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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Question about the Green's function for a conducting sphere

In Jackson's Classical Electrodynamics, he gives the Green's function for a conducting sphere with Dirichlet boundary conditions as $$ G(\mathbf{x},\mathbf{x}^\prime) = \frac{1}{|\mathbf{x} - \mathbf{...
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Sign in the equation to find Green's functions in QFT

Studying in an introductory course to field theory I came across the equation useful for determining a green function given the equation of motion of the theory dealt with. In the appendix of the book ...
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Physical interpretation of the spectral function in a superconductor

Starting from the standard mean-field Hamiltonian of a superconductor, $$H = \sum_{\mathbf{k},\sigma} \epsilon(k) \; c^\dagger_{\mathbf{k}\sigma} c_{\mathbf{k}\sigma} + \Delta \sum_\mathbf{k} (c^\...
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Should not the propagator depend on the interaction term? [closed]

The propagator being the Green's function of the Euler-Lagrange operator corresponding to the Lagrangian of some QFT, should not depend on the interaction term. But shouldn't the probability amplitude ...
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Origin of the factor of $i$ in the photon propagator

I'm following Peskin and Schroeder and am having trouble tracking down a particular factor of i that is persistently used in the definition of Green's functions. For example, equation 9.52 states that ...
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Klein-Gordon equation multiple Green's functions

I am trying to understand Green's functions for a Klein-Gordon equation: $ (\frac{\partial^2}{\partial t^2} - \nabla^2 +m^2) \phi(\vec{x},t) = 0$ and $ (\frac{\partial^2}{\partial t^2} - \nabla^2 +...
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Greens function derivative [duplicate]

I´ve given a green's function defined as $$ G_N(\vec{x},\vec{x}') = \frac{1}{4\pi} \frac{1}{|\vec{x}-\vec{x}'|}+\frac{1}{4 \pi}\frac{1}{|\vec{x}-\vec{y}_s'|} $$ with $$y_s=(y_1,y_2,-y_3)$$ on $$H\...
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Getting Feynman propagator using path integral

In QM using Feynman path integral(FPI) we derive the propagator of free particle which comes out to $$(f(t))e^{iS_{cl}/\hbar}$$ But in QFT the Feynman propagator is derived using the differential ...
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Can the real-time Green's function be written in the form of path integral on the real axis?

In every textbook, the path integral of the Green's function is written in imaginary-time. I wonder whether we could write real-time green function in the path integral form.
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Real-time Green Function in finite temperature

As in standard many-body textbook (at least in my class), we use real-time green function when temperatures is zero, and we use imaginary-time green function when the temperature is finite. My ...
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How to connect Green function to propagator?

I know that there has already been many questions related to this question, such as in Differentiating Propagator, Green's function, Correlation function, etc. However, that question mainly ...
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Correlation function of single annihilation/creation operator vanishes

I could not find anything on that on google, or here on physics stack exchange, which surprises me. My problem is, that I do not see, why exactly $<a> = <a^{\dagger}> = 0$ where <...> ...
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Propagator for a $x^3$ potential in momentum space

Is there a way of calculating the propagator/Green's function K(p,t;p',t'), allowing me to calculate \begin{equation} \Psi(p,t) = \int dp' \Psi(p',t') K(p,t;p',t') \end{equation} of a massive ...
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In electron transport calculations, should lead self-energies always be diagonal in a mode energy eigenbasis?

In electron transport calculations for semiconductor devices, the non-equilibrium Green's function method is often used. The Green's function takes the form $$G = \left[EI-H-\Sigma_L-\Sigma_R\right]^{-...
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How to unify the electrons and holes in Feynman Diagram?

As we know, following diagram describes the electron-hole pairs excitation. Namely, the right-going line describes creating electrons propagating with energy $(\omega_n+\nu_n)$, and the left-going ...
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Trace of the Operator

I want to ask a question about the fundamental knowledge of trace of the an operator. The operator $A$ is $$A = v (G_r-G_a)$$ where v is the velocity operator of the Hamiltonian ($v=dH/dk$); $G_r$ and ...
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Renormalization conditions of the Callan-Symanzik equation

Assume we have a massive $\phi^4$ theory the exact two-point correlation function is given as $$G=\frac{iZ}{p^2-m_r^2}+\text{terms regular at } p^2=m_r^2 $$ and if I want to apply renormalized ...
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Confusion between Green function and solution of equation of motion in Witten's paper on holography and AdS

I was going through Witten's paper on AdS and holography , and am confused in section 2.4. He starts by considering a massless scalar action in Euclidean AdS spacetime, with a boundary value $\phi_0$. ...
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Evaluating integral for Friedel oscillation using branch cuts

I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts: $$\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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Proof for getting delta function on $t \to t_0 $ from the equation of the propagator for the free particle in 1 dimension

From Sakurai's quantum mechanics equation 2.5.16 give propagator for a free particle in 1 dimension. Equation 2.5.16 is $$K (x^",t;x',t_0)=\sqrt {m\over {2\pi i\hbar (t-t_0)}} \exp \Biggl [{im (x^"...
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Are Green functions Nonperturbatively infrared finite?

Are Green functions Nonperturbatively infrared finite? In other words imagine one had the final form of the 4-point function for spin-1/2 field, do we still need infrared radiation correction? In ...
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Green's Function fo the Optical Path Difference

In this paper (1), the authors claim that the solution of the following Poisson equation $$ k\nabla^2 T(\mathbf r) = P\delta(\mathbf r) $$ is $$ T(r)=T_0+\frac{P_0}{4\pi k r}=T_0+\frac{P_0}{4\pi k }...
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Cyclic invariance of trace of fermions

Consider the Green's function of fermion operators with imaginary time, $$\mathcal{G}(\nu, \nu', \tau) = - \langle T_\tau c_{\nu}(\tau) c_{\nu'}^\dagger(0)\rangle~~~~~~~(1)$$ To show it satisfies the ...
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How to understand the internal structure for composite fermion?

There exists a concept of "composite fermion" when refer to heavy fermion system, which means spin-flip and conducting electron can be combined together to form a new "composite fermion". In details,...
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Microscopic theory of superconductivity in the language of the vertex function

In Chapter 7 of Abrikosov, Gorkov, and Dzyaloshinski (AGD), the authors cover a microscopic overview of superconductivity, with an emphasis on the poles of the vertex function $\Gamma$. Despite the ...
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Green's Functions

I am having trouble understanding how to apply Green's function to an impulsive forcing function. I am having trouble understanding how to use Green's function to solve for the motion of a driven ...
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Expectation value through spectral theorem in s-f Model

The model that i am studying is the s-f model. I wrote some post about it, then, in order to understund better my notation go to this question Now, I am computing some parameters that emerge out of ...
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How to understand the negative sign in poor man's scaling for Kondo model?

For the poor man's scaling for Kondo model $$H_{K}=\sum_{k, \sigma} \xi_{k} c_{k, \sigma}^{\dagger} c_{k, \sigma}+\sum_{k, k^{\prime}, \alpha, \beta} J c_{k, \alpha}^{\dagger} \sigma_{\alpha, \beta} ...
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Calculating the Kallen-Lehmann representation of conserved currents

Let $J^u$ be a complex conserved current. I want to obtain the Kallen-Lehmann representation of the vacuum expectation value $<T(J^u J^v)>_{0}$ where $T$ stands for the time ordered product. I ...
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Change sign for response function

There is a argument about response function: according to the Kramers-Kronig relation$$G(\omega)=\int_{-\infty}^{+\infty}d\omega' \frac{A(\omega')}{\omega+i0_+-\omega'}$$ response function will change ...
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Propagators and Green functions for general fields

In my QFT class we have defined the Feynman propagator of a field $\phi^r$ (where $r$ could be a vector or spinor index, or even a multiindex if $\phi$ is a tensor field etc.) as $$ \Delta^{rs}_F(x - ...
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Propagator in Path Integral Quantum Mechanism as Green Function of Schrodinger Equation

I'm studying in Ryder's book of QFT. I'm dealing with QM in the path integral approach and he is trying to prove that the propagator $K(x_f t_f;x_i t_i)$ is the Green function of the Schrodinger (S.) ...
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How to calculate the Fock exchange interaction self-energy of a system in momenttum space

I have a Hamiltonian in the momentum space which has a strong non-local electron-electron interaction. I know that I have to find its corresponding exchange self-energy and solve the Dyson equation ...
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How to find the functional self-energy?

I have a system with $e$-$e$ interaction. After using the mean field approximation, my Hamiltonian has the following form: $H = H_0 + H_I$, which $H_0$ is the non-interacting and $H_I$ is the ...
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Green's function for the screened Poisson equation

Assuming we are given a Lagrangian \begin{equation} \mathcal{L}(\phi(r),\partial^i\phi(r)) = \frac{1}{2} \partial_i\phi \partial^i\phi + \frac{m^2}{2} \phi^2 + \lambda \phi, \end{equation} the ...
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Iterative Greens function calculation

I have a Hamiltonian which has an interactive and non-interactive parts. $H = H_0 + H_I$ $H_I$ comes from the non-local electron-electron interaction and must be calculated self-consistently. I ...
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Correlation functions and non-bravais lattices

Consider a many-body system characterized by some Green's function $G(\vec{x},\vec{x}',t)$. In the presence of translation invariance, it's natural to work with the Fourier transform of this: $$G(\...
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Potential for a finite cylinder inside grounded sphere

I am looking at some problems on electrostatics and expansion of potential in spherical harmonics. I have been given a problem on finding the potential of inside a grounded sphere, which also has a ...
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How to deal with the poles in imaginary axis when applying Matsubara Sums?

The start point of Matsubara Sums is: $$\frac{1}{\beta}\sum_{\omega_n}F(i\omega_n)=\frac{1}{2\pi}\oint_C dz F(z)n(z)$$ where $n(z)$ is the bosonic/fermionic distribution function with the pole $z=i\...
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Keldysh Field Theory: Self Energy Structure in RAK basis

Consider the Keldysh formulation of electrons interacting with each other through the Coulomb potential. Suppose that we've switched to the RAK (Retarded-Advanced-Keldysh) basis of Larkin & ...
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What's the corresponding energy dispersion of Green function?

I want write a "toy" Green function witch can describe the electrons in the band with a width of $±W$ with uniform density of states (DOS). The reference gives an explicit expression of imaginary-time ...
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Green function of three band crossing topological material in energy coordinate representation do not converge

Recently I am studying the properties of Green function of three diamentional topological materials with three band crossings, whose Hamiltonian can be written as $\hat H= v_f \ \vec k \cdot \vec S $, ...
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How many-body density $n(\vec{r},t)$ can be viewed as a kind of correlation function?

I am reading Martin's book: interacting electrons. In chapter five about the definition of the correlation function, some points about density as correlation function confused me. The author adopted ...
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Green function for Dirac operator from Laplace Green function

I know the modified Green function of Laplace-Beltrami operator $\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)$ on some 2d-manifold (namely torus surface). Is there a general way to obtain ...
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Green function for two layer system

I deal with system consisting of two layers, $$H=\begin{cases}H_1,z<0;\\ H_2,0<z<a,\end{cases}$$ where $a$ is width of the second layer. What should I do to find Green functions of this ...
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Green's function on torus

I have a question about the Green's function $G(z,w)$ on torus which takes the form (for example the first equation in the paper https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p03-...
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Propagator for Dirac spinor field

I am currently trying to learn Quantum Field Theory through David Tong's notes which only talk about canonical quantisation for the scalar field and Dirac spinor field. In Chapter 2, the propagator ...
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Are the points on a closed surface included in the volume enclosed by it?

Green's theorem is used when finding a general solution for Poisson's or Laplace's equation. The Dirichlet and Neumann boundary conditions assure unique potential. While finding a general formula that ...
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Reference request: Hall current in heterojunctions

I am interested in Hall current derivation for a heterojunction via Green functions formalism. Any papers with specific systems will be useful for me. I expect that this derivation should be based on ...
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Doubt about the derivation of the Callan-Symanzik equation

I was reading about the Callan Symanzik equation from Peskin and Schroeder. On page 411, they assume that since $G^{(n)}$, the connected Green's function is renormalized, the $\beta$ and $\gamma$ ...

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