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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What is the recursive relation for three-particle Green's functions?

In condensed matter physics, one often choose to study the many-body Green's functions (GF) with the diagram (perturbation) expansion technique. In what follows only two-body interaction is concerned. ...
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42 views

Feynmann Propagtor and the Green's Function for a Free Field

I'm going through Mark Srednicki's Quantum Field Theory. Chapter 8 on The Path Integral for the Free Field Theory includes the following: In the presence of a classical source, $J$, the ground state-...
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93 views

Which inverse of $-(\partial^2 + m^2)$ should be used in the path integral?

The partition functional for free scalar field is $$Z=\int D\varphi e^{i\int d^4x[-\frac{1}{2}\varphi (\partial^2+m^2)\varphi+J\varphi]}.\tag{1}$$ To evaluate this functional integral, we usually ...
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Viscous accretion disk, green formula

I have this problem. I am studying a the viscous accretion disk problem and in the assumption of constant viscosity the equation of diffusion of the superficial density becomes: $$ \frac{\partial}{\...
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4answers
142 views

The analytical result for free massless fermion propagator

For massless fermion, the free propagator in quantum field theory is \begin{eqnarray*} & & \langle0|T\psi(x)\bar{\psi}(y)|0\rangle=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i\gamma\cdot k}{k^{2}+i\...
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58 views

Fourier transform in two dimensions, Green's function for Schrodinger equation

I want to calculate this Fourier transform: $$ \int\limits_0^{\infty} \mbox{d}k \int\limits_0^{2\pi} \mbox{d}\varphi~ k \frac{e^{i \vec{k} \cdot(\vec{x}-\vec{x}')}}{k^2+B} $$ which will be 2D Green's ...
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83 views

Fluctuation-dissipation theorem in QFT

If I understand correctly, the fluctuation-dissipation theorem (fdt) in QFT technically arises because of $\pm i\epsilon$ - infinitesimally small summand in the denominator of spectral representation ...
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61 views

Physical interpretations of the generating functions $Z[J]$ and $W[J]$ (or $E[J]$)

In quantum field theory, the generator of all Green's functions $Z[J]$ and that of the connected Green's functions $E[J]$ are related as $$Z[J]=\exp[-iE[J]]=\int D\phi\exp[i\int d^4x(\mathcal{L}(\phi)+...
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67 views

Propagator from Path integral

In class we have proved something like: $$ \frac{\partial^2 Z(J,\bar{J})}{\partial J(x) \partial \bar{J}(x')}\frac{1}{Z}|_{J=\bar{J}=0}=\Delta(x-x').$$ That by introducing source terms to path ...
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89 views

Simple, physical explanations for Hadamard behaviour of two-point functions

The two-point function of local quantum fields on a curved spacetime exhibits a singularity of a very particular form, known as Hadamard form, for null separated points $(x,y)$ (including the ...
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148 views

Physical interpretation of the retarded vs. Feynman propagators?

We calculate the real-space propagator $\Delta(x)$ for a free real scalar field $\varphi(x)$ with mass $m$ by performing the Fourier transform (using sign convention +---) $$\Delta(x) = \int \frac{d^...
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47 views

Solving Laplace's equation using Green's functions

Consider a potential in cylindrical coordinates, $\Phi(\rho,\phi,z)$ that satisfies Laplace's equation $\nabla^2\Phi=0$. I am looking to determine $\Phi(0,0,z)$ for $z>0$ using Green's functions. ...
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58 views

How to calculate the contour integration with branch point? [closed]

The question come from a Mutusbara Sum like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ it equal a contour integral around Imaginary ...
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1answer
45 views

Notation of a vector containing equation in a paper

I'm trying to implement Coulomb long range interactions into a molecular simulation program using a particle-particle/particle-mesh Ewald solver. The following equation from the paper "How to mesh up ...
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193 views

What's the Propagator in the Free Particle Case? (Path Integrals with Source Term)

If I take the Lagrangian to be, $$L(t)=\frac{1}{2}m \dot q(t)^2$$ The Euclidean Path Integral is supposed to be, $$K=\int D[q(t)] \ e^{-\int L(\dot q) d \tau}$$ If I add a source term $J(\tau)$ we ...
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It it true that the Fourier transform of the return probability can be written as the product of the retarded propagator & the advanced propagator?

I know the retarded green's function $G(r,r; E)$ is expressed as $$G(r,r; E)=\sum_n \frac{|\mu_n(r)|^2}{E-E_n+i\eta}$$ with the eigenvectors/eigenvalues $\mu_n(r)/E_n$. This expression of $G(r,r; E)$ ...
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31 views

Can you have a problem with a Dirichlet boundary condition but with waves that reflect off the boundary?

