Quantized meson fields in and out of equilibrium. I : Kinetics of meson condensate and quasi-particle excitations
We formulate a kinetic theory of self-interacting meson fields with an aim to describe the freezeout stage of the space-time evolution of matter in ultrarelativistic nuclear collisions. Kinetic equations are obtained from the Heisenberg equation of motion for a single component real scalar quantum field taking the mean field approximation for the non-linear interaction. The mesonic mean field obeys the classical non-linear Klein-Gordon equation with a modification due to the coupling to mesonic quasi-particle excitations which are expressed in terms of the Wigner functions of the quantum fluctuations of the meson field, namely the statistical average of the bilinear forms of the meson creation and annihilation operators. In the long wavelength limit, the equations of motion of the diagonal components of the Wigner functions take a form of Vlasov equation with a particle source and sink which arises due to the non-vanishing off-diagonal components of the Wigner function expressing coherent pair-creation and pair-annihilation process in the presence of non-uniform condensate. We show that in the static homogeneous system, these kinetic equations reduce to the well-known gap equation in the Hartree approximation, and hence they may be considered as a generalization of the Hartree approximation method to non-equilibrium systems. As an application of these kinetic equations, we compute the dispersion relations of the collective mesonic excitations in the system near equilibrium.
keywords:meson condensate; Vlasov equation
Theoretical study of the space-time evolution of matter in high energy nuclear collisions has a long history since the early pioneering works of Fermi and Landau employing thermodynamics and fluid mechanics [Fer50, ]. More recent works Bjo83  have been motivated by the prospect of studying a new form of matter experimentally by means of very energetic collisions of heavy nuclei as those currently underway at Brookhaven with Relativistic Heavy-Ion Collider (RHIC) and planned at CERN with Large Hadron Collider (LHC). The data taken from the RHIC experiments has shown existence of a strong anisotropic (elliptic) collective flow of hadrons in non-central collisions RHICWP05 , indicating early local equilibration of dense matter produced by the collision Flow . This supports the hydrodynamic picture of matter evolution and gives us a hope to learn information of the equation of state of dense matter from systematic analyses of the data.
Much attention has been paid to the early stage of matter evolution where a dense plasma of unconfined quarks and gluons has been expected to be formed through some complex non-equilibrium processes Bay84  . One anticipates also a breakdown of the hydrodynamic behavior of the system when it is diluted sufficiently and the collision time exceeds the characteristic time scale of expansion. This stage of the matter evolution is usually refered to as the freezeout stage. The aim of this work is to present a kinetic theory which is designed to describe the freezeout stage of the expanding hadron gas.
The freezeout of the expanding hadronic matter would proceed in several steps; chemical freezeout of the relative abandunce of hadron spieces may occur before the kinetic freezeout of the momentum distribution of hadrons. Observed relative abandunce of hadrons fit very nicely to a simple picture that the chemical freezeout occurs at a certain temperature and baryon chemical potential BS99 , RHICWP05 . Some of the other observables, such as two particle momentum correlations, the pion analogue of the Hanbury Brown and Twiss two photon intensity interferometry, may be very sensitive to the dynamics of the final kinetic freezeout as indicated by the “HBT puzzle” found in the recent data analysis HBT .
If the initial state of the dense matter formed in the collision is a plasma of deconfined quarks and gluons, freezeout of the color degrees of freedom should proceed these processes when the quark-gluon plasma hadronizes. Chiral symmmetry NJL61 , an approximate global symmetry of QCD which becomes exact in the limit of vanishing quark masses and is considered to be broken spontaneously in the QCD vacuum, may also play an important role in the freezeout dynamics.
