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We have constructed a consistent system of equations for the Yang-Mills quantum-wave fluctuations in the classical Yang-Mills condensate based on canonical quantization in the Heisenberg representation. Such a quasi-classical system has been thoroughly analyzed in the conformal limit in the linear and quasi-linear approximations, both analytically and numerically. We have found that interaction between waves and condensate triggers a significant transfer or swap of energy from the condensate to the wave modes in the SU(2) gauge theory. Remarkably, a similar energy swap effect has been found in the maximally-supersymmetric N=4 Yang-Mills theory, as well as in the two-condensate SU(4) gauge theory. Such a generic feature of Yang-Mills dynamics opens up vast phenomenological implications in ultra-relativistic Yang-Mills plasma physics.

A consistent non-perturbative theory of the Yang-Mills (YM) vacuum responsible e.g. for spontaneous chiral symmetry breaking and color confinement phenomena in quantum chromodynamics (QCD) [

The major goal of our paper is to study dynamical properties of the spatially-inhomogeneous wave modes in the homogeneous YM condensate (YMC) incorporating interactions between the waves and the condensate in the simplest one-condensate

We work in Hamilton gauge―the only known gauge which allows to formulate the YM theory in the Heisen- berg representation beyond the perturbation theory (for more details, see e.g. Reference [

Let us mention a few aspects of the YM theory in Hamilton gauge which are important for our analysis. Typically, the YM theory is formulated in terms of the functional (or path) integral [

In our analysis, we take the following theoretically consistent pathway^{1} which has certain methodological advantages for interacting systems of YM waves and YM condensates compared to the standard functional integral formulation:

• First, one starts with the YM Lagrangian in Hamilton gauge and writes down the Lagrange equations of motion in the operator form following to the Bohr’s correspondence principle;

• Second, since the zeroth component of the YM field

• Third, from the Lagrange formulation of the YM theory one can turn to the Hamilton (or canonical) formulation and canonical quantization in a theoretically consistent way.

Let us discuss the third point of the above scheme in more detail. In the case of free YM field (without its interactions with the condensate), free longitudinal YM mode does not have any proper frequency and dispersion, thus in the Minkowski space without the condensate this mode is aperiodic. The latter has three consequences: 1) it does not contribute to the Hamiltonian; 2) it is impossible to calculate its contribution to the YM propagator as a vacuum expectation value of time-ordered operator product; 3) there is a problem with canonical quantization. The first consequence is essentially one of the advantages of the Hamilton gauge which excludes any non-physical degrees of freedom from the Hamiltonian. The other two consequences point out to the fact that the YM theory in Hamilton gauge cannot be constructed according to standard algorithms of a non- degenerate field theory. We found a simple alternative way to resolve the latter issue: in a system without the YMC we introduce extra “virtual” infinitesimal terms, which are proportional to an infinitesimal parameter, into the YM Lagrangian in Hamilton gauge. These terms are chosen in such a way that the longitudinal mode acquires a small dispersion proportional to the infinitesimal parameter which allows to perform canonical quantization in the standard way and to calculate a contribution of this mode to the YM propagator as a vacuum expectation value of time-ordered operator product. After canonical quantization procedure and construction of the YM propagator, the infinitesimal parameter is safely turned to zero leading to exactly the same S-matrix as the one defines in the standard functional integral formulation. This methodological trick therefore leads to theoretically consistent results and allows to realize the scheme described above in practice.

Consider now physically interesting case of the YM wave modes interacting with the YMC. Here, the situa- tion changes significantly, namely, in this case an important physical effect of dynamical generation of the lon- gitudinal plasma waves as collective excitations (also known as plasmons) of macroscopic medium takes place. The latter effect is well-known in physics of ordinary plasma [