Say we are looking for a solution to the Helmholtz equation $$(\Delta + k^2) u = 0,$$ in in the upper half space ($y > 0$) in 2D with a Dirichlet boundary condition on the $x$-axis, that is, $u(x, ...
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105 views

Two-point function of a free massless scalar field in Euclidean space-time

Let $\phi(x)$ be a free massless scalar field on $d$-dimesnional space-time with Euclidean metric. I am interested in the operator formalism, i.e. $\phi(x)$ is an operator satisfying $\Delta \phi=0$ ...
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48 views

What's the connection between the pole contours of propagators and their causality?

Wikipedia distinguishes between three kinds of propagators for a scalar field: The Retarded propagator's contours have $\mathrm{Im}(k^0)>0$ on both poles, so its limit is completely in the first ...
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46 views

Linear Response And path integral

I'm following Wen's book on Quantum field theory, and I'm struggling with section 2.2.1 on linear response and response functions. Specifically I'm unable to reproduce equation 2.2.7 in which the ...
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46 views

Charged conducting wire ring and Greens function

ρ(r) = q 2πa2 δ(r − a)δ(cos θ). If this is my charge distribution on a conducting wire ring. radius=a. I am attempting to show my electrostatic potential as an integral along the z-axis I started by ...
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1answer
20 views

Conductance of disordered conductor

I'm struggling with a rather advanced problem. Consider a conductor placed between two leads. The conductor is not completely clean but contains all kinds of impurities. The goal is to find the ...
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126 views

Conditions to determine the Green's function for scattering phenomena

Consider the elastic scattering of particles by a potential $V$ in Quantum Mechanics. In the zone of influence of the potential the Hamiltonian may be written as $$H = H_0 + V,$$ being $H_0$ the ...
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177 views

Why is the propagator the Green's function for Schrodinger equation? [duplicate]

Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying $$\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+V(\mathbf{x''})-ih\frac{\partial }{\...
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36 views

Can anyone suggest any good texts on Green's functions in quantum mechanics?

I am currently learning about Green's functions and want to write an essay on their use in quantum mechanics as part of an assessment. I have seen that they can be used in describing the probability ...
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25 views

Nonequilibrium Green's functions weakly interacting two-component Bose gas

I am planing to describe time evolution of two-component BEC. I was thinking about non-equilibrium Green's functions, but I don't if the method can be applied to the problem describe below. I know ...
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91 views

Renormalisation group equation for Green's functions

The renormalization group equations for the $n$-point Green’s function $$\Gamma(n) = \langle \psi_{x_1} \dots \psi_{x_n}\rangle $$ in a four-dimensional massless field theory are $$\mu \frac{d}{d \mu}...
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122 views

Help with understanding Green's Functions

I. The Green's Function Method The Green's function is immensely useful as a tool in Solid State Physics. Using a Green's function, one can compute all relevant data from a physical system. For ...
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74 views

Green's function in Hamiltonian vs. Path Integral QFT

For a spacetime $M$, the Green's function for the operator $\Delta+m^2$ is the following distribution on $M\times M$: $$G(x,y):=\langle \phi(x)\phi(y)\rangle=\int_{C^\infty(M)}\mathcal D\phi\,\phi(x)\...
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Non-equivalence between $\omega \to \omega \pm i\varepsilon$ and Cauchy principle value

I am looking to gain a more rigorous and deeper understanding as to how an $i\varepsilon$ prescription actually changes the end result of a divergent integral, specifically in regards to Green's ...
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theoretical echo from a point scatterer

How can I compute an echo coming back from a point scatterer? Let's say I know the excitation signal (plane wave), scatterer position, medium properties, what else do I need to see, how the echo will ...
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60 views

Representing propagators as Dirac delta functions [closed]

I have found online, in particular on the wolfram site, http://mathworld.wolfram.com/DeltaFunction.html, certain identities that allow one to represent a delta function as limits. Of particular ...
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103 views

Green's Function in the Lippmann Schwinger Equation

When deriving the scattering cross section using the Lippmann-Schwinger equation we need to calculate the Green's function defined by $$G(\mathbf{r},\mathbf{r'},E)=\langle\mathbf{r}|\frac{1}{E-H+i\...
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70 views

Green's functions and spectral function

I'm struggling to understand something in the book by Fetter & Walecka, p.295, relating to the causal ($G$), advanced ($G^A$) and retarded ($G^R$) Green's functions, and the spectral function ($\...
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36 views

Poisson's equation inside a sphere with linear charge distribution

Consider the Poisson equation \begin{equation*} \nabla^2\Phi = -4\pi \rho \end{equation*} inside a sphere of radius $b$. The charge density $\rho(x)$ corresponds to an evenly distributed charge $Q$ ...
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Relation between the reduced Green's function and the full Green's function

Let us assume that we have some Hamiltonian and we know its spectrum $$H_0 \psi_n = E_n \psi_n .$$ We define the Green's function in as $$ G(x,y,E) =\sum_m \frac{\psi_m^*(x)\psi_m(y)}{E-E_m}, $$ and ...
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Can the occupation of Floquet bands be calculated from the Keldysh Green's function?

