Linked Questions

3
votes
1answer
3k 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 }{\...
25
votes
2answers
8k views

How to interpret correlation functions in QFT?

I'm fairly new to the subject of quantum field theory (QFT), and I'm having trouble intuitively grasping what a n-point correlation function physically describes. For example, consider the 2-point ...
21
votes
1answer
4k views

Propagators, Green’s functions, path integrals and transition amplitudes in quantum mechanics and quantum field theory

I’m trying to make a simple conceptual map regarding the things in the title, and I'm finding that I’m a little perplexed about a couple of items. Let me summarize a few things I regard as being true, ...
6
votes
3answers
8k views

Susceptibilities and response functions

It is often confusing whether a susceptibility is the same as a response function, specially that often they are used interchangeably, in the context of statistical mechanics and thermodynamics. Very ...
13
votes
2answers
2k views

Differential equation (Greens function) satisfied by the kernel using path integrals

I'm reading Feynman and Hibbs, Quantum Mechanics and Path Integrals. How do I show that the kernel $$\tag{2-25} K(x_2 ,t_2;x_1, t_1)=\int_{x=x_1}^{x=x_2}\mathcal{D}x~ e^{\frac{i}{\hbar}S[2,1]}$$ ...
14
votes
1answer
2k views

Dirac Delta in definition of Green function

For a inhomogeneous differential equation of the following form $$\hat{L}u(x) = \rho(x) ,$$ the general solution may be written in terms of the Green function, $$u(x) = \int dx' G(x;x')\rho(x'),$$ ...
7
votes
3answers
4k views

What's the difference between "boundary value problems" and "initial value problems"?

Mathematically speaking, is there any essential difference between initial value problems and boundary value problems? The specification of the values of a function $f$ and the "velocities" $\frac{\...
22
votes
1answer
12k views

How exactly is the propagator a Green's function for the Schrodinger equation

Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves $$\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\...
7
votes
1answer
3k views

What is different between resolvent and green function

I bumped into a book, where Resolvent $R^{\pm}(E)$ is defined as $e^{\mp iHt/\hbar}=\pm\frac{i}{2\pi}\int_{-\infty}^{\infty}dER^{\pm}(E)e^{\mp iEt/\hbar}$ and $R^{\pm}(E)=\frac{1}{\pm i\hbar}\int_0^{...
11
votes
2answers
2k views

Where is the Feynman Green's function in quantum mechanics?

In quantum field theory, the Feynman/time ordered Green's function takes the form $$D_F(p) \sim \frac{1}{p^2 - m^2 + i \epsilon}$$ and the $i \epsilon$ reflects the fact that the Green's function is ...
5
votes
1answer
1k views

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^...
6
votes
1answer
736 views

Are propagators in QFT really Green functions?

In many textbooks the notions Green function and propagator are used interchangeably. But are they really the same thing? This popular answer argues that a retarded propagator function $D_R(x,t,x',t')...
18
votes
0answers
294 views

Definition of vacua in QFT in generic spacetimes

I have been learning QFT in curved spaces from various sources (Birrell/Davies, Tom/Parker, some papers), and one thing that confuses me the most is the choice of vacua in various spacetimes, and the ...
3
votes
1answer
569 views

Single-particle Green's function

Define a single-particle Green's function as \begin{equation} i\hbar G(xt;x't') = \langle x| e^{-iH(t-t')/\hbar} | x'\rangle. \end{equation} By inserting the completeness relation, we have \begin{...
1
vote
0answers
480 views

What's the difference between a Green's function and a fundamental solution?

The Wikipedia article on fundamental solution says In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the ...

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