# Numerical real-space, real-time green's functions?

## Main Question or Discussion Point

Is there any numerical work on Green's functions in real-space and real-time?

The reason I ask is that it seem the self-energy diagrams correspond to much simpler expressions in space-time than in energy-momentum. However, I suspect the space-time greens functions are sort of pathological.

eg. (retarded) free particle Green's function in energy-momentum

$$G\left(k,E\right)=\frac{1}{E-E_{k}+i\eta}, E_{k}=\frac{\hbar^{2}k^{2}}{2m}$$

seems nicely behaved, could be conceivably be numerically represented on a rectangular mesh of E and Ek values, for nonvanishing eta (suppose there's some scattering in the system).

But in space-time, it becomes a numerical abomination:

$$G\left(r,t\right)=\Theta\left(t\right)e^{-\eta t}\left(\frac{m}{2\pi i\hbar t}\right)^{3/2}\exp\left(-\frac{mr^{2}}{2i\hbar t}\right)$$

Maybe the above expression is okay for analytical work, but how on earth does one represent it numerically? The space coordinate seems to require a t-dependent grid to avoid aliasing in space... but maybe this is doomed anyway because the oscillation frequency in space is infinite for as t approaches zero.

I'm aware that real-space, *imaginary*-time methods are popular with some models, but my own googling hasn't been fruitful on results for real-time. Any thoughts would be appreciated.