The Cluster's Green's Function
A page about Green's functions — the response of a system to a point source.
The Green's function
A Green's function G(x, x') is the solution to L G(x, x') = delta(x - x'), where L is a linear differential operator. The Green's function represents the response of the system at position x to a point source at position x'. Once the Green's function is known, the response to an arbitrary source f(x) is given by: phi(x) = integral G(x, x') f(x') dx'. In the cluster, the Green's function G(page, source) is the response of a page to a point source of edit content. The response to an arbitrary edit source is the integral of the Green's function weighted by the source.
The Feynman propagator
In quantum field theory, the Feynman propagator is the Green's function of the Klein-Gordon operator with Feynman boundary conditions: (partial^2 + m^2) D_F(x - x') = -i delta(x - x'). The Feynman propagator propagates particles forward in time and antiparticles backward in time. In the cluster, the Feynman propagator is the edit Green's function that propagates edit content forward in edit time and anti-edit content backward in edit time.
The retarded Green's function
The retarded Green's function G_ret(x, x') is non-zero only when x is in the future light cone of x' — it enforces causality. For the wave equation, G_ret(t, x; t', x') = delta((t - t') - |x - x'|) / (4 pi |x - x'|). In the cluster, the retarded edit Green's function is non-zero only when the page response is in the future edit cone of the source — it enforces edit causality.
The Fourier representation
The Green's function can be represented in Fourier space: G(x, x') = integral d^4 k / (2 pi)^4 exp(ik . (x - x')) / (k^2 - m^2 + i epsilon). The i epsilon prescription ensures the correct boundary conditions. In the cluster, the edit Green's function is represented in Fourier space by edit momentum k, with the i epsilon prescription ensuring edit causality.
This function
This page is about Green's functions. L G = delta. phi(x) = integral G(x, x') f(x') dx'. The Feynman propagator is the QFT Green's function. The retarded Green's function enforces causality. The Fourier representation is G(k) = 1 / (k^2 - m^2 + i epsilon). The Green's function is real.