As a dense hadronic matter formed by ultrarelativistic nuclear collision is diluted by the expansion, one expects that the system undergoes a phase transition associated with the spontaneous breakdown of the chiral symmetry which is restored temporalily after the collision by the formation of a quark-gluon plasma. As the quark-gluon plasma hadronizes and the system turns into the confining phase, the system would gradually develop a vacuum chiral condensate and remaining excitaions would expand in the influence of the growing chiral condensate. This physical picture has been elaborated in terms of a classical equation of motion for the chiral condensate; effects of excitations were described in terms of statistical fluctuations in the classical fields Bjo87 , RW93 . The fluctuating condensate described by classical pion field has been termed Disoriented Chiral Condensate (DCC), emphasizing the symmetry aspect of the problem RW93 . What was missing in these classical treatments of the meson fields is the existence of particle excitations in addition to the condensate. Inclusion of particle excitations requires the quantization of the fields. The effect of the quantum fluctuations of meson fields has been studied by Tsue, Vautherin and one of the present authors (TM) TVM99  by the functional Schrödinger picture formalism KV89 , EJP88 .
The physical picture we have described is very similar to what happens when the dilute gas of magnetically trapped atoms of alkali metals cools down by evapolation PS02 . Some of the atoms condense into the lowest single particle level in the trapping external potential forming a Bose-Einstein condensate. The dynamics of such a system may be described by the coupled equations of motion for the condensate, or the Gross-Pitaevskii equation GP61  , and the kinetic equation, or the Boltzmann-Vlasov equations, for the phase space distribution of the excitations in the presence of the condensate ZNG99 , ITG99 .
In this paper we will show that a similar set of equations can be derived for a system of interacting mesons described by the relativistic quantum field theory by the mean field approximation. This approximation corresponds to a neglect of all correlations in the system KB62 . We will make no attempt in this work to justify this approximation and leave it as an open problem for further study.
In this context we note that field theoretical derivation of a Boltzmann-type
kinetic equation has been given by many others CH88 .
Most of these works focuses on the derivation of the collision terms.
The present work is distinguished from these works in the emphasis of
the role of the mean field in the evolution of the system coupled with the
quasi-particle excitations in the same spirit as in EJP88 , TVM99 .
We omit the effect of the quasi-particle collisions in this work. This procedure
may be reasonable, at least as a first step, for describing dynamical aspect of
the freeze-out process: even in the absence of collisions, interactions between
the quasi-particles and the evolving condensate would affect the final
But the present approach is not adequate, however, for the early thermalization
problem, where the collision terms play essential role Bay84 .
In the next section we formulate a quantum kinetic theory for quantum scalar meson field, starting from the Heisenberg equations for quantum field of one-component real scalar. The mean field approximation replaces the products of quantum fields by the products of classical fields and the statistical average of bilinear forms of the quantum fluctuations. The latter is expressed in terms of the Wigner functions which is reduced to the single particle distribution function in the classical Boltzmann equation. Our theory contains another forms of the Wigner functions which have no classical counter parts and arises due to the coherent pair creation and annihilation processes in non-uniform systems. Only in uniform systems, these off-diagonal components of the fluctuation can be eliminated by suitable redefinition of the particle mass. Appearance of the off-diagonal Wigner functions is reminiscent of the anomalous propagators in the microscopic theory of superconductivity FW71 , Gor58 , Nam60 . Similar structure also appears in the theory of Bose-Einstein Condensate ITG99 . Some details of the mean field calculation is given in Appendix.
In section 3, we apply our method to uniform systems and show that each mode characterized by the particle momentum obeys non-linear forced oscillatory motion. In section 4 we show that in equilibrium these kinetic equations are reduced to a gap equation DJ74  which determine the equilibrium amplitude of the condensate and the mass parameter as a function of the temperature. In this paper, we ignore the effect of the divergent vacuum polarizations which requires a subtle renormalization procedure in the mean field approximation BG77 . The solution exhibits characteristic features of the first order phase transition.
In section 5, we study slowly varying non-uniform systems. Taking long wavelength approximation, the equations of motion of the diagonal components of the Wigner functions are reduced to a Vlasov equation in a form generalized by Landau for quasi-particles excitations in quantum Fermi liquid Lan57 , BP91  with the quasi-particle energy given in the mean field approximation. Non-vanishing off-diagonal components of the Wigner functions generate extra terms in the Vlasov equation which may be interpreted as particle source and sink terms.
In section 6, we compute the dispersion relations of excitations
of the system near equilibrium. Solving the coupled kinetic equations by
linearizing the equations with respect to small oscillatory deviations from
equilibrium solution we obtain dispersion relation of the excitation
modes in the system near equilibrium.
