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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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Calculating a space and time dependant temeprature profile of a very long pipe in an infinite medium

I am trying to calculate the time and space depenadant temperature profile of an infinite medium after inserting a very long pipe into it using Green's function. The pipe serves as a heat source with ...
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Given Green's function, can I find the corresponding operator? [migrated]

Green's function is the solution to the equation $L G(x;x') = \delta(x-x')$, where $L$ is a linear differential operator. Usually, we want to find the Green's function of a given $L$. Instead, if we ...
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Photon propagator in path integral vs. operator formalism

I am self-studying the book "Quantum field theory and the standard model" by Schwartz, and I am really confused about the derivation of the Photon propagator on page 128-129. He starts ...
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Are there exactly solvable problems in curved space, except for cases of constant curvature of space?

I have two questions. I know the expressions for geodesic distance in Minkowski, de Sitter and anti de Sitter space-time and their Euclidean analogues $R^n$, $S^n$ and $H^n$ [1]. For what other curved ...
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Retarded Green's function in Peskin & Schroeder

In an Introduction to Quantum Field Theory by M. E. Peskin & D. V. Schroeder (eq. 2.56 on page 30) the following relation for the retarded Green's function was established: $$(\partial^2 + m^2) ...
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Questions about fundamental solutions and propagators for the Dirac operator

I thought that propagator is a synonym for fundamental solution. But that seems not to be the case since in this answer it is said that an expression with delta function on a surface has to be ...
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I don't understand Green's derivation of the Laplace/Poisson equation inside an electrically charged body

I am currently reading George Green's "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism"Green's essay to gain some insight into the original ...
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Intuition for imaginary time Greens function

I understand that $$G^M(0,0^+) = \operatorname{tr}\{\rho O_2 O_1\}$$ (I am not putting hats on the operators here because they don't render in the correct position) is simply the expectation value of ...
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Green function in scattering theory

I'm having a bit of trouble with a step in scattering theory. Context: The Schrödinger equation for a two-body scattering problem can be written as: $$ (E - H_0) |\psi\rangle = V |\psi\rangle. $$ Here,...
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Keldysh rotation and Langreth theorem

Given a (Green) function of two time arguments on the Keldysh contour $g(\tau_1, \tau_2)$, we can distinguish between four cases, depending on whether each contour time lies on the forward or reverse ...
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General Solution to Maxwell's Equations with Duhamel's Principle

In one dimension, it is easy to prove that if two solutions $\{u_1, u_2\}$ are known to $\mathcal{L}u(t) = 0$ where $\mathcal{L} \equiv{a(t)\partial_t^2+b(t)\partial_t+c(t)}$, the general solution to ...
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Fourier transform of spin Matsubara Green function

Is the spin Matsubara Green function of a generic spin operator (or product of spin operators) bosonic? How can one obtain its frequency decomposition?
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How to study regularity of a Green's function when solving field equations perturbatively?

Preliminaries Consider a nonlinear differential operator $\mathcal{O}$ acting on a field $\phi$, with source $\rho$ $$\mathcal{O}(\phi)=\rho$$ Let's say the charge density is small, so we can define $\...
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Bulk-to-bulk propagator in 3-point function in AdS-CFT correspondence. Trouble solving a PDE

I have encountered an issue in a PDE (A Green's function actually). I am solving it in $(d+1)$-dimensions and I use Poincare coordinates in AdS spacetime, meaning I have a dimension $z$ and I also ...
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Photo-excited Radiation/Electric Current with Green Function Method

Suppose there is an incident photo with a specific frequency hitting the material; the incident photo is absorbed by this material. Meanwhile, the material is excited by this incident photo and ...
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Relation between time-ordered propagator in condensed matter and Feynman propagator

In particle physics I am used to the Feynman propagator being decomposed into positive and negative frequency Wightman functions. For example, this is the representation used in Eq. (6.2.13) of ...
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Self Consistency of Wave Function Given By Green Function ($\psi(r,t) = \psi^0(r,t) + \int dr'\int dt G_0(r,r',t,t') V(r')\psi(r',t') $)

In "Introduction to Many-body Quantum theory in condensed matter physics" by Bruus and Flensberg there is an exercise regarding Green functions. We want to solve the time dependent ...
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Different definitions of resolvent in matrix model

When I study the matrix models, I get confused of different definations of resolvent. After we define the partition function as $$Z=\int[dM]e^{-NTrV(M)},$$ where $V(M)$ is a matrix valued function of $...
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Green Function equation

I have a doubt on how to compute the green function $G$ of an operator $O$. By definition the green function of an operator $O$ is, $$\int dx_2 O(x_1,x_2)G(x_2,x_3)=\delta(x_1-x_3)$$ my operator is, $$...
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Interpretation of spectral function for Anomalous Green Function

This question assumes we have quasi-particles which can be diagonalized by a BV transformation (here I am using imaginary time and I am in position space): $$ \hat{c}_{i\sigma}(\tau) = ...
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Two-particle Green's function, possible typo in the book referred?