A complete theory of quark-gluon plasma accounting for interactions between waves is very complicated and is not yet constructed. In this paper, we consider a simple one-condensate toy-model as our starting point where interactions between wave modes and condensate are taken into account only while interactions between differ- ent wave modes are not included. Physically, this situation corresponds to a YM system in the beginning of its time evolution with a few wave modes interacting with the condensate such that the interactions between waves are negligibly small compared to interactions of the waves with the condensate. Noticeably enough it turns out that even in this model the longitudinal modes acquire proper frequencies providing their periodic dynamics in agreement with previous considerations in the literature. Moreover, it turns out that as soon as one extracts the homogeneous condensate in the initial YM Lagrangian in Hamilton gauge, the canonical quantization of longi- tudinal modes automatically appears to be natural and theoretically consistent without introducing any “virtual” infinitesimal terms discussed above. Of course, the longitudinal (plasma) waves in the condensate get excited together with transverse ones so they contribute to observable quantities and must be taken into consideration on the same footing to the transverse ones. Based on the canonical framework we present in this paper and play out in the simplest one-condensate model, one may further extend this study incorporating effects of mutual interac- tions between different wave modes. Certainly, the latter have to be taken into account in a complete theory of ultra-relativistic gluon plasma.

The effect of dynamical generation of longitudinal modes in a YM medium and their dynamical role described above is well-known in the literature [^{2}. Such a specific energy swap effect between the two vacuum subsystems may have serious consequences to the theory of non-perturbative YM vacuum and, in particular, may have important phenomenological implications e.g. in the theory of QCD phase transition in early Universe and in particle production mechanisms in the hot cosmological plasma.

To start with, we have constructed the exact quasi-classical equations for the wave modes and the condensate, and investigated them in the linear approximation (in small wave amplitudes limit). In this case, the equations of motion have a characteristic form of Mathieu equations having certain regions of parametric resonance instabil- ity which leads to an increase of amplitude of the waves. As was argued above, the constraints written for the system of interacting homogeneous YMC and inhomogeneous waves do not allow to exclude the longitudinal modes, such that these extra d.o.f. acquire their own dynamical properties due to interactions between the two subsystems.

As a consequence of energy conservation, an increase of the YM waves energy reflected in a corresponding increase of their amplitudes has to be accompanied by a corresponding decrease of the YMC energy. In particu- lar, this fact must be taken into account in derivation of the quasi-linear YMC equation of motion where the “back reaction” effect of the wave modes to the YMC is consistently incorporated. The numerical analysis of the resulting system of equations has indeed revealed the energy swap effect satisfying the energy conservation: a decrease of the YMC energy is exactly compensated by an increase of energy attributed to the wave modes, which is an important test of our calculations. In addition, we have investigated the energy spectrum of the free YMC steady-state solutions of the corresponding Schrödinger equation. Interestingly enough, it has been found that its energy spectrum corresponds to a potential well of the fourth power.

Further, we have generalized our study to the maximally

The paper is organized as follows. In Section II we derive the equations of motion for the YMC and YM wave modes in the first (linear) approximation. An extension to the quasi-linear case accounting for the leading-order “back reaction” effect of the wave modes to the YMC has been performed and thoroughly investigated in Section III. In Section IV we apply our quasi-classical approach to the

For a comprehensive introduction to the theory of YM fields we refer to the standard quantum field theory textbooks [

The Lagrangian of a pure YM field is

where

is the YM stress tensor as usual. Here, we work with the

Imposing the Hamilton (or Weyl) gauge

we end up with nine equations of motion

and three constraints in the form of first integrals of motion

The total time derivative here can be removed in degenerate case, i.e.

It has been demonstrated in Refs. [^{3}

Thus, a trivial projection of the Yang-Mills vector field

Here, the spatially-homogeneous time-dependent scalar field

The representation (8) enables us to rewrite the YM equation of motion (4) through the YMC,

The constraint Equation (6) provides an extra condition:

The Equation (9) is separable by averaging over the Heisenberg state vector. To the leading (zeroth) order in

which has to be fulfilled in order to find the equations of motion for the free YM wave modes in the first (linear) approximation. It is convenient to turn to Fourier transforms for

Then, we expand the Fourier transforms of antisymmetric

and

respectively, where the coefficients satisfy the following conditions

Thus, instead of nine components of the

and

where we introduced the following shorthand notations:

Therefore, one arrives at the system of nine equations of motion for nine d.o.f. Finally, the equations of constraint (10) can be conveniently transformed to the following explicit form in terms of new d.o.f.

It is straightforward to check that these two constraints are automatically satisfied for a solution of the system of YM Equations (16) and (17). Note, in a degenerate YM theory, these constraints do not explicitly contain time derivatives, i.e.

which are thus the first integrals of motion in this case.

