# nLab fundamental solution

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.)

## Examples

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

Green functions for the Klein-Gordon operator on a globally hyperbolic spacetime:

propagator$\phantom{AA}$$\phantom{AA}$ primed wave front seton Minkowski spacetimegenerally
causal propagator$\array{\Delta \coloneqq \Delta_R - \Delta_A }$ $\array{- \\ \phantom{A} \\ \phantom{a}}$ $\array{\Delta_S(x,y) = \\ \langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$Peierls-Poisson bracket
advanced propagator$\Delta_A$$\array{\Delta_A(x,y) = \\ \Theta((y-x)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$
retarded propagator$\Delta_R$$\array{\Delta_R(x,y) = \\ \Theta((x-y)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$
Dirac propagator$\Delta_D = \tfrac{1}{2}(\Delta_A + \Delta_R)$ $\array{+ \\ \phantom{A} \\ \phantom{a}}$
Hadamard propagator\begin{aligned} \omega &= \tfrac{i}{2}\Delta + H \\ & = \omega_F - i \Delta_A \end{aligned}$\array{\omega(x,y) = \\ \langle vac \vert \Phi(x) \Phi(y) \vert vac \rangle }$normal-ordered product (2-point function of quasi-free state)
Feynman propagator$\array{\omega_F & = i \Delta_D + H \\ & = \omega + i \Delta_A}$$\array{E_F(x,y) = \\ \langle vac \vert T(\Phi(x)\Phi(y)) \vert vac \rangle }$time-ordered product