link ...
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Kallen-Lehmann representation and branch cuts at threshold masses

Let us consider the Kallen-Lehmann representation for the two-point function of scalar fields $$ \langle \Omega | T\left\{\phi(x) \phi(y)\right\}|\Omega\rangle = \int \frac{d^4 p}{(2\pi)^4} e^{ip\...
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Literature recommendations for superconductivity

What text books / review papers give an easy introduction to superconductivity? The literature should be suitable for Master students with some background knowledge in theoretical solid state physics ...
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Evaluation of Green Function for Helmholtz Equation - Phillips and Panofsky

I'm reading Phillips & Panofsky's textbook on Electromagnetism: Classical Electricity and Magnetism. At chapter 14, section 2, we are presented with a solution of the wave equations for the ...
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No bound states if propagator is everywhere infinite?

Assuming the energy spectrum is discrete, the propagator for the time independent Schrodinger equation can be represented as $$G(x,y,E)=\sum_n\frac{\psi_n(x)\psi_n^*(y)}{E-E_n}.$$ The propagator's ...
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Green's function in a region between a conductor sphere and two charged spheres inside, with point charges inside of each

Please, help me. I have to find the Green's function in the following region, but I don't have any idea how to find it: I have a conducting spherical shell of radius a; in the center there are 2 ...
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How to understand the construction of the retarded Green function?

I'm trying to make clear of arxiv:1608.05392v1, where the expression of retarded Green function in eqation (2.10) is hard to understand. I've read many references about Green function, but I didn't ...
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Green's function for infinite square well

The Green's function can be given in terms of left and right solutions. $G(x,x';k) = \frac{1}{W}\left(\Psi_{L}(x_{<})\Psi_{R}(x_{>})\right)$ But I don't understand how to determine these left ...
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124 views

What happens in the Hartree and Fock diagrams?

I am trying to understand the Hartree and Fock diagram shown in the picture. To understand it a assume there is an electron entering and leaving at the tail of the tadpole (Hartree diagram) and an ...
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A confusing point in linear response theory on the ground state

Information about a quantum system could be drawn from its response to a small perturbation. This is formulated in what is known as linear response theory. In second quantization, consider a ...
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56 views

Constant Density of States and Perturbation theory

Given a constant Density of States (DOS) corresponding to a one-electron hamiltonian, $\text{DOS}(\omega)=\dfrac{1}{2D} \chi_{[-D,D ]}(\omega),$ where $\chi$ is the indicator function, I want to know ...
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68 views

Greiner's Green's function for diffusion

I am reading Greiner's "Quantum Electrodynamics". In example 1.5 he derives the Green's function for diffusion. I am stuck on a step in the derivation. He has the defining differential equation as $$ ...
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70 views

Green's Function Method for a Spring and mass system [closed]

I think I've done part a) correctly and I have a general solution. However, I now have two unknown constants in my general solution and, as far as I can see, only one condition ($x(0)=-1$) with which ...
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287 views

Why the imaginary part of green function is the spectral function?

In the zero temperature Green's function, $$G_{\alpha\beta}(xt,x't')=-i\langle \Psi_H|\hat{T}[\psi_{H\alpha}(xt)\psi^{\dagger}_{H\beta}(x't')]|\Psi_H\rangle .$$ In Lehmann Representation, $$G_{\...
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Are vacuum-fluctuations a consequence of causality?

I'n new to QFT, and recently lerned about the propagator of a free scalar field theory in Minkowski-space, which according to our lecture notes looks like $$G(p, q) = \frac{1}{q^\mu q_\mu + M^2} \...
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1answer
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How to derive the retarded Green's function matrix for a quadratic Hamiltonian?

Start with the quadratic Hamiltonian for fermion: $$\hat{H}=\sum_{ij}H_{ij}\hat{c}_i^\dagger \hat{c}_j$$ and the definition of retarded Green's functon in time domain: $$G_{i,j}^r(t_1,t_2)=-i\theta(...
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Deriving Ward identity directly from a given formula for the conserved current only using the equal-time canonical commutation relation

I have a very technical question on deriving a Ward identity directly from a given explicit form of the "conserved current". Let me emphasize that I do not start with an apriori knowledge on the ...
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61 views

What's the physical meaning of the kinetic Green's function?

I'm struggling to understand the physical meaning of some of the Green's functions relations. Especially the relation known as the Kinetic Green's function. Which by definition is the sum $ G^{K} = G^{...
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184 views

How to calculate conductivity / electron mobility from theory?

Is there a way to make quantitative statements about the conductivity of materials with band theory? If not I should still be able to get information about the conductivity from Green-Kubo relations ...
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Switching between advanced and retarded solutions mid-integral?

In wave mechanics, an advanced solution can be thought of as a wave that propagates until it is "caught" and stopped by the forcing function, and a retarded solution can be thought of as a wave that ...
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Fourier transforming a Dyson equation

I have a Dyson equation for a Green's function that comes in this form: $$ G[t,x_f;0,x_i]=G_0[t,x_f;0,x_i]+i\int_\Omega\int_0^t\ dx\ d\tau\ G_0[t,x_f;\tau,x]xG[\tau, x;0, x_i] $$ For convenience, I'...