For further considerations and consistency checks, it is instructive to represent quadratic Lagrangian and Hamiltonian densities of the

It is straightforward to check that the system of Equations (16) and (17) can be obtained directly from Equations (23) or (24) in usual way. Finally, the complete effective

which will be used below in studies of the dynamical properties of the “waves + condensate” system below.

The equation of motion which determines dynamical properties of the YMC,

where

The energy spectrum of quasi-harmonic YMC fluctuations can be found in standard way from the Schrö- dinger steady-state equation and is shown in

one arrives at the Schrödinger equation

It straightforward to show that the free YMC spectrum corresponds to a potential well of the fourth power. Numerical calculation provides us with the first few energy levels in the spectrum (see also, Reference [

For practical use, it is convenient to come up with an approximate analytic formula for the lower end of this

spectrum, e.g. in the following form

The maximal error of this formula for the first thirty energy levels does not exceed 4%.

Now, consider dynamics of the wave modes in the linear approximation (without taking into account for the “back reaction” of wave modes to condensate) encoded in the system of YM Equations (11), (16) and (17). In fact, Equation (16) is a closed system of two equations for two functions

Further, introducing superpositions

Equation (16) in the above basis falls apart into two independent equations for

which are recognized as Mathieu equations. Here,

where

An analytical analysis of remaining Equation (17) is less feasible due to the presence of quadratic term in YMC,

As an example, in

Let us consider the limiting case of free YM field without taking into account its interactions with the YMC, i.e. setting

and two constraints

since the considered case with

Thus, in the considering limiting case the constraints reduce the number of physical d.o.f. from nine down to six transverse ones. Note, such a reduction is not possible for non-zeroth interactions with the YMC, e.g. when

Due to energy conservation the growth of energy of the wave modes observed in the previous Section has to be followed by a certain redistribution of energy between YMC and wave modes. In order to take into account this effect consistently it is necessary to incorporate second-order contributions to the YMC equation for

Here, the averaging

The effective second-order Hamiltonian density incorporating the “back reaction” effect of the wave modes to the YMC can be represented as a sum of three components corresponding to the free YMC,

where dots stand for omitted contributions from other modes. It can be seen from these expressions that interaction term

In our numerical analysis and in all the plots in this paper we consider the complete system of all nine wave d.o.f. and YMC including interactions between them. We found that wave-condensate interactions lead to a decrease of amplitude of the YMC oscillations in time as is seen in

In addition, interactions between the YMC and wave modes (particles) lead to a redistribution of energy between the modes with different impulses which rather strongly depends on particles momentum due to para- metric resonance-like instability of YM solutions. The latter happens because the interaction strength, and hence the energy transfer intensity, depends on amplitude of impulses which is different for different modes and particles momenta (for a given mode). As an illustration of this effect, in

tum dependence (in dimensionless units) of the ratio of averaged absolute value of the wave amplitude at a fixed final

As the main physical result to be emphasized here, we have found the significant energy swap effect between the YMC and particle-like modes of the ultrarelativistic YM plasma due to their interactions in quasilinear approximation. Due to energy conservation it is clear that the parametric resonance for the

So far we have considered the

It has been demonstrated above that the interactions between the wave modes and the condensate lead to an increase of energy accumulated by the wave modes at expense of a corresponding decrease of YMC energy. This means that as some point in evolution of the system the amplitude of wave modes becomes too large so that the initial approximation

singular and unbounded. The latter anomaly is due to breakdown of the quasi-linear approximation. In what follows, we show that inclusion of the principal part of the higher-order terms into the linear Equations (16) and (17) for the wave modes allows to eliminate such anomalies and reveals qualitative stability of the energy swap effect under consideration.

The equations of motion for the wave

where

Now let us multiply

due to Equation (32). Here each fourth-order term can then be transformed as follows

Omitting

We notice here that the third-order terms have transformed to effective mass terms given by averages of various two operator products. In particular,

Certainly, this simplified analysis is not complete and is aimed only at illustrating that inclusion of the major part of higher-order terms significantly improves and stabilizes the results of the quasi-linear model. In a fully consistent model one has to take into account contributions from all the higher-order terms both in the wave equations of motion and in the YMC equation simultaneously, which will be done elsewhere.

