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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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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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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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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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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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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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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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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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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'...
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Integrating a function defined in Matsubara frequencies

I am writing a code for the numerical evaluation of susceptibilities. The formalism is explicitly written on the Matsubara axis (fermionic case) and in the heart of the procedure lie multiple ...
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Stress state in an elastic half space with uniform constant body force

Is there a closed form expression for the stress state in an elastic halfspace subject to a uniform constant body force? I know that the Green's function for this problem is given by Mindlin's ...
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Property of surface Green function in electrostatic field

Let's consider a 2D-square with 4 equal subsquares containing different dielectrics. Inside the square domain, the unknown electric potential function $\Phi$ satisfies the Laplace equation: $$\nabla^...
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S-matrix and Green's function

I'm considering one paper about electron recombination and there is an expression for S-matrix that confuses me $${S_{fi}} = i\mathop {\lim }\limits_{t' \to \infty \atop t \to - \infty } \left\langle ...
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Reflection and transmission of general electromagnetic wave

Given is a source $S$ which produces an electromagnetic wave $E(x,y,z)$. The source is in vacuum. At z=0 there is an interface between vacuum and a perfect dielectric with $\epsilon$. The electric ...
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Where are the poles of the one-particle Green's function located in the complex plane?

This post is a followup question to: How to get an imaginary self energy? In the cited post, the two following representations for the one-particle Green's function are shown: $$G(k,\omega) = \...
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Method of Images for the diffraction integral with Neumann and Dirichlets conditions

In the solution of the wave equation $$[\nabla^2-\frac{1}{v^2} \frac{\partial^2}{\partial t^2}]U=0$$ and consequently the Helmholtz equation $$[\nabla^2+k^2]G(R,\tau)=\delta(R)=\delta(\vec{x}-\vec{...
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Green's function: relation to DOS and analyticity

It is not uncommon to see the (retarded) Green's function being defined in terms of the DOS: $$ G^R(\omega) := \int \frac{\rho(\varepsilon)}{\varepsilon-(\omega+i0^+)} \mathrm d \varepsilon. $$ I ...
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Functional derivative for the same function expressed before and after Wick rotation

This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf On page 17, they ...
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Truncated $N$-Point Functions

In Quantum Field Theory, truncated N-Point functions (or truncated Green's functions) are the N-Point functions of diagrams with their external legs chopped off. I was told that the truncated N-Point ...
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interpretation of the retarded many-body Green's function as particle propagator

Recently I realized that I might have overlooked, or I misinterpret, something about the retarded Green's Function in the context of many-body theory. Let's consider the simple case with: $$G_R(\vec ...
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Causal propagator and Feynman propagator

I have some questions about the Green’s function of the Klein-Gordon operator and the Feynman propagator. The first is about retarded Green’s function: \begin{eqnarray} \int_{-\infty}^\infty\frac{d^...
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Delta function from poles of Green's function

In quantum mechanical scattering theory, we often use Green's functions which contain poles. For example, in Schroedinger quantum mechanics the free Green's function is given by $$ G_0(\vec{p}) = \...
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Green's function regularization and delta distribution

I have a free Green's function which is proportional to a $2\times 2$ matrix: $$ G_0 = \frac{1}{E^2-E_k^2}\begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ The total Green's function after ...
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Density of states after scattering

I need some help with a probably simple question because I'm not sure whether my approach is correct. Let the free Green's function of a system on a discrete lattice described by a massless Dirac ...
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Formulating BCS theory in the functional integral with a real order parameter

Often in BCS theory, people take the order parameter $\Delta$ to be real. I tried to construct a BCS theory with a real order parameter from the start and ran into some trouble. I'd be interested to ...
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What is the reason for the extra minus sign in $(\Box_x - m^2)G_\mathrm{F}(x,y) = - \delta^{(4)}(x-y)$?

The Feynman propagator is given by the expectation value of two time-ordered (scalar) field operators (evaluated in the vacuum): $$ G_\mathrm{F}(x,y) \equiv \langle 0 | \mathcal{T}\big( \hat{\phi}(x) \...
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Dyson Equation and scattering amplitude

For scattering theory in quantum mechanics, one can use the Dyson Equation which states that the Green's function which is a solution to the equation $$ (E - H_0 - V)G = 1$$ is given by $$ G = ...
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Is $\delta(r-ct)/4\pi r$, the 3D wave equation elementary solution, a transverse or longitudinal wave?

