Turbulence: a nonequilibrium field theory

Resummation of perturbation series in non-equilibrium scalar field theory

Resummation of perturbation series in non-equilibrium scalar field theory

The problem of renormalization of the closed--time-path Green's function in nonequilibrium statistical field theory is studied. Under some reasonable assumptions on the high-energy behavior of the initial correlation functions, it is found that the same counterterms which eliminate the ultraviolet divergences in the usual field theory can also make the closed--time-path Green's functions free of ultraviolet divergences. The renormalization-group equation satisfied by the closed--time-path vertex functions is obtained and the Callan-Symanzik coefficient functions are shown to be the same as in the usual field theory.

In this paper we discuss a Schwinger-Dyson [SD] approach for determining the time evolution of the unequal time correlation functions of a non-equilibrium classical field theory, where the classical system is described by an initial density matrix at time $t=0$. We focus on $\lambda \phi^4$ field theory in 1+1 space time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation [BVA], is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for 1/N expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of $\phi$ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations. We find that both approximations based on the SD equations yield good agreement with exact numerical simulations and cure the late time oscillation problem of the Hartree approximation. We also discuss the relationship between the quantum and classical SD equations.

The emergence of an effective field theory out of equilibrium is studied in the case in which a light field—the system—interacts with very heavy fields in a finite temperature bath. We obtain the reduced density matrix for the light field, its time evolution is determined by an effective action that includes the influence action from correlations of the heavy degrees of freedom. The non-equilibrium effective field theory yields a Langevin equation of motion for the light field in terms of dissipative and noise kernels that obey a generalized fluctuation dissipation relation. These are completely determined by the spectral density of the bath which is analyzed in detail for several cases. At T = 0 we elucidate the effect of thresholds in the renormalization aspects and the asymptotic emergence of a local effective field theory with unitary time evolution. At new ‘anomalous’ thresholds arise, in particular the decay of the environmental heavy fields into the light field leads to dissipative dynamics of the light field. Even when the heavy bath particles are thermally suppressed this dissipative contribution leads to the thermalization of the light field which is confirmed by a quantum kinetics analysis. We obtain the quantum master equation and show explicitly that its solution in the field basis is precisely the influence action that determines the effective non-equilibrium field theory. The Lindblad form of the quantum master equation features time dependent dissipative coefficients. Their time dependence is crucial to extract renormalization effects at asymptotically long time. The dynamics from the quantum master equation is in complete agreement with that of the effective action, Langevin dynamics and quantum kinetics, thus providing a unified framework to effective field theory out of equilibrium.

Derivation of non-Markovian transport equations for trapped cold atoms in nonequilibrium thermal field theory

The generation of a baryon asymmetry via leptogenesis is usually studied by\nmeans of classical kinetic equations whose applicability to processes in the\nhot and expanding early universe is questionable. The approximations implied by\nthe state-of-the-art description can be tested in a first-principle approach\nbased on nonequilibrium field theory techniques. Here, we apply the\nSchwinger-Keldysh/Kadanoff-Baym formalism to a simple toy model of\nleptogenesis. We find that, within the toy model, medium effects increase the\nvertex contribution to the CP-violating parameter. At high temperatures it is a\nfew times larger than in vacuum and asymptotically reaches the vacuum value as\nthe temperature decreases. Contrary to the results obtained earlier in the\nframework of thermal field theory, the corrections are only linear in the\nparticle number densities. An important feature of the Kadanoff-Baym formalism\nis that it is free of the double-counting problem, i.e. no need for real\nintermediate state subtraction arises. In particular, this means that the\nstructure of the equations automatically ensures that the asymmetry vanishes in\nequilibrium. These results give a first glimpse into a number of new and\ninteresting effects that can be studied in the framework of nonequilibrium\nfield theory.\n

Renormalization of spatially inhomogeneous nonequilibrium field dynamics

The evolution of hard probes in a medium is a complex multiscale problem that significantly benefits from the use of Effective Field Theories (EFTs). Within the EFT framework, we aim to define a series of EFTs in a way that addresses each energy scale individually in separate steps. However, studying hard probes in a medium presents challenges. This is because an EFT is typically constructed by formulating the most general Lagrangian compatible with the problem’s symmetries. Nevertheless, medium effects may not always be encoded adequately in an effective action. In this paper, we construct an EFT that is valid for studying the evolution of a heavy quark in a QCD plasma containing few other heavy quarks, where degrees of freedom with an energy of the order of the temperature scale are integrated out. Through this example, we explicitly demonstrate how to handle the doubling of degrees that arise in non-equilibrium field theory. As a result, we derive a Fokker–Planck equation using only symmetry and power counting arguments. The methods introduced in this paper will pave the way for future developments in the study of quarkonium suppression.