Now we would like to extend our analysis to the

Here

Also, we perform the corresponding tensor decompositions for the YM field

Next, let us rewrite the supersymmetric part of the Lagrangian density (37) in terms of new Fourier modes

and the corresponding Hamiltonian density (24) has a form

The equations of motion for the extra supersymmetric d.o.f.

The equations for pseudoscalar modes

Taking into consideration only additional scalar and pseudoscalar fields in numerical analysis we notice that the qualitative picture of YMC dynamics shown in

As has been pointed out in the beginning of Section II, the YMC is a dynamical vacuum object which can be introduced for the

The generators of

The generators

In the Hamilton gauge, two different YMCs

where

The equations of motion are given by general formula from the classical YM theory (4). The equations for free (non-interacting) condensates

Note, these equations do not contain mixed terms like

The linear equations of motion for the Fourier transformed wave modes

One can show by a direct calculation that equations for the wave modes corresponding to each of the

Now let us investigate the quasi-linear “back reaction” effect of the wave modes to YMCs. Including next (second) order in waves, the equation for the

The corresponding equation for the second condensate

In general, time evolution of wave modes of the

Starting from the basic idea about an important dynamical role of the YMC (8), we have constructed a consistent quasi-classical approach based on Hamilton formulation and canonical quantization of the wave modes in the classical YMC. This approach has been applied in analysis of the system of YM wave modes (or particles after quantization) in the ultra-relativistic plasma interacting with the YMC (in the limit of small interactions between waves).

Namely, we have derived the YM equations of motion for the waves in condensate in linear approximation (16), (17) and (11) in the

It has been shown that dynamics of waves and condensate in the extended

As the main result of this paper, the energy redistribution effect from the YMC to the YM wave modes has been found and investigated from the first principles of quasi-classical YM theory in one- and two-condensate cases. This effect can be of major importance for cosmological processes in the early Universe, in particular, in the processes of particle production during the preheating period after cosmic inflation which is planned for further studies. In addition, an extension of the quasi-classical approach to a full quantum field theory formalism (including fermion modes) could become one of the next important steps in further theoretical understanding of dynamics of the wave modes interacting with the condensate.

This work was supported in part by the Crafoord Foundation (Grant No. 20120520). R. P. is grateful to the “Beyond the LHC” Program at Nordita (Stockholm) for support and hospitality during completion of this work.

The major difficulties of the canonical quantization of the free YM field (without the YMC) arise due to the presence of its time-retarded zero component in the Lagrangian (1). One of the ways to resolve this issue is based upon the method of expansion in configuration space elaborated in Refs. [

Let us construct the YM propagator in the form of chronologically ordered vacuum average of operator product. In a realistic case, consider a system of a YM field and a color-charged multiplet of fermions. The Lagrangian and Hamiltonian densities of such a system are

respectively. Then, the corresponding Lagrange equations of motion read

The canonical quantization procedure is based upon the (anti)commutation relations between the field operators

where generalized momenta conjugated to the fields

respectively. Here we kept color index

as well as in the interaction representation

After Fourier transformation, the equations for longitudinal

respectively. The equation for longitudinal mode does not have a wave solution, so it is impossible to take into account its contribution in the YM propagator constructed as a vacuum average of the chronologically ordered operator product. In order to resolve this problem we can modify the Hamiltonian density by means of adding an extra small “virtual” term depending on an infinitesimal parameter

After such modification the equation of motion for the longitudinal component becomes

such that it acquires an infinitesimal frequency. This modified equation enables us to incorporate the longitu- dinal mode into the YM propagator which is given by (in the limit

where

As an important test of the proposed method of infinitesimal parameter, the Formulas (55) and (56) turn out to coincide with the corresponding Green functions constructed for initial (non-modified) Equation (52).

Let us now perform canonical quantization of the YM wave modes in the classical YMC and therefore construct the quasi-classical YM theory. For this purpose, as the matter of the Bohr’s correspondence principle we introduce operators instead of field functions in the Hamiltonian density of, for example, the

Commutation relations for pseudoscalar modes

Finally, quantum Hamilton equations in commutators can be constructed in the standard way. For example, for the

Such equations written for all wave modes coincide with the corresponding equations of motion which were constructed previously (16), (17) and (40)-(43). The latter is an important validation of our calculations.