Register to reply

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

Share this thread:
icwchan
#1
Jun7-11, 12:18 PM
P: 6
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

[tex] G\left(k,E\right)=\frac{1}{E-E_{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 space-time, 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 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.
Phys.Org News Partner Physics news on Phys.org
Step lightly: All-optical transistor triggered by single photon promises advances in quantum applications
The unifying framework of symmetry reveals properties of a broad range of physical systems
What time is it in the universe?

Register to reply

Related Discussions
No real purpose in space-time? Special & General Relativity 37
Metric space of all bounded real functions is complete Calculus & Beyond Homework 6
Fabric of space-time real? Astronomy & Astrophysics 9