The large density of gluons, which is present shortly after a nuclear collision at very high energies, can lead to the formation of a condensate. We identify a gauge-invariant order parameter for condensation based on elementary non-perturbative excitations of the plasma, which are described by spatial Wilson loops. Using real-time lattice simulations, we demonstrate that a self-similar transport process towards low momenta builds up a macroscopic zero mode. Our findings reveal intriguing similarities to recent discoveries of condensation phenomena out of equilibrium in table-top experiments with ultracold Bose gases.
Within the closed-time-path formalism of nonequilibrium QCD, we derive a Slavnov-Taylor (ST) identity for the gluon polarization tensor. The ST identity takes the same form both in Coulomb and covariant gauges. Application to quasi-uniform quark-gluon plasma (QGP) near equilibrium or nonequilibrium quasistationary QGP is made.
Gluons may converge to a stable state at a critical momentum in hadrons. This gluon condensation is predicted by a nonlinear QCD evolution equation. We review the understanding of the gluon condensation and present a clear physical picture that produces the gluon condensation from the colour glass condensate. We summarize the applications of the GC effect in the $p-p(A)$ collisions and predict that the $p-Pb$ and $Pb-Pb$ collisions at the LHC are close to the energy region of the gluon condensation. We warn that for the next generation of hadron colliders with the increasing of the collision e…
We study the transport equation describing a dense system of gluons, in the small scattering angle approximation, taking into account medium-generated effective masses of the gluons. We focus on the case of overpopulated systems that are driven to Bose-Einstein condensation on their way to thermalization. The presence of a mass modifies the dispersion relation of the gluon, as compared to the massless case, but it is shown that this does not change qualitatively the scaling behavior in the vicinity of the onset.
We study the thermalization of gluons far from thermal equilibrium in relativistic kinetic theory. The initial distribution of gluons is assumed to resemble that in the early stage of ultrarelativistic heavy ion collisions. Only elastic scatterings in static, nonexpanding gluonic matter are considered. At first we show that the occurrence of condensation in the limit of vanishing particle mass requires a general constraint for the scattering matrix element. Then the thermalization of gluons with Bose-Einstein condensation is demonstrated in a transport calculation. We see a continuously increa…
We consider a black hole solution with a non-trivial dilaton from IIB super gravity which is expected to describe a strongly coupled hot gauge plasma with non-vanishing gluon condensation present. We construct a rotating and moving baryon to probe the screening and phases of the plasma. Melting of the baryons in hot plasma in this background had been studied previously, however, we show that baryons melt much lower temperature than has been suggested previously.
The accumulation of gluons inside nucleons, i.e., the gluon condensation, may lead to a characteristic broken power-law gamma-ray spectrum in high-energy nucleon collisions. Here we show that the observed spectra of at least 25 sources in the second Fermi Large Area Telescope Catalog of Gamma-ray Pulsars can be well fitted by such a broken power-law function that has only four free parameters. It strongly indicates that the gamma-ray emission from these pulsars is of hadronic origin, but with gluon condensation inside hadrons. It is well known that the quark-gluon distribution in a free nucleo…
We consider gluon condensation (GC) at finite temperature using AdS/CFT. We first show that in the presence of regular horizon, the GC is forbidden in high temperature. Then we consider gravity back-reaction to dilaton coupling and show that the back-reaction develops an singularity, and non-vanishing value of gluon condensation is allowed. We also study thermodynamic quantities and the trace anomaly in the presence of the GC. We discuss how to define a temperature in the presence of the singularity which forbids Hawking temperature. Finally we describe the thermodynamics of the gluon condensa…
The very high energy gamma-ray emissions from pulsars are usually considered to be dominated by leptonic scenario since the hadronic flux is weak. We point out that the gluon condensation predicted by a nonlinear QCD evolution equation may greatly enhance the cross sections of proton-target interactions and give rise to a characteristic broken power law in the gamma-ray spectra. The result is used to explore the gluon condensation signature in the observed gamma-ray spectra from pulsars.
The gluon condensation effect in cosmic proton-proton collisions at high energy is used to explain an excess in the positron spectrum observed by AMS. We find that this excess may originate from the GC-effect in Tycho's supernova remnant.