Dissipative effects, such as the relaxation of quasiparticle occupation numbers, arise from absorptive parts of Green's functions, which typically appear first at the second order of perturbation theory. Within the closed-time-path formalism, it is shown, using a generalized renormalization technique, that these absorptive parts may be approximately resummed so as to appear in unperturbed propagators. In this way, it becomes possible to study, in low-order perturbation theory, the evolution in time of a field theory which is driven away from thermal equilibrium by the presence in its Hamiltonian of explicitly time-dependent parameters. Particular attention is given to a scalar field with time-dependent mass, which is relevant to the dynamics of phase transitions in the very early Universe. Under favorable conditions, the analysis leads to a kinetic equation of the Boltzmann type, and an approximate numerical solution of this equation is presented for illustrative purposes.

Breakdown of time-reversal symmetry is a defining property of nonequilibrium systems, such as active matter, which is composed of units that consume energy. We employ a formalism that allows us to derive a class of identities associated with the time-reversal transformation in nonequilibrium field theories, in the spirit of Ward-Takahashi identities. We present a generalization of the fluctuation dissipation theorem valid for active systems as a particular realization of such an identity, and consider its implications and applications for a range of active field theories. The field theoretical toolbox developed here helps to quantify the degree of nonequilibrium activity of complex systems exhibiting collective behavior.

We derive quantum kinetic equations for scalar fields undergoing coherent evolution either in time (coherent particle production) or in space (quantum reflection). Our central finding is that in systems with certain space-time symmetries, quantum coherence manifests itself in the form of new spectral solutions for the dynamical 2-point correlation function. This spectral structure leads to a consistent approximation for dynamical equations that describe coherent evolution in presence of decohering collisions. We illustrate the method by solving the bosonic Klein problem and the bound states for the nonrelativistic square well potential. We then compare our spectral phase space definition of particle number to other definitions in the nonequilibrium field theory. Finally we will explicitly compute the effects of interactions to coherent particle production in the case of an unstable field coupled to an oscillating background.

Chapter 6 discusses the extension of field-theoretic representations, models, and numerical methods to treat the dynamical evolution of systems that are out of equilibrium. For Bose fluids and magnets, this entails the use of coherent state fields on complex contours with real and imaginary time segments. Complex Langevin sampling of such non-equilibrium field theories provides access to finite temperature dynamics with full quantum fluctuations, both near and far from equilibrium. For classical polymers, approximations are required to address chain entanglement and time-dependent couplings between chain conformations and collective field variables. Current approaches to mesoscopic field-based polymer dynamics are described, along with their numerical implementation.

The possibility of a friction term $\ensuremath{\eta}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}$ in the equation of motion for a scalar field is investigated in nonequilibrium field theory. The central result is that this equation of motion does not possess a time-derivative expansion, and so the friction coefficient $\ensuremath{\eta}$ is not well defined. It is nevertheless possible to obtain an effective friction coefficient numerically. In $\ensuremath{\lambda}{\ensuremath{\Phi}}^{4}$ theory, this differs greatly from existing estimates based on linear response theory, and is actually negative for some values of temperature and coupling constant, indicating that the breakdown of the time-derivative expansion may be quantitatively rather significant.

Coarsening of an isolated far-from-equilibrium quantum system is a paradigmatic many-body phenomenon, relevant from subnuclear to cosmological length scales and predicted to feature universal dynamic scaling. Here, we observed universal scaling in the coarsening of a homogeneous two-dimensional Bose gas, with exponents that match analytical predictions. For different initial states, we reveal universal scaling in the experimentally accessible finite-time dynamics by elucidating and accounting for the initial-state-dependent prescaling effects. The methods we introduce allow direct comparison between cold-atom experiments and nonequilibrium field theory and are applicable to any study of universality far from equilibrium.