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

[icwchan]

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.

 

[eljose79]

I can offer some insights on the numerical work that has been done on Green's functions in real-space and real-time. While there is definitely more research and literature available on Green's functions in energy-momentum, there have been some efforts to study and use real-space and real-time Green's functions in certain systems.

One approach that has been used is to discretize the space and time coordinates and then solve the resulting equations numerically. This can be done using various numerical techniques such as finite difference methods, finite element methods, or spectral methods. The choice of method will depend on the specific system being studied and the desired accuracy.

Another approach is to use Monte Carlo simulations to compute the Green's function in real-space and real-time. This method has been applied to study quantum systems with strong interactions, where analytical solutions are not available. Monte Carlo simulations can be computationally intensive, but they offer a powerful tool for studying complex systems.

It is also worth mentioning that there are some techniques for numerically representing the Green's function in real-space and real-time without discretizing the coordinates. For example, the spectral representation method has been used to study the Green's function in systems with translational symmetry.

In general, the numerical representation of the Green's function in real-space and real-time can be challenging due to the oscillatory behavior and singularities that can arise. However, with the advancements in numerical methods and computing power, it is possible to accurately compute and study the Green's function in these domains.

In conclusion, while there may not be as much numerical work on Green's functions in real-space and real-time compared to energy-momentum, there have been efforts to study and use them in various systems. The choice of numerical method will depend on the specific system being studied and the desired accuracy. I hope this helps answer your question and provides some insight into the current state of research on this topic.