Abstract
I review the status of the Complex Langevin method, which was invented to make simulations of models with complex action feasible. I discuss the mathematical justification of the procedure, as well as its limitations and open questions. Various pragmatic measures for dealing with the existing problems are described. Finally I report on the progress in the application of the method to QCD, with the goal of determining the phase diagram of QCD as a function of temperature and baryonic chemical potential.
Highlights
1.1 Sign problemIt is well known that in many instances the functional measure in Euclidean Quantum Field Theory is not positive, making standard importance sampling impossible
Plenary Lecture given at Lattice 2017, the 35th International Symposium on Lattice Field Theory, Granada, Spain, 18–24 June 2017
Stamatescu [23], which showed some success in computing correlation functions of a quantum field theory at real time
Summary
It is well known that in many instances the functional measure in Euclidean Quantum Field Theory is not positive, making standard importance sampling impossible. In such cases the functional measure is described by a complex (or nonpositive) density ρ on a real configuration space M. The following motto describes a strategy to deal with this ‘sign problem’: When the measure is complex, complexify the fields. This means the following: for holomorphic observables O, one tries to represent the complex measure by a measure on the complexified configuration space Mc which is either nonnegative or at least only mildly oscillating, making the sign problem less severe. If that measure is described by a density P, consistency requires that for all holomorphic observables O.
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