Background: https://en.wikipedia.org/wiki/Longitudinal_wave 'Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the ...
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Lippmann-Schwinger Equation in Dirac theory

Consider a scattering process of some particle which is described by a Dirac Equation. We use the Lippmann-Schwinger Equation for the total scattering wave function in representation-free form up to ...
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Asymptotic relation of Green's function for diverging self energy

I am considering the derivation on pages 64 to 66 of Zagoskin's Quantum Theory of Many-Body Systems. They consider a Green's function in the Lehmann representation: $$ G(p,\,\omega)=(2\pi)^3 \sum_s \...
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Electrostatics: Induced Boundary Dipole Layer

In Jackson's classical electrodynamics 3ed eq. 1.36, the electric potential in a region $\mathcal{R}$ (that is, $V(\vec{r})$ for $\vec{r} \in \mathcal{R}$) is given by the sum of three terms, i) the ...
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Relativistic scattering off Dirac delta potential

Consider the case of a relativistic electron on a graphene lattice described by the Hamiltonian $$ \mathcal{H} = v\begin{pmatrix} 0 & p_x+ip_y \\ p_x-ip_y & 0 \end{pmatrix}, $$ where $v$ is ...
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Formal definition of Green function

The formal definition of a Green's function is: \begin{equation} L(\mathbf{r})G(\mathbf r,\mathbf r^\prime) = \delta(\mathbf r-\mathbf r^\prime), \tag 1 \end{equation} where L is a time linear ...
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Topological Classification of Band Insulators (in terms of Green's functions)

I am currently reading Topological Classification and Stability of Fermi Surfaces by Y. X. Zhao and Z. D. Wang (PRL 110, 240404 (2013)). They remark that the Green's function (along the complex ...
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Green's function in Frequency Domain

I am learning some basics of Green's functions applied in physics from the article https://arxiv.org/abs/1604.02499 I am struck at equation no (23) which is said to be derived from equation (22) by ...
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Help with Correlation/Green's Function of Rotated Variables (Keldysh Rotation)

I'm working through this paper, and have encountered "a little algebra shows that...", yet I'm not familiar enough with the topic at hand to figure this out. Here is the paper: https://arxiv.org/abs/...
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Show that Propagator satisfies Schrödinger equation

I want to show that $$K=K(x,x',t-t')=\sum_{\beta}\exp\left[\frac{-iE_{\beta}}{\hbar}(t-t')\right]$$ satisfies the Schrödinger equation $$ H|\psi\rangle = i\hbar\partial_t|\psi\rangle$$ with respect ...
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D'Alembertian Green's Function and Wick Rotation

Consider the wave equation: $$ \square A(t,x^i) = S(t,x^i) , $$ where $\square = -\partial_\mu \partial^\mu = \partial_t ^2 - \nabla^2 $, $S(t,x^i)$ is the source term and $A(t,x^i)$ is the field of ...
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Closed form of Iterated integrals arising in Fredholm's integral equation solution in the context of Nonequilibrium Quantum field Theory?

While solving a Non-equilibrium quantum field theory problem I came across this class of $2n_{}^{\text{th}}$ order iterated integral : $$F(T_{}^{},T_{0}^{},\epsilon)=\int_{T_{0}^{}}^{T_{}^{}}dt_{1}^{}...
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the effects of initial condition on Green function

In literature, for proving the existence of Green function for linear systems, it is argued that if for a linear differential equation like $\mathcal{D}[y] = \sum_{n=0}^N {a_n y^{(n)}}$ $y(0)=y_0, \...
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Is the replacement $i\omega_n\rightarrow \omega+i\eta$ in Matsubara Green function valid?

In many-body theory we know that to find the retarded Green function in frequency space $G^R(\omega)$ we first find the Matsubara time-ordered Green function $\mathcal{G}(i\omega_n)$ and then replace $...
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QFT Propagator across spacelike separation

I have this general formula for the propagator. $$ D(x)=-i\iiint\dfrac{ d^3k}{\big( 2\pi \big)^32\omega_k}[e^{-i[\omega_kt-\vec k\cdot\vec x]}\Theta(t)+e^{i[\omega_kt-\vec k\cdot\vec x]}\Theta(-t)] $...
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How is 'nonlinear optics' related to 'nonequilibrium'?

According to what I found, nonlinear optical process is related to nonequilibrium physics - nonequilibrium green's function (Keldysh green's function/formalism) appears in nonlinear optics. However, ...
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Calculation of conductivity, some integral question

I have some questions of the integral when I calculate the conductivity. The integral actually is as the following form: $$\frac{1}{L^3}\sum_{p} p^2 G_{p}(\epsilon^-) G_{p}(\epsilon^+ +w) = \int \...
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Doubt while completing the square in the generating functional of free scalar theory

Consider the free massive real scalar field in 4 Euclidean dimensions. The generating functional for Correlation functions is $$ Z[J]=\int[d\phi]\exp\bigg(\int d^4x\,[-\frac{1}{2}\phi(-\partial^2+m^2)\...
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Determining the Greens function

I understand the basics of greens function. But here in this paper, i am not able to understand how have they solved for the Greens function. In particular, I don't get what $G_{R}$ and $G_{A}$ are....
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Method of Images and Green functions in Quantum Field Theory

Traditional Laplace equation solutions techniques for a system of conductors rely on the fact that the Green function for solving Laplace's equation represents the potential due to an image charge ...