Using a simple kinetic model we study whether a pion condensate can be formed during the hadronization of the quark-gluon plasma in ultrarelativistic heavy-ion collisions. If hadronization proceeds fast and the pion effective mass is close to zero, some pions condense into the zero momentum state. On the other hand, for the pion effective mass equal to the vacuum one or larger no onset of the Bose-Einstein condensation is observed. The constraints on the condensation process coming from the requirement of the entropy increase are discussed in detail.
Hardening of cosmic proton- and nuclei-spectra is explained by using the gluon condensation (GC) model, which states that a large amount of gluons in proton may condense near the high energy threshold. The results present the GC-effect as common origin of a series of anomalous astrophysical phenomena including the broken power-law in gamma-ray spectra, the excess in positron- and electron-spectra and hardening of proton- and nuclei-spectra.
The problem of the gluonic quasiparticle excitations in QCD is considered under the aspect of the condensation of gluon pairs in the ''squeezed'' vacuum. The present approach is a field theoretical generalization of the Bogoliubov model which successfully reproduced the Landau spectrum in the microscopic theory of superfluidity. We construct a gauge invariant QCD Hamiltonian by formally solving the Gauss equation such that the physical variables are separated by a non-Abelian projection operator, instead of fixing a gauge. By using Dirac quantization we show that the Bogoliubov condensation of…
A new type of gamma ray spectrum is predicted in a general hadronic framework if considering the gluon condensation effects in proton. The result presents the power-law with a sharp break in the gamma ray spectra at the TeV-band. We suggest to probe this GC-signature in the Earth limb gamma-ray spectra using the DArk Matter Particle Explorer and the CALorimetric Electron Telescope on orbit.
In this paper, we study the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. In the present study we ignore the effect of the longitudinal expansion, i.e., we restrict ourselves to spatially uniform systems, with spherically symmetric momentum distributions. Furthermore we take into account only elastic scattering, i.e., we neglect inelastic, num…
Gluons dominate the proton behavior at high energy collisions, they can be condensed at ultra high energy. The collisions of the accelerated high energy protons with interplanetary matter in cosmic rays will produce a huge number of secondary particles at the gluon condensate energy region, which break the primary power-law of cosmic rays. The above predictions seem to be consistent with the recent DAMPE data concerning the electron plus positron spectra. We find that the smoothly broken power-law at $\sim 0.9 TeV$ and $3\sim 4 TeV$ in the DAMPE data can be understood as the gluon condensation…
Hypothesis of quark binding through condensation of gluons inside hadrons is formulated in the context of a renormalization group procedure for effective particles (RGPEP) in the light-front (LF) Hamiltonian approach to QCD. At the momentum scales of relative motion of hadronic constituents that are comparable with Lambda_QCD, the hypothetical boost-invariant constituent dynamics is identified using gauge symmetry. The resulting picture of mesons and baryons closely resembles constituent quark models with harmonic oscillator potentials, shares some features of AdS/QCD, and can be systematicall…
We perform real-time lattice simulations of nonequilibrium quark production in the longitudinally expanding QCD plasma. Starting from a highly occupied gluonic state with vacuum quark sector, we extract the time evolution of quark and gluon number densities per unit transverse area and rapidity. The total quark number shows after an initial rapid increase an almost linear growth with time. Remarkably, this growth rate appears to be consistent with a simple kinetic theory estimate involving only two-to-two scattering processes in small-angle approximation. This extends previous findings about t…
We report that the saturation/CGC model of gluon distribution is unstable under action of the chaotic solution in a nonlinear QCD evolution equation, and it evolves to the distribution with a sharp peak at the critical momentum. We find that this gluon condensation is caused by a new kind of shadowing-antishadowing effects, and it leads to a series of unexpected effects in high energy hadron collisions including astrophysical events. For example, the extremely intense fluctuations in the transverse-momentum and rapidity distributions of the gluon jets present the gluon-jet bursts; a sudden inc…
The collinear color electric and magnetic fields have been discussed to be produced immediately after high energy heavy ion collisions. We discuss anomalous gluon production under the background gauge fields. The gluons are Nielsen-Olesen unstable modes. The production rate of the modes by Schwinger mechanism has recently been found to be anomalously larger than the rate of quarks or other stable gluons. Analyzing classical evolutions of the modes with initial conditions given by vacuum fluctuations, we find that their production makes the color electric field decay very rapidly. The life time…
An upper limit to the critical density for the transition to the deconfined phase, at zero temperature, has been evaluated by analyzing the behavior of the gluon condensate in nuclear matter. Due to the non linear baryon density effects, the upper limit to the critical density, ρ_c turns out about nine times the saturation density, rho_0 for the value of the gluon condensate in vacuum <(α_s/π) G^2>=0.012 GeV^4. For neutron matter ρ_c \simeq 8.5 ρ_0. The dependence of the critical density on the value of the gluon condensate in vacuum is studied.