I'm trying to follow a computation in some QFT book, p64. The goal is to derive the equation of motion for the lesser Green's function $G^<$ defined as $$ G^< = \mp i {\rm Tr}\left(\rho \Psi^\...
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Analytic continuation Matsubara/imaginary-time to retarded function in complex time domain

In linear response theory, one may either use the real-time retarded correlation function, or analytically continue to imaginary time/frequency to use the Matsubara Green's function instead. While ...
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Green's function solution in 2D for the potential of solenoids in the Lorenz gauge

My main goal is to find a general expression for the potential in the Lorenz gauge of some solenoidal (not necessarily circular) current density using the Green's function. I assume that the current ...
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The relation between spectral function, Green function and particle number in many-body systems

The spectral function is defined as the imaginary part of Green function multiplied by $-2$ (Ref. Mahan. Many-Particle Physics 3ed. Kluwer Academic, 1990.), $$ A(\mathbf{k},\mathrm{i} \omega_n) = -2 \...
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Invert operator to integrate heavy fields

We have a Lagrangian $$\mathcal{L}=\frac{1}{2} \partial_\mu \Phi \partial^\mu \Phi - \frac{1}{2} M^2 \Phi^2- \frac{\lambda}{4}\phi^2 \Phi^2 - \frac{g}{2} \Phi \phi^2+\cdots $$ where $\Phi$ denotes a ...
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Fermionic propagator [closed]

Given the fermionic generating functional $$Z[\eta]=\ det^{\frac{1}{2}}(K_{ij})e^{-\frac{i}{2}\eta_{i}G^{ij}\eta_{j}},\tag{1}$$ where $$G^{ij}=K^{-1}_{ij}$$ is the Green function of our theory, then ...
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Non-homogenous Helmholtz equation in 3+1D: Green's function and solution

I've been reading Jackson's Chapter 8.10 and trying to find the Green's function for a non-homogenous Helmholtz equation. The problem is in cylindrical coordinates. I first made a Fourier transform to ...
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Linewidth from correlation functions

I work on cavity modes interacting with matter inside the cavity and I want to determine the linewidth of the light. I do not have excess to the state of the system directly but in a steady state, I ...
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How is the Schrodinger kernel also a propagator? [duplicate]

Let $e^{-it\hat{H}/\hbar}$ be the time evolution operator for a Hamiltonian $\hat{H}$ and $K(x,t)$ its associated integral kernel, i.e. $$\varphi(x,t) = e^{-it\hat{H}/\hbar}\varphi_0(x) = \int_{\...
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The problem with derivation of an equation in a many-body book

In the corrected version (14 January 2016) of the book "Many-Body Quantum Theory in Condensed Matter Physics: An Introduction (Oxford Graduate Texts)" chapter 8, for the following time ...
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Understanding Green's Function Derivation for Inhomogeneous Wave Equation

I am currently reading through Zangwill's Modern Electrodynamics. In Chapter 20, Zangwill derives the Green's function for the wave equation $$[\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}]\...
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Why are 2-point functions Green's functions?

I asked a question about this earlier but I think it was unfocused so I have rephrased it and asked it again. The propagator/two-point function $\langle \phi(x_1)\phi(x_2)\rangle$ for any theory can ...
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Is there any intuitive reason why 2-point functions are inverse operators to the free Lagrangian? [duplicate]

To compute $n$-point functions in quantum field theory we use Wick's theorem to reduce this problem to computing 2-point functions. In many textbooks, such as Peskin & Schroeder, the 2-point ...
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Why does a singularity imply the need for a distribution?

I am following Section 11 of Prof. Etingof's MIT OpenCourseWare notes on "Geometry And Quantum Field Theory" in which he says: ...for $d = 1$, the Green's function $G(x)$ is continuous at $...
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4 votes
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Wrong solution for Green function of one-dimensional Poisson equation

An old electrodynamics exam question asks: "Find the Green function (for the one-dimensional Poisson equation) that solves the equation $$ \frac{d^2}{dx^2}G(x,x') = -\delta(x-x'). $$ Choose the ...
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Renormalization condition for field strength renormalization

I am studying $\phi^4$ theory and so far I understand the mass and coupling constant renormalizations. In these theories, once we expand a diagram in perturbation theory we "cancel" the ...
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2 votes
2 answers
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How does the Green's function related the wavefunctions at different space-time points in Schrödinger's equation?

