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?


1 Answer 1


I don't count as a lattice expert, just a student - if I get something wrong please correct me. But I can give some thoughts:

  • Firstly, the algorithm you present (a 'single-flip' type algorithm) where you update only locally, is not used in practice for Euclidean time simulations. Instead, algorithms such as HMC globally updates the field values. (And, there are modern developments like using Machine Learning to speed up sampling)
  • The simplest procedure to simulate real-time dynamics is just to do a Hamiltonian formulation. This has the problem that the hilbert space blows up in size and it gets (at present day) impractical for d >= 4, for reasonably sized lattices.
  • 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.
  • For some other things people try, path integral deformations try to deform the path integral into complexified variables, to make the action real again. Also there is Complex Langevin and Lefschetz Thimbles, neither of which I am very familiar with, maybe someone else can comment.

My impression of the status of lattice QCD is that the most promising near-term approach is the Hamiltonian formalism. Reweighting seems too difficult practically, and the theoretical methods like deformations and complex langevin I believe need a lot more theoretical work before they are applicable to 3+1D lattice QCD.

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    $\begingroup$ I thought about the Hamiltonian formalism, but there time remains continuous. How suitable is this for relativistic field theories where time and space must be on the same footing? $\endgroup$
    – Peter
    Oct 3, 2022 at 14:18
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    $\begingroup$ Any lattice discretisation invariably breaks Lorentz symmetry to a subgroup (in fact, for hypercubic discretisation of space+realtime, I don't think you even get any exact discrete boost symmetries on the lattice(?)) What matters is that you recover the correct symmetries as you take the continuum limit, which is the case with the Hamiltonian formalism. (Also, on a computer, the Hamiltonian formalism still requires a discretised time anyway. And just to mention - people do euclidean lattice QCD on anisotropic lattices to probe thermal effects as well) $\endgroup$ Oct 3, 2022 at 14:54

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