The gluon condensation (GC)-effects in high energy cosmic rays are investigated. After a brief review of the GC, several examples including gamma-, electron-, and positron-spectra in a broad GeV$\sim $TeV region can be explained by the GC-effects. We find that the GC may break the power-law of the cosmic ray spectra if the energy of accelerated protons exceeds the GC-threshold. The GC is a new phenomenon that is not yet known, it provides a new window to understand the high energy cosmic ray spectra.
In the present paper, we return to the problem on a spontaneous generation of the $A_0$-background field in QCD at finite temperature and include in addition a quark chemical potential, $μ$. We reproduce the known expressions in terms of Bernoulli's polynomials for the gluons and quarks using the same formalism. Then we calculate the $μ$-dependence, both for small $μ$ as expansion, and numerically for finite $μ$. One result is that the chemical potential only weakly changes the values of the condensed field, but quite strongly deepens the minima of the effective potential. The gauge dependence…
I reexamine the phenomena of the chromomagnetic gluon condensation in Yang-Mills theory. The extension of the Heisenberg-Euler Lagrangian to the Yang-Mills theory allows to calculate the effective action, the energy-momentum tensor and demonstrate that the energy density curve crosses the zero energy level of the perturbative vacuum state at nonzero angle and continuously enters to the negative energy density region. At the crossing point and further down the effective coupling constant is small and demonstrate that the true vacuum state of the Yang-Mills theory is below the perturbative vacuu…
In Phys. Rev. Lett. 114 (2015) 6, 061601, we reported on a new universality class for longitudinally expanding systems, encompassing strongly correlated non-Abelian plasmas and $N$-component self-interacting scalar field theories. Using classical-statistical methods, we showed that these systems share the same self-similar scaling properties for a wide range of momenta in a limit where particles are weakly coupled but their occupancy is high. Here we significantly expand on our previous work and delineate two further self-similar regimes. One of these occurs in the deep infrared (IR) regime of…
In this work, the role of inelastic processes in the formation of a transient Bose-Einstein condensation (BEC) is investigated based on kinetic theory. We calculate the condensation rate for an overpopulated gluon system which is assumed to be in thermal equilibrium and with the presence of a BEC. The matrix elements of the inelastic processes are chosen as the isotropic one and the gluons are considered to have a finite mass. Our calculations indicate that the inelastic processes can hinder the formation of a BEC since the negatively infinite net condensation rate can destroy any BEC instantl…
Gluons may converge to a stable state at a critical momentum in nucleon. This gluon condensation will greatly increase the proton-proton cross section provided that the collision energies exceed the gluon condensation threshold. Based on the analyses of cosmic gamma-ray spectra, we find that the $p-Pb$ and $Pb-Pb$ collisions at the LHC are close to the energy region of the gluon condensation effect. We warn that for the next generation of hadron colliders increasing the collision energies, the extremely strong gamma-rays will be emitted in a narrow space of the accelerator due to the gluon con…
Direct photon production from a longitudinally expanding quark-gluon plasma (QGP) at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies is studied with a real-time kinetic description that is consistently incorporated with hydrodynamics. Within Bjorken's hydrodynamical model, energy nonconserving (anti)quark bremsstrahlung q(\bar{q})\to q(\bar{q})γand quark-antiquark annihilation q\bar{q}\to γare shown to be the dominant nonequilibrium effects during the transient lifetime of the QGP. For central collisions we find a significant excess of direct photons in the rang…
The Equation of State and the properties of matter in the high temperature deconfined phase are analyzed by a quasiparticle approach for $T> 1.2~T_c$. In order to fix the parameters of our model we employ the lattice QCD data of energy density and pressure. First we consider the pure SU(3) gluon plasma and it turns out that such a system can be described in terms of a gluon condensate and of gluonic quasiparticles whose effective number of degrees of freedom and mass decrease with increasing temperature. Then we analyze QCD with finite quark masses. In this case the numerical lattice data for …