I have been trying to study Quantum Field Theory and have come across Green's Functions for the first time. While referring to Tom Lancaster's book Quantum Field Theory for the Gifted Amateur, the ...
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2 answers
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Why $n-1$ point function vanishes in $D=0$ scalar theory?

If we consider a $D=0$ theory with the Lagrangian: $$\mathcal{L}[\phi]=g\phi^n+J\phi$$ And its Green functions: $$G_n=\langle\phi^n\rangle_{J=0}=\frac{1}{Z[0]}\frac{\delta^nZ[J]}{\delta J^n}|_{J\...
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Yukawa potential as the time integral of 4D retarded Green's function

I am attending an advanced QFT course, and trying to verify the instructor's claim that the retarded Green's function $$ G_{\text{ret}}^{(4D)}(t,\mathbf{x}) = \theta(t) \left[ \frac{1}{2\pi}\delta(\...
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Dirac equation: Green's function specified for only one dimension

Normally, the Dirac equation for the Green's function reads: $$(i\gamma^\mu\partial_\mu - m)S_F(x,y) = \delta^{(4)}(x-y)$$ Is it possible to define a Green's function describing the propagation ...
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How to derive the graviton propagator in curved spacetime?

Is it possible to derive the graviton propagator in curved spacetime from the graviton propagator in Minkowski spacetime?
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Question about impurity scattering of multiband Hamiltonians in $T$-matrix approximation

Imagine, we have a 2-by-2 Hamiltonian: $$ H = H_0+H_{imp}. $$ The impurity Hamiltonian $H_{imp}$ is $$H_{imp}=u_{imp}I_2\sum_{r_{imp}}\delta(\vec r - \vec r_{imp}).$$ $I_2$ is identity matrix of order ...
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6 votes
1 answer
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Feynman propagator as a sum over eigenfunctions

I often read something like "the Feynman propagator is the Green's function of the Klein-Gordon equation", so I try to write it as a sum over eigenfunctions, as should be possible for any ...
Proto's user avatar
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Amputated connected 2-point function is inverse to connected 2-point function

Let $D_n$ denote the $n$-point correlation function consisting of only connected diagrams. We may decompose this as an integral of two products. The first factor consists of a product over the $n$ ...
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Green's function of screened Coulomb interaction using partial Fourier transform

The solution of the differential equation (DGL) $$ (-\epsilon_0\nabla^2 + l^{-2} )G(\vec{r},\vec{r}') = \delta(\vec{r},\vec{r}') $$ is given by a screened Coulomb potential $$ G(\vec{r},\vec{r}') = \...
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Green function and probability amplitude

Consider the following Green function: $$G_{2}(x,t,x',t') = \langle \Omega_{0}, e^{itH_{0}}a_{x}e^{-itH_{0}}e^{it' H_{0}}a_{x'}^{*}e^{-it' H_{0}}\Omega_{0}\rangle$$ for $t' > t$. Here, $a_{x}^{*}$ ...
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How does one rigorously define two-point functions?

Let $\mathscr{H}$ be a complex Hilbert space, and $\mathcal{F}^{\pm}(\mathscr{H})$ be its associated bosonic (+) and fermionic (-) Fock spaces. Given $f \in \mathscr{H}$, we can define rigorously the ...
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Why do different contours give different answers in the limit $\epsilon \rightarrow 0$ when calculating propagators?

Let $\phi$ denote the Klein-Gordon field. Then its propagator $\langle 0 \mid [\phi(x), \phi(y)] \mid 0 \rangle$ can be calculated as $$\int \frac{d^4}{(2\pi)^3} \frac{-e^{-ip(x-y)}}{p^2 -m ^2}. \tag{...
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How small is $\eta$ when we say $\eta\to 0^+$ in Green's functions

When we convert Matsubara's imaginary time Green's function to the retarded Green's function, we perform an analytical continuation by substituting $i\omega_n$ with $\omega + i\eta$, with $\eta\to0^+$....
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Difference of the Transmission Coefficient between Thermal and Charge Conductance by Nonequilibirum Green Function Method

The equation 57 in the reference [Jian-Sheng Wang, Jian Wang and J. T. Lu, Quantum thermal transport in nanostructures, Eur. Phys. J. B 62, 381 (2008)] explains the the transmission coefficient for ...
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