A periodically driven band structure can be semiclassically described by Floquet theory, resulting in photon-dressed Floquet bands (non-equilibrium steady states). Usually, for non-equilibrium systems,...
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72 views

Confusion with poles of single particle green's function / propagator

On p22 of "Green's Functions for Solid State Physicists" by Doniach and SondHeimer, there is the following definition: $$G^0(\omega)=\frac{1}{2M\Omega_0}\left( \frac{1}{\omega-\Omega_0+i\eta} - \frac{...
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78 views

Transition Amplitude in Field Theory

I am currently reading the "Quantum Field Theory" by Lewis Ryder. In chapter 5 he is talking about path integrals and says that the transition amplitude $ \langle q_f t_f \vert q_i t_i\rangle $ is $$ \...
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24 views

Green function of squared chiral pseudoscalar in QCD

I need to compute the Green function $$ G(0) \equiv \int d^{4} x\int D[\text{QCD}]\bar{q}\gamma_{5}Mq(x)\bar{q}M\gamma_{5}q(0)e^{iS_{QCD}} \equiv $$ $$ \tag 1 \equiv \int d^{4}x\langle 0|T( \bar{q}\...
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134 views

Time-ordered product vs path integral

Suppose we have the Green function $$ G(k) \equiv \tag 1\int d^4x e^{ikx}\langle 0| T\left(\partial^{x}_{\mu}A^{\mu}(x)B(0)\right)|0\rangle , $$ which in path integral approach is equal to $$ \tag 2 ...
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What is the meaning of thermal spectral function and thermal decay width in thermal field theory?

In Kallen-Lehmann spectral representation of 2-point correlation function \begin{equation} \langle 0|T\phi(x)\phi(0)|0\rangle=\int_0^\infty \frac{dM^2}{2\pi}\rho(M^2)D_F(x-y;M^2),\quad (a) \end{...
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99 views

Physical meaning of Ward Identity and computing vertex functions

Following the derivation of Ward Identity by Weinberg book, you get it in the form $$ (l-k)_\mu S'(k)\Gamma^\mu(k,l)S'(l) = i S'(l) - iS'(k) $$ Can anyone explain the physical meaning of this ...
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Convolution of 1D Greens functions

So I need some help calculating the convolution of two 1D Greens functions, where the specific formula reads $$\int dk G_k(w+i\eta)G_{k+q}(w+i\eta) = \int dk\left[\Im\left[ \frac{1}{(\omega+i\eta+k)(\...
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1answer
61 views

Wightman function for massless vector fields in Coulomb gauge

I've been looking for quite some time an expression for the Wightman functions for a massless vector field in the Coulomb gauge $\nabla\cdot\mathbf{A}=0$ (I think it is equivalent to the Feynman gauge ...
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38 views

Greens function for forward propagating waves

I would like to find a form of a Green's function which accounts for the propagation of spherical waves expanding out from the spatial point $r$, but restricted to the forward propagating direction ...
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27 views

References for the non-Abelian gauge covariant Laplace equation?

Is there a standard reference which discusses solutions to the non-Abelian gauge covariant Laplace equation $D_{\mu} D^{\mu} \phi = 0$, where $D_{\mu} \phi = \partial_{\mu} + ig[A_{\mu}, \phi]$? Note ...
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42 views

locator equation of motion

I strugle with folowing problem. I do start with the locator equation of motion: $$G_{i j} = g_i \delta_{i j} + g_i \sum\limits_{k \ne i} W_{i k} G_{k j}$$ where $G_{i j}$ are matrix elements of ...
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226 views

Propagator and probability amplitude that a particle propagates

My QFT knowledge has very much rusted and i got confused by these few lines from Peskin and Schroeder: p.27: " [..] the amplitude for a particle to propagate from $y$ to $x$ is $\langle 0| \phi(x) ...
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39 views

The $i\epsilon$ in non-relativistic scattering theory

When doing quantum mechanical scattering theory, we obtain the Lippman-Schwinger equation $$|\psi\rangle=|\psi_0\rangle+(E-H_0)^{-1}V|\psi\rangle$$ Here $\psi_0$ is the unperturbed wavefunction, $\...