I read the Wiki page about the reweighting procedure as a way to use Monte Carlo methods with the sign problem, but I'd like to know more about how this could be implemented in the Metropolis algorithm. Can anyone describe an implementation of this procedure using a $\phi^4$ lattice field theory in real time as an example? I want to simulate this action: $$S=\sum_{x}\sum_{μ=0}^3η_{μμ}\frac{(\phi_{x+μ}-\phi_x)^2}{2}-\frac{(m^2+iε)}{2}\phi_{x}^2-\frac{λ}{4}\phi_{x}^4$$ with Minkowski metric $\eta_{00}=+1$, $\eta_{ii}=-1$
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$\begingroup$ If you’re asking about how lattice QFT approaches this, it is almost exclusively done in the Euclidean setting. Lattice computations deal with the decaying exponential rather than an oscillatory term. You can then extract predictions about the Minkowski space QFT from Euclidean correlation functions. $\endgroup$Prof. Legolasov– Prof. Legolasov2025-10-27 14:13:33 +00:00Commented Oct 27 at 14:13
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$\begingroup$ @Prof.Legolasov I know that most calculations are performed in the Euclidean signature, but my question is how to adapt the Metropolis algorithm to the reweighting procedure so that calculations in the Minkowski signature are possible. I tried generating configurations using the method of separating the weighting function into absolute value and a phase and calculating the ratio of expectation values exactly how it was described in wiki but failed. $\endgroup$Peter– Peter2025-10-27 14:32:31 +00:00Commented Oct 27 at 14:32
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$\begingroup$ The way you asked it, the question is hard to answer, because you asked a very general question about a procedure and linked sources, that already should well describe how to apply that procedure in general. Then you state that you failed to implement it in a concrete example, but you did not provide any information apart from the key word “field theory in real time” on what exactly you want to achieve. $\endgroup$Zaph– Zaph2025-10-27 16:38:33 +00:00Commented Oct 27 at 16:38
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$\begingroup$ @Zaph I added an action to the question that i want to simulate on the lattice. $\endgroup$Peter– Peter2025-10-27 17:13:21 +00:00Commented Oct 27 at 17:13
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Reweighting is based on the identity $$ \langle O \rangle = \frac{\int D\phi\, O \, e^{-(S_0+ S_I)}} {\int D\phi \, e^{-(S_0+S_I)}} = \frac{\int D\phi\, \left( O \, e^{-S_I}\right) e^{-S_0}} {\int D\phi \, \left( e^{-S_I}\right) e^{-S_0}} = \frac{\langle O \, e^{-S_I}\rangle_0}{\langle e^{- S_I}\rangle_0} $$ where $\langle .\rangle_0$ is the average with respect to the measure defined by $S_0$. This is useful only if $S_0$ is real, and $S_I$ is reasonably small, so that the average in the denominator is not too small.
The issue is that we need a useful reference action $S_0$. For example, in QCD at finite chemical potential only $i\mu \psi^\dagger\psi$ is imaginary in euclidean space, the rest of the action is real. Then, for small $\mu$ we can reweight in the chemical potential. What is unclear about the Minkowski space path integral is how to find a useful reference action.
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$\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$2025-11-02 16:47:56 +00:00Commented Nov 2 at 16:47