nLab
fundamental solution

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Functional analysis

Contents

Idea

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

Pf=δ P f = \delta

is called the fundamental solution for PP; alternative names like Green function and function of influence are also used. A particular solution of the nonhomogeneous equation for some other gg 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 ff such that PfδP 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:

namesymbolwave front setas vacuum exp. value
of field operators
as a product of
field operators
causal propagatorΔ S =Δ +Δ \begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}
A\phantom{A}\,\,\,-
iΔ S(x,y)= [Φ(x),Φ(y)]\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 propagatorΔ +\Delta_+ iΔ +(x,y)= {[Φ(x),Φ(y)] | xy 0 | yx\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 propagatorΔ \Delta_- iΔ (x,y)= {[Φ(x),Φ(y)] | yx 0 | xy\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 propagatorΔ H =i2(Δ +Δ )+H =i2Δ S+H =Δ FiΔ \begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned} Δ H(x,y) =Φ(x)Φ(y) =:Φ(x)Φ(y):=0 =+[Φ ()(x),Φ (+)(y)]\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 propagatorΔ F =i2(Δ ++Δ )+H =iΔ D+H =Δ H+iΔ \begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned} Δ F(x,y) =T(Φ(x)Φ(y)) ={Φ(x)Φ(x) | xy Φ(y)Φ(x) | yx\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 DD is a differential operator with constant coefficients in nn-dimensional real space R n\mathbf{R}^n then the fundamental solution exists in the Schwarz space 𝒮(R n)\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.

References

Last revised on May 28, 2018 at 04:54:22. See the history of this page for a list of all contributions to it.