We find the continuum of quasi-particle excitations in the entire space-like
energy-momentum region in addition to the continuum in time-like region
due to the (thermally induced) pair creation.
We found that in the low temperature phase the meson pole shifts due to
the coupling to the quasi-particle continua.
The effective meson mass vanishes at the edge of the spinodal instability
line of the first order transition.
A short summary of the paper is given in section 7 with remarks on the remaining problems.
2 Kinetic equations for the meson condensate and quasi-particle excitations
In this section we derive quantum kinetic equations which describes the time evolution of the meson condensate coupled with mesonic quasi-particle excitations. We use the natural unit throughout this paper.
2.1 Quantized real scalar field in the Heisenberg representation
We first take a simple model of a self-interacting real scalar field in the Heisenberg picture. The Hamiltonian is given by
Here the scalar field and its canonical conjugate momentum field are quantized by the equal-time commutation relations:
The Heisenberg equation of motion of the quantum field is given by
while the equation of motion of the canonical conjugate field becomes
Elimination of the field momentum from these equations yields a modified Klein-Gordon equation for the quantum scalar field :
2.2 Density matrix and Gaussian Ansatz
We are interested in the time evolution of the system described by the density operator
where are a set of normalized wave functions and is the probability distribution for a mixed state described by this density matrix so that it satisfies
This density matrix can be also expressed in terms of some complete set of the wave functions of our Hilbert-Fock space as
All physical information of the system to be described are contained in a specific form of the density matrix. In thermodynamic equilibrium, the density matrix is given by
where gives the Helmholtz free energy of the system at temperature .
In the Heisenberg picture the density matrix is time-independent since the wave functions are time-independent; all time dependence arises from the time dependence of an operator:
For example, we define the classical condensate fields by the statistical average of the quantum fields,
In the following we take a Gaussian Ansatz for the density matrix:
where and are shifted field operators defined by
The shifted field operators obey the same equal-time commutation relations as the original fields:
2.3 The Wigner functions
We introduce the one-particle Wigner function by
Here the particle creation and annihilation operators may be defined in terms of the Fourier transforms of the shifted fields
These relations are rewritten as:
with which we may interpret () as annihilation (creation) operator of “particle excitation” with momentum .
We also introduce the following other forms of the Wigner functions:
These Wigner functions are not independent but are related to each other. The complex conjugate of the Wigner functions are given by
where the asterisk () stands for the complex conjugate. The commutation relations imply also that and are even functions of
The four Wigner functions may be grouped together to form a matrix form of the Wigner function
The appearance of the “off-diagonal” components of the Wigner functions is reminiscent of the anomalous propagators in the BCS theory of superconductivity which arises due to the presence of fermion pair condensate FW71 , Gor58 , Nam60 . Note that our definition of the matrix components of the Wigner function is slightly different from those for the propagators.
We write the Fourier transforms of the Wigner functions as
while (37) implies that and are complex conjugate to each other:
Here we have chosen the particle “mass” to be different from the mass parameter in the original Hamiltonian. Physical particle mass for interacting fields is generally different from the mass parameter in the Hamiltonian or Lagrangian due to the effect of interaction, e.g. renormalization with or without the spontaneous symmetry breaking. It may also depend on the physical conditions described by the statistical average with the density matrix . Since we are interested in non-equilibrium time-evolution of the system where the physical particle mass may not have a definite meaning in the intermediate states, we consider here just as a parameter to be chosen at our discretion, for an appropriate choice of the initial conditions specified by the Gaussian density matrix. A different choice of this mass parameter would give different definition for “particle excitations”; the Wigner functions thus depend on the particular choice of the mass parameter.
