# Real time lattice simulation

A typical simulation of lattice quantum chromodynamics uses imaginary time, so the weight function of the path integral is positive definite $$e^{-S_E}$$.

In the case of metropolis-hastings, the algorithm is basically:

1. Initialize the field with random numbers from a normal distribution
2. Pick a random point on a 4-dimensional lattice
3. Generate candidate from distribution and calculate the action difference $$ΔS$$.
4. Generate a uniform random number $$u \in [0,1]$$ Accept configuration if $$ΔS<0$$ or $$u <= e^{-ΔS}$$
5. Repeat 2,3,4 for all lattice points.
6. Repeat 5 for $$n$$ iterations;

So let's say we want to simulate it in real time , how to do it using Monte Carlo importance sampling? The weight function is a complex exponential $$e^{iS_M}$$.

I have read that it is possible to carry out a reweighing procedure in order to make the weight function positive definite. But this leads to large cancellations of contributions to path integral and sign problem. Anyway i want to hear a lattice experts.

Also how would you account a time-slicing in the algorithm?

• As you mention, you can perform reweighting, where the idea is that instead of sampling from $$e^{iS}$$, you sample from a euclidean action that is hopefully "close" in some sense, and then reweight your samples. Also as you point out - the sign problem is particularly bad. I'm not aware of any studies that actually try real-time dynamics via reweighting, but people do this for example to study finite chemical potential which adds an imaginary part to the action.