Inertial Mechanism:
Dynamical Mass as a Source of Particle Creation

A.V. Filatov, A.V. Prozorkevich,
S.A. Smolyansky and V.D. Toneev
Saratov State University, RU-410026, Saratov, Russia
Bogoliubov Laboratory for Theoretical Physics,
Joint Institute for Nuclear Research,
RU-141980, