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0
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1answer
31 views

Good values for gravitational potential [on hold]

In the context of a project, I had to solve numerically Poisson equation with cylindrical coordinates. I put here results for z = 0 on a 3D mesh 256x256x256. Below 1 figure representing the final ...
8
votes
3answers
520 views

Two definitions of Green's function

In literature, usually two types of definition exist for Green's function. $\hat{L}G=\delta(x-x')$. This equation states that Green's function is a solution to an ODE assuming the source is a delta ...
0
votes
2answers
72 views

Self-teaching Green's function approach to quantum many-body systems

My question is where can I find a good book, review, online course, or all of them for self-teaching Green's function in quantum many-body problems (if it has problems with solutions for ...
1
vote
2answers
79 views

Klein-Gordon Green's function: derivative of delta distribution?

In Peskin/Schroeder there is an explicit calculation showing that the retarded Green's function of the real Klein-Gordon field $$D_R(x-y) ~\equiv~ \theta(x^0 - y^0) \langle 0 | [\phi(x), \phi(y)] ...
3
votes
1answer
130 views

How is Lippmann-Schwinger equation derived?

I'd like to know the derivation of Lippmann-Schwinger equation (LSE) in operator formalism and on what assumptions it is based. I consulted the Ballentine book as advised in this Phys.SE post, but I ...
1
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0answers
37 views

Matsubara Green function of anderson impurity model

I am currently having trouble computing the imaginary-time Green's functions of a model similar to the single-impurity anderson model. The hamiltonian is given as: \begin{equation} H = ...
4
votes
2answers
162 views

Green's function for the Klein-Gordon equation diverging?

I'm trying to work out the propagator for the free scalar field theory (i.e., the Green's function for the Klein-Gordon equation). On pages 23 and 24 of Zee's Quantum Field Theory in a Nutshell (you ...
1
vote
0answers
66 views

Poisson-like green functions

How can I verify that equation $$\nabla ^2 f (r) = - \frac{e}{4 \pi \epsilon ^2} \delta (r-\epsilon)$$ in 3D has a solution of the form $$f (r) = a - \frac{e}{4 \pi r} \theta (r-\epsilon) ...
1
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0answers
26 views

Density of states for the diffusion

For the wave equation, the propagator in the Fourier domain is written as $$G(\mathbf{k},\omega)=-\frac{1}{\frac{\omega^2}{c^2}-\mathbf{k}^2+\mathrm i\epsilon}.$$ When $\omega/c$ is close to ...
2
votes
1answer
98 views

How is Green function in many-body theory introduced?

Normally, for a (linear) operator $L$ and a DE $$ Lu(x) = f(x) $$ the Green function is defined as $$ LG(x,s) = \delta(x-s) $$ and it is found that $$ u(x) = \int G(x,s) f(s) ds $$ is the ...
1
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0answers
58 views

Why are the integral form of the GR equations problematic?

I have heard that working with the integral form of the GR equations is problematic - relative to determining a Greens function. Can someone explain the details as why?
2
votes
1answer
165 views

Retarded and advanced Green's function

Is there a use of advanced Green's functions? If yes then when or in which context? Why in quantum field theory, we always use Feynman's prescription for finding the propagator and not the retarded ...
1
vote
1answer
131 views

How to compute the density of state from the Green function?

I'd like to plot the density of state (DOS) for a specific system, say an s-wave BCS superconductor, the Green function of which is ...
2
votes
1answer
143 views

Green's function for 1 D hubbard model?

Consider the 1D two-site Hubbard model at half filling $H=-t\sum _{\sigma} (c_{1\sigma} ^{\dagger}c_{2\sigma}+h.c.)+U\sum_i(n_{i\uparrow}-\frac{1}{2})(n_{i\downarrow}-\frac{1}{2})$ where the sum is ...
1
vote
1answer
89 views

Evaluation of Green function for two site system?

Let's consider I have two site system whose hamiltonian has $2\times2$ matrix form. In general we can write the Green function for above Hamiltonian as $G^{-1}=i \omega-H $ or $G=[i\omega-H]^{-1}$ and ...
0
votes
1answer
65 views

Fourier transform of random variables

My question is concerning Fourier transforms of random variables. So if the question itself is too heavy a task but you know of any good resources to learn this topic that would also be very much ...
1
vote
0answers
32 views

Neumann Green's function inside semi-infinite conductor [closed]

Consider a semi-infinite conductor with uniform conductivity $s$ occupying the space $z>0$. What is the Green's function with Dirichlet and Neumann boundary conditions inside the region $z>0$? ...
0
votes
1answer
37 views

Green function in non-relativistic quantum mechanics for particles with complicated isotropic spectrum

Let's consider a free particle with some non-trivial isotropic spectrum. What I mean is that Hamiltonian of the particle depends only on the square of the momentum: $$\hat{H}=f(\mathbf{\hat{p}}^2)$$ ...
1
vote
1answer
112 views

Green's Functions from Gell-Mann and Low Theorem

What I want to do: $\newcommand{\ket}[1]{\left|#1\right\rangle}$ $\newcommand{\bra}[1]{\left\langle#1\right|}$ $\newcommand{\braket}[1]{\left\langle#1\right\rangle}$ The Gell-Mann Low Theorem tells ...
4
votes
0answers
124 views

Grand canonical Hamiltonian

How to explain introducing "grand canonical" Hamiltonian $$ \hat{H'}= \hat{H}-\mu \hat{N} $$ when we study a quantum system with fixed chemical potential? I understand such a substitution in a ...
0
votes
1answer
82 views

What's the meaning of the propagator in QM?

Yesterday I was solving some exercises, and after solving the time evolution I was asked to find the probability of the system to some state. In specific: $$|\Psi(t)\rangle = ...
2
votes
1answer
114 views

Solving inhomogeneous differential equation with Green function

I'm not sure if this question is for physics forum, but my book's title is "Green's Functions in Quantum Physics", so I ask here. The book says that the Green's function defined as $$ (z-L( ...
2
votes
1answer
99 views

Why are Green Functions/(Correlation Functions) not on the mass shell?

The difference between Green Functions and the S-matrix in Quantum Field Theory is whether the momentum is on the mass shell. Why are the Green Functions/(Correlation Functions) not on the mass shell? ...
0
votes
1answer
190 views

Green function solutions in electrostatics

I have a conducting plate on $x$-$y$ plane. So I have a boundary condition at $z=0$ $\Phi=0$ but, for $z>0$ I have a point charge at z=a which is expected to create a potential. $$ ...
6
votes
1answer
549 views

Mathematically, what is the kernel in path integral?

Mathematically, what is the kernel in path integral? At first, I thought that it is the kernel in the integral transform because when we use the (physical) kernel to transform the wave function (Eq ...
0
votes
0answers
25 views

What does a completely negative Greens function in frequency mean?

What can a Greens function of frequency mean when it is always negative? The Greens function is for the photons as the following: (It's derived by Matsubara method to enter the thermal effects and the ...
1
vote
1answer
89 views

Understaning Euclidean Green's function

Consider a scalar field coupled to a source $$(\Box - m^2)\phi(x) = -J(x)\tag{1}.$$ Then, the response of the source is determined by the Green's function $G(x-y)$, which satisfies $$(\Box - ...
0
votes
0answers
121 views

Green function for single impurity

I am working on the first problem on self-consistent T-matrix approximation in Chapter 5 of Condensed Matter Field Theory by Altland and Simons. This is on page 234 of the textbook. I have some ...
0
votes
1answer
71 views

Green function for interacting system

If we can diagonalize our interacting Hamiltonian then can we write a Green's function like we do for a non-interacting system? Green's function here means Matsubara in frequency-momentum space, ...
0
votes
0answers
48 views

How can I determine the convergence of self energy in Green's function

I want to solve for the Green's function (in the context of many body theory) but I have a question. After the determination of the retarded Green's function and the lesser Green's function we ...
2
votes
1answer
109 views

1-particle non-interacting Green function

At $T=0$ in the non-interacting case the $1$-particle Green function for an electron in the excited state $\lambda$ (empty band) is of the form \begin{eqnarray} G^{(0)}(\lambda,t-t') = -i \theta(t-t') ...
0
votes
0answers
59 views

Retarded Green function and the gravitational field of a point particle

I'm trying to understand a calculation by Aichelburg and Sexl of the gravitational field of a point particle. Linearizing the Einstein field equations in the usual way (that is, supposing a metric of ...
1
vote
1answer
76 views

Why do we have different signs before the delta on the Klein-Gordon and the Dirac Green's function equation?

Let's read equation (2.56) on Peskin & Schroeder $$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y).$$ Let's look now to equation (3.118) $$(i\gamma^{\nu}\partial_{\nu}-m)S_R(x-y)=i\delta^4(x-y).$$ ...
0
votes
0answers
46 views

Electron transmission in Landauer formalism: why just imaginary part?

In Landauer formalism the electron transmission function is defined as $T = Tr(G_M^\dagger\Gamma_LG_M\Gamma_R)$ where $G$ are Green's function of subsystems, index $L$ and $R$ means subsystem of ...
4
votes
0answers
141 views

Green's function for a dielectric with a charge [closed]

Suppose there are two infinite planes, one in $z=a$ and the other in $z=b$, with $a<b$. Between the planes, there is a dielectric medium with constant $\epsilon_1$. The differential equation for ...
2
votes
1answer
181 views

How to choose the Correct Green's Function?

In order to solve the Green’s function of the Helmholtz operator $$(\nabla^2+k^2)G(\vec r-\vec r’)=\delta^{(3)} (\vec r-\vec r’)$$ one can obtain four different Green’s functions corresponding to four ...
1
vote
0answers
53 views

Physical meaning behind causal massless scalar propogator

I know that for a scalar massless field in $(3+1)$ spacetime that: $$ \langle 0 | \phi(x) \phi(y) | 0 \rangle \propto |x-y|^{-2} = (-(t_x-t_y)^2 + (\vec{x}-\vec{y})^2)^{-1}. $$ I also know that $$ ...
0
votes
1answer
390 views

Green function for simple harmonic oscillator

I'm interested in examples on how to use Green function (GF)for simple harmonic oscillator (SHO)? I am from undergrad physics, so I need a fundamental math and quantum mechanical application of GF ...
0
votes
0answers
92 views

Greens function/resolvent of hydrogen Hamiltonian

Let $H$ be the Hamiltonian for the nonrelativistic hydrogen atom, i.e. $$H=-\frac{1}{2}\Delta-\frac{1}{r}$$ I am searching for an asymptototic expansion of the Greens function or respectively the ...
1
vote
2answers
86 views

How to get the relation for dependence of anomalous dimension on regularization?

Here is the anomalous dimension: $$ \gamma_{\Gamma}(t, g) = \left[\frac{\partial }{\partial t}\ln \left(Z_{\Gamma}(t , g) \right)\right]_{t = 1}, $$ where $Z_{\Gamma}$ is renormalization factor which ...
2
votes
0answers
62 views

Benefit of using Matsubara Green function

Physicists often calculate Matsubara Green function and then perform an analytic continuation $i\omega_n \rightarrow \omega +i\eta$ to obtain the retarded Green function. Why is doing so better than ...
2
votes
1answer
166 views

Integral over a product of two Green's functions

Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral/summation as a product of a retarded and advanced ...
3
votes
1answer
116 views

Are constant terms in second-quantization relevant?

I have a rather broad question and a specific problem. Let's take a orthonormal single-particle basis $\{ \vert i \rangle \}$, a simple single-particle Hamiltonian $$\tilde{H} = \sum_{i, j} h_{i j} ...
0
votes
1answer
60 views

Can someone show me how Green's function would apply for this simple case?

I'm reading up on some stuff on basic electrostatic here: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/laplace.html Can someone use Green's function to show me the form of $V$? Update: I ...
5
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0answers
73 views

Green's function for moving solidification front

Consider a liquid solid interface $z =\zeta(x,t)$ moving at constant speed $v$, for a two dimensional problem. Due to solidification interface is changing it position. For simplicity heat ...
3
votes
0answers
121 views

Variations of S-matrix functional and Feynman diagrams in Weinberg QFT

Weinberg on p. 287 of his QFT vol. 1 introduces the extended interaction operator: $$ \tag 1 \hat{V}(t) \to \hat{V}(t) + \sum_{a}\int d^{3}\mathbf x \hat{o}_{a}(\mathbf x ,t)\varepsilon_{a}(x). $$ ...
0
votes
0answers
47 views

simplification of Green's intergral for diffusion

\begin{eqnarray} I_1 = \int^{\infty}_{0} \frac{1}{\tau}d\tau \int^{+\infty}_{-\infty} \exp\left[-\frac{p}{2\tau}(x-x')^2 + (z-z' + \tau)^2\right]dx' \end{eqnarray} where, $p$, $x$ $z$ and $\tau$ are ...
1
vote
1answer
144 views

The n-point Green functions and Heisenberg picture

Let's have the S-matrix: $$ S_{\beta \alpha} = \langle \beta | \hat{S} | \alpha\rangle . $$ Here $|\alpha \rangle , | \beta \rangle$ are $t \to \mp \infty$ limit of the free states, $\hat {S} = ...
2
votes
0answers
89 views

How to calculate the 2-point function of gravitons?

I'm curious about how to calculate the 2-point function of graviton, but there are no textbooks of general relativity covering this problem. So how to calculate it? In which book can I find the ...
1
vote
0answers
73 views

Sum of Green's functions in Condensed Matter

I am working on the Ginzburg-Landau model for Charge density waves, and I am carrying out the sum of Green's functions to calculate the terms in the GL model. I have the following question: Is the ...