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We analyze to what extent the complex Langevin method, which is in principle capable of solving the so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and then point out various mathematical loopholes. The detailed study of some simple examples leads to practical suggestions about the application of the method.

In lattice QCD, the Maximum Entropy Method can be used to reconstruct spectral functions from euclidean correlators obtained in numerical simulations. We show that at finite temperature the most commonly used algorithm, employing Bryan's method, is inherently unstable at small energies and give a modification that avoids this. We demonstrate this approach using the vector currentcurrent correlator obtained in quenched QCD at finite temperature. Our first results indicate a small electrical conductivity above the deconfinement transition.PACS numbers: 12.38.Gc Lattice QCD calculations, 12.38.Mh Quark-gluon plasmaIn the deconfined, high-temperature phase of Quantum Chromodynamics, the behaviour of spectral functions of conserved currents at small energies is of intrinsic interest due to its relation with transport properties of the quark-gluon plasma (QGP). According to the Kubo formulas [1], transport coefficients, such as the shear and bulk viscosities and the electrical conductivity, are proportional to the slope of appropriate spectral functions at vanishing energy. The success of e.g. ideal hydrodynamics in heavy ion phenomenology, assuming vanishing viscosities and requiring early thermalization [2], has lead to the notion [3] that the QGP created in relativistic heavy ion collisions at RHIC is strongly coupled and that the ratio of shear viscosity to entropy density in this sQGP may be close to the conjectured lower bound [4] reached in thermal field theories that admit a gravity dual [5].In order to put these ideas on firm footing, it is important to have a first-principle calculation of transport coefficients in the strongly coupled regime of hot QCD. As is well known [6,7], a nonperturbative calculation using lattice QCD is difficult due to the necessity to perform an analytic continuation from imaginary to real time. The most common approach used to obtain spectral functions from euclidean correlators is the Maximum Entropy Method [8], employing Bryan's algorithm [9]. Our first aim in this Letter is to discuss this method in some detail and point out a source of numerical instabilities present in most finite-temperature studies available to date. We show how the algorithm can be modified to avoid this problem. Our second aim is to apply the new method to the vector current-current correlator, obtained in quenched lattice QCD simulations at finite temperature, using staggered quarks. We study the behaviour at small energies and argue that it allows us to extract a value for electrical conductivity in the strongly coupled regime above the deconfinement transition.Maximum Entropy Method -The relation between the euclidean correlator G(τ ) = d 3 x J(τ, x)J † (0, 0) at zero momentum and the corresponding spectral function ρ(ω) readswhere the kernel is given byWe consider (local) meson operators of the form J(τ, x) =q(τ, x)Γq(τ, x), where Γ depends on the channel under consideration. The temperature T is related to the euclidean temporal extent N τ by 1/T = aN τ , where a is the (temporal) lattice spacing. Th...

The complex Langevin method is a leading candidate for solving the socalled sign problem occurring in various physical situations. Its most vexing problem is that in some cases it produces 'convergence to the wrong limit'. In the first part of the paper we go through the formal justification of the method, identify points at which it may fail and identify a necessary and sufficient criterion for correctness. This criterion would, however, require checking infinitely many identities, and therefore is somewhat academic. We propose instead a truncation to the check of a few identities; this still gives a necessary criterion, but a priori it is not clear whether it remains sufficient. In the second part we carry out a detailed study of two toy models: first we identify the reasons why in some cases the method fails, second we test the efficiency of the truncated criterion and find that it works perfectly at least in the toy models studied.

We derive the nonequilibrium real-time evolution of an O(N)-invariant scalar quantum field theory in the presence of a nonvanishing expectation value of the quantum field. Using a systematic 1/N expansion of the 2PI effective action to next-to-leading order, we obtain nonperturbative evolution equations which include scattering and memory effects. The equivalence of the direct method, which requires the resummation of an infinite number of skeleton diagrams, with the auxiliary-field formalism, which involves only one diagram at next-to-leading order, is shown.