Suppose we take a different particle mass to define the particle creation and annihilation operators
These new particle creation and annihilation operators should also obey the commutation relations,
so that they are related to the original ones by the Bogoliubov transformation:
where the real parameter is determined by requiring that they describe the same fields:
and this gives
The new Wigner functions defined by replacing the creation and annihilation by the new ones are related to the original Wigner functions by:
For a small change of the mass parameter the Wigner function will change by
In uniform equilibrium system the mass parameter may be chosen to ”diagonalize” the one-body mean field Hamiltonian. As will be shown later, this procedure will lead to the well-known gap equation, a self-consistency condition to determine . If the system is slowly changing in time, one may still use such procedures adjusted to slowly varying quasi-equiliblium conditions, introducing a time dependent effective mass as a dynamical parameter to describe such adiabatic process. For such calculations, the relation (60) may be used to describe the adiabatic change of the Wigner functions by the change of the mass parameter.
In more general non-equilibrium situations as we expect to encounter at the freeze-out stage of expanding matter, however, there may be no such appropriate condition to determine the mass parameter. The situation could even be worse: if the system goes through unstable state with respect to small fluctuation of the field, then the adiabatically determined mass parameter would become pure imaginary, reflecting the extremum, instead of the minimum, of the effective potential. In such case we may keep the value of at a certain real value reflecting initial conditions. The instability would then show up as appearance of a growing solution to our kinetic equations; which would eventually be stabilized by the non-linear interaction.
To extract the physical information, such as the particle distribution in the final asymptotic state, we should use the Wigner function defined with the physical particle mass in the vacuum. But these asymptotic physical Wigner functions may be calculated from the Wigner functions with a different choice of the mass parameter by the relation (56).
2.4 Equation of motion in the mean field approximation
The equation of motion of the classical mean field is obtained by taking the quantum statistical average of the field equation (7) . With the Gaussian Ansatz for the density matrix, we find
This equation corresponds to the non-linear Schödinger equation (also called the Gross-Pitaevskii equation PS02 ) in the theory of Bose-Einstein condensates. So we may call this equation non-linear Klein-Gordon equation. The non-linearity arises due to the self-interaction of the classical field (condensate) and also due to the interaction with fluctuations which also depends on implicitly. The latter may be interpreted as due to “particle excitations”, since the fluctuation can be expressed by the Wigner functions as
The time-evolution of the classical mean field is thus coupled with the time-evolution of the Wigner functions.
To derive the equation of motion of the Wigner functions, we need to compute the time-derivative of the bilinear forms of the operators and which in turn requires computation of the commutators of these operators with the hamiltonian. We decompose the original hamiltonian as
where is the classical hamiltonian obtained from by replacing the quantum fields by their classical expectation values and contain the -thrth power of the quantum fluctuation (or and ). A straightforward calculation yields
The commutators of bilinear forms of and with vanish and the commutators with would give either a linear term or the third power of the fluctuation, both of which may vanish when taking the average with the Gaussian density matrix. What remain to be computed are then the commutators with and with . They will give either the bilinear form of and or the fourth power of the fluctuations. The Gaussian average of the resultant equations of motion of the bilinear field operators would give the desired equations of motion of the Wigner functions. Details of this computation is given in Appendix A.
The resultant equation of motion of the Wigner functions may be obtained more easily by introducing the mean field Hamiltonian defined by
In the momentum representation this mean field Hamiltonian may be written as
The commutator of a bilinear operator product of and with this mean-field Hamiltonian is given by
We show in Appendix that the quantum statistical average of this commutator with the Gaussian density matrix gives precisely the same result for the same statistical average of the commutator with the original Hamiltonian:
Therefore one can compute the equations of motion of the Wigner functions using this effective Hamiltonian,
Using this we find,
Equations of motion of other three Wigner functions , , can be also computed from the commution relations of the product operators , , , with the mean field Hamiltonian , respectively: We obtain
and the equation of motion of can be obtained from (LABEL:eomF) by the substitution :
These equations form a closed system of coupled differential equations with the non-linear Klein-Gordon equation (61) which may be rewritten as
These four equations of motion of the Wigner function may be combined into a single matrix form as
3 Uniform system
For a uniform system, we expect that the classical mean field and the self-energy become functions only of time:
Thus the non-linear Klein-Gordon equation (80) becomes
and the mean-field Hamiltonian is reduced to
In this case the Wigner functions contain non-vanishing components only for the diagonal elements () so that they may be written as
Then the equations of motion of the Wigner functions become
These coupled equations have a time-independent solution of the form