
#1
Jun711, 12:18 PM

P: 6

Is there any numerical work on Green's functions in realspace and realtime?
The reason I ask is that it seem the selfenergy diagrams correspond to much simpler expressions in spacetime than in energymomentum. However, I suspect the spacetime greens functions are sort of pathological. eg. (retarded) free particle Green's function in energymomentum [tex] G\left(k,E\right)=\frac{1}{EE_{k}+i\eta}, E_{k}=\frac{\hbar^{2}k^{2}}{2m} [/tex] 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 spacetime, it becomes a numerical abomination: [tex]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) [/tex] 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 tdependent 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 realspace, *imaginary*time methods are popular with some models, but my own googling hasn't been fruitful on results for realtime. Any thoughts would be appreciated. 


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