We study the deep inelastic scattering (DIS) of a proton-targeted lepton in the presence of gluon condensation using gauge/gravity duality. We use a modified $AdS_5$ background where the modification parameter $c$ corresponds to the gluon condensation in the boundary theory. Firstly, when examining the electromagnetic field, we find that non-zero $c$ can increase the magnitude of the field. Our goal is to find the acceptable value of $c$ for this scattering and our method is based on setting the mass of the proton as an eigenvalue of the baryonic state equations of the DIS to find the acceptab…
The deconfinement transition and the hadronization mechanism at high energy are related to the quark-antiquark string breaking and the corresponding temperature depends on the string tension $σ$. In the Unruh scheme of hadron production it turns out $T = \sqrt{σ/2π}$, with $σ\simeq \E$, the vacuum energy density. In heavy ion collisions at lower energy, i.e. large baryonchemical potential, $μ_B$, the dynamics is dominated by Fermi statistics and baryon repulsion. However one can still consider $\E$ as the relevant physical scale and its evaluation as a function of the baryon density, in a nucl…
We investigate the infrared (strong-coupling) regime of SU(N)-Yang-Mills theory on a self-dual background. We present an evaluation of the full effective potential for the field strength invariant F_{μν}F^{μν} from non-perturbative gauge correlation functions and find a non-trivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the beta function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators. Consistency between both gauges in the infrared …
In ultra-relativistic heavy ion collisions, the matter formed shortly after the collision is a dense, out of equilibrium, system of gluons characterized by a semi-hard momentum scale $Q_{\rm s}$. Simple power counting arguments indicate that this system is over-occupied: the gluon occupation number is parametrically large when compared to a system in thermal equilibrium with the same energy density. On short time scales, soft elastic scatterings tend to drive the system towards the formation of a Bose--Einstein condensate that contains a large fraction of the gluons while contributing little t…
We show that a certain type of color magnetic condensation originating from magnetic monopole configurations is sufficient to provide the mass for off-diagonal gluons in the SU(2) Yang-Mills theory under the Cho--Faddeev--Niemi decomposition. We point out that the generated gluon mass can cure the instability of the Savvidy vacuum. In fact, such a novel type of magnetic condensation is shown to occur by calculating the effective potential. This enables us to explain the infrared Abelian dominance and monopole dominance by way of a non-Abelian Stokes theorem, which suggests the dual superconduc…
$Q\bar{Q}$ potential is studied in the SU(2) gauge theory. Based on the nonlinear gauge of the Curci-Ferrari type, the possibility of a gluon condensation $\langle A_μ^+A_μ^-\rangle$ in low-energy region has been considered at the one-loop level. Instead of the magnetic monopole condensation, this condensation makes classical gluons massive, and can yield a linear potential. We show this potential consists of the Coulomb plus linear part and an additional part. Comparing with the Cornell potential, we study this confining potential in detail, and find that the potential has two implicit scales…
Relativistic hydrodynamics simulations of quark-gluon plasma play a pivotal role in our understanding of heavy ion collisions at RHIC and LHC. They are based on a phenomenological description due to Mueller, Israel, Stewart (MIS) and others, which incorporates viscous effects and ensures a well-posed initial value problem. Focusing on the case of conformal plasma we propose a generalization which includes, in addition, the dynamics of the least damped far-from-equilibrium degree of freedom found in strongly coupled plasmas through the AdS/CFT correspondence. We formulate new evolution equation…
We report recent progress on understanding the thermalization of the quark-gluon plasma during the early stage in a heavy ion collision. The initially high overpopulation in the far-from-equilibrium gluonic matter ("Glasma") is shown to play a crucial role. The strongly interacting nature (and thus fast evolution) naturally arises as an {\em emergent property} of this pre-equilibrium matter where the intrinsic coupling is weak but the highly occupied gluon states coherently amplify the scattering. A possible transient Bose-Einstein Condensate is argued to form dynamically on a rather general g…
We review recent progress in open quantum system approach to the description of quarkonium in the quark-gluon plasma. A particular emphasis is put on the Lindblad equations for quarkonium and its numerical simulations.
