nLab fundamental solution

Idea

Given a linear differential operator (ordinary or partial) $P$ on a domain $M\subset\mathbb{R}^n$ or a manifold $M$, one can consider both the homogeneous differential equation $P f = 0$ and the nonhomogeneous equation of the form $P f = g$ where $g$ is a given nonhomogeneous term. If $g$ is a delta distribution and the boundary conditions are given, then the generalized solution of the nonhomogenous equation

$$P f = \delta$$

is called the fundamental solution for $P$; alternative names like Green function and function of influence are also used. A particular solution of the nonhomogeneous equation for some other $g$ can be obtained by calculating the convolution with the fundamental solution. (Compare the fact that the delta distribution is the identity element for convolution.)

As a first step one may also considers searching for $f$ such that $P f - \delta$ is a smooth function, the solution to the latter problem is called a parametrix.

Examples

Propagators for free fields

The Green functions for the wave operator/Klein-Gordon operator are known as the propagators for free fields in field theory:

propagators (i.e. integral kernels of Green functions)
for the wave operator and Klein-Gordon operator
on a globally hyperbolic spacetime such as Minkowski spacetime:

name	symbol	wave front set	as vacuum exp. value of field operators	as a product of field operators
causal propagator	$\begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}$	$\phantom{A}\,\,\,-$	$\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned}$	Peierls-Poisson bracket
advanced propagator	$\Delta_+$		$\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned}$	future part of Peierls-Poisson bracket
retarded propagator	$\Delta_-$		$\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}$	past part of Peierls-Poisson bracket
Wightman propagator	$\begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned}$		$\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & = \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned}$	positive frequency of Peierls-Poisson bracket, Wick algebra-product, 2-point function $\phantom{=}$ of vacuum state $\phantom{=}$ or generally of $\phantom{=}$ Hadamard state
Feynman propagator	$\begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned}$		$\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x } \right.\end{aligned}$	time-ordered product

(see also Kocic‘s overview: pdf)

Properties and results

(Malgrange-Ehrenpreis theorem) If $D$ is a differential operator with constant coefficients in $n$-dimensional real space $\mathbf{R}^n$ then the fundamental solution exists in the Schwarz space $\mathcal{S}'(\mathbf{R}^n)$. The proof is using Fourier transform which sends the equation for the fundamental solution into an algebraic equation for the Fourier transform of the solution seeked; the algebraic equation requires inversion and one checks that the conditions for finding the inverse in the Schwarz space are satisfied.

Related concepts

• Feynman propagator

• Dirac propagator

• parametrix

References

• Lars Hörmander, section 3.3 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

• Sergiu Klainerman, chapter 4 of Lecture notes in analysis, 2011 (pdf)

• Christian Bär, Nicolas Ginoux, Frank Pfäffle, section 2 and 3 of Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (arXiv:0806.1036)

• Nicolas Ginoux, Linear wave equations, Ch. 3 in Christian Bär, Klaus Fredenhagen, Quantum Field Theory on Curved Spacetimes: Concepts and Methods, Lecture Notes in Physics, Vol. 786, Springer, 2009

• Igor Khavkine, section 2 of Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv.1402.1282)

• Wikipedia, Fundamental solution

• Wikipedia, Green’s functions