We study the evolution of a gluon system under conditions of density and temperature similar to those explored in the early stage of ultra-relativistic heavy-ion collisions. We first describe the implementation of Relativistic Boltzmann-Nordheim (RBN) transport approach that includes in the collision integral the quantum effects of Bose-Einstein Statistics. Then, we describe the evolution of a spatially uniform gluon system in a box under elastic collisions solving the RBN for various initial conditions. We discuss the critical phase-space density that leads to the onset of a Bose-Einstein con…
For a moving heavy quark antiquark ($Q\bar{Q}$) in a quark gluon plasma (QGP), we use gauge/gravity duality to study both real and imaginary parts of the potential (Re$V_{Q\bar{Q}}$ and Im$V_{Q\bar{Q}}$ respectively) in a gluon condensate (GC) theory. The complex potential is derived from the Wilson loop by considering the thermal fluctuations of the worldsheet of the Nambu-Goto holographic string. We calculate Re$V_{Q\bar{Q}}$ and Im$V_{Q\bar{Q}}$ in both cases where the axis of the moving $Q\bar{Q}$ pair is transverse and parallel with respect to its direction of movement in the plasma. Usin…
We present a kinetic description of Bose-Einstein condensation for particle systems being out of thermal equilibrium, which may happen for gluons produced in the early stage of ultra-relativistic heavy-ion collisions. The dynamics of bosons towards equilibrium is described by a Boltzmann equation including Bose factors. To solve the Boltzmann equation with the presence of a Bose-Einstein condensate we make further developments of the kinetic transport model BAMPS (Boltzmann Approach of MultiParton Scatterings). In this work we demonstrate the correct numerical implementations by comparing the …
In the framework of the functional renormalization group, we reexamine the stability of the Yang-Mills vacuum with a chromomagnetic condensation. We show that the Nielsen-Olesen instability of the Savvidy vacuum with a homogeneous chromomagnetic condensation disappears in the $SU(2)$ Yang-Mills theory. As a physical mechanism for maintaining the stability even for the small infrared cutoff, we argue that dynamical gluon mass generation occurs due to a BRST-invariant vacuum condensate of mass dimension-two, which is related to two-gluon bound states identified with glueballs. These results supp…
Photon production from a thermalized quark-gluon plasma of finite lifetime is studied directly in real time with a nonequilibrium formulation that includes off-shell (energy nonconserving) effects. To lowest order we find that production of direct photons form a quark-gluon plasma of temperature T ~ 200 MeV and lifetime t ~ 10-20 fm/c is strongly enhanced by off-shell (anti)quark bremsstrahlung q(\bar{q})->q(\bar{q})γ. The yield from this nonequilibrium finite-lifetime effect dominates over those obtained from higher order equilibrium rate calculations in the range of energy E > 2 GeV and fall…
We study the confinement physics in QCD in the maximally abelian (MA) gauge using the SU(2) lattice QCD, based on the dual-superconductor picture. In the MA gauge, off-diagonal gluon components are forced to be small, and the off-diagonal angle variable $χ_μ(s)$ tends to be random. Within the random-variable approximation for $χ_μ(s)$, we analytically prove the perimeter law of the off-diagonal gluon contribution to the Wilson loop in the MA gauge, which leads to abelian dominance on the string tension. To clarify the origin of abelian dominance for the long-range physics, we study the charged…
We study the nonequilibrium photon production in the quark-hadron phase transition, using the Friedberg-Lee type solitons as a working model for quark-hadron physics. We propose that to search for nonequilibrium photons in the direct photon measurements of heavy-ion collisions may be a characteristic test of the transition from the quark-gluon to hadronic phases.
We consider various special cases of gluon condensates at finite temperature. The gluon condensate for an ideal gas of gluons with a given vacuum expectation value is introduced for the sake of comparison with that calculated using the recent finite temperature lattice gauge simulations for a pure Yang-Mills SU(3) gauge theory at the known critical temperature. We extend this comparison using the high precision lattice data for two light dynamical quarks. The investigation of these three cases show some interesting differences arising from the strong interaction alone and in the presence of qu…
We apply classical gluodynamics to early stages of ultrarelativistic heavy-ion collisions. We start by going through the space-time evolution of ultrarelativistic heavy-ion collisions in the color glass condensate framework and the basics of real-time gluodynamics on the lattice in the temporal gauge. We study the plasmon mass scale in three- and two-dimensional systems by comparing three different methods to measure the mass scale. The methods are a formula which can be derived from Hard Thermal Loop effective theory at leading order (HTL), the effective dispersion relation (DR) and measureme…
We present real-time lattice simulation results for nonequilibrium quark production from an over-occupied gluon plasma in longitudinally expanding geometry. The quark number density per unit transverse area and rapidity shows almost linear growth in time, and its growth rate appears to be consistent with a simple kinetic theory estimate involving only two-to-two scattering processes in small-angle approximation. We also find that quarks produced at early times satisfy a nonequilibrium scaling law.
We discuss the implementation of leading-order photon production in nonequilibrium partonic transport simulations. In this framework photons are produced by microscopic scatterings, where we include the exact matrix elements of Compton scattering, quark-antiquark annihilation, and bremsstrahlung processes. We show how the hard-thermal loop inspired screening of propagators has to be modified such that the microscopic production rate agrees well with the analytically known resummed leading-order rate. We model the complete quark-gluon plasma phase of heavy-ion collisions by using the partonic t…
A hard $γ$-ray spectrum of supernova remnant G106.3+2.7 can be explained by using the hadronic model with the gluon condensation effect. This implies that not only PeVatrons but also EeVatrons generally exist in the universe including our galaxy, and they can accelerate protons to beyond "ankle" ($10^{19}~eV$). Although these proton beams are very weak in the galaxy and cannot be observed individually on the earth, the gluon condensation effect may greatly enhance the proton-proton cross section, which can compensate for the weak proton flux and produce the observed $γ$-rays. We also show that…
We propose a vacuum condensate of mass dimension 2 consisting of gluons and ghosts in the framework of the manifestly covariant gauge fixing of the SU(N) Yang-Mills theory. This quantity is both BRST and anti-BRST invariant for any gauge. It includes the ghost condensation $C^a \bar{C}^a$ proposed first in the modified Maximal Abelian gauge and reduces to the gluon condensates $(\mathscr{A}_μ)^2$ of mass dimension 2 proposed recently in the Landau gauge of the Lorentz gauge fixing. The vacuum condensate of dimensions 2 can provide the effective mass for gluons and ghosts. The possible existenc…
This thesis is devoted to studying aspects of real-time nonequilibrium dynamics in quantum field theory by implementing an initial value formulation of quantum field theory. The main focus is on the linear relaxation of mean fields and quantum kinetics in nonequilibrium multiparticle quantum systems with potential applications to ultrarelativistic heavy ion collisions, cosmological phase transitions and condensed matter systems. We first study the damping of fermion mean fields in a fermion-scalar plasma with a view towards understanding baryon transport phenomena during electroweak baryogenes…
We show that the QCD vacuum (without dynamical quarks) is a dual superconductor at least in the low-energy region in the sense that monopole condensation does really occur. In fact, we derive the dual Ginzburg-Landau theory (i.e., dual Abelian Higgs model) directly from the SU(2) Yang-Mills theory by adopting the maximal Abelian gauge. The dual superconductor can be on the border between type II and type I, excluding the London limit. The masses of the dual Abelian gauge field is expressed by the Yang-Mills gauge coupling constant and the mass of the off-diagonal gluon of the original Yang-Mil…
We review applications of Dyson-Schwinger equations at nonzero temperature, T, and chemical potential, mu, touching topics such as: deconfinement and chiral symmetry restoration; the behaviour of bulk thermodynamic quantities; the (T,mu)-dependence of hadron properties; and the possibility of diquark condensation.
Using the general framework of nonequilibrium statistical mechanics for relativistic quantum field systems we derive the basic equations of quantum field kinetics. The main aim of the approach is calculation of observables associated with quark-gluon plasma which is out of thermal and chemical equilibrium. We show that in the regime of high rate of the phase and statistical mixing the perturbation theory for many-body quantum field system should have the form which much differs from the standard expansion in powers of coupling constant. ( Figures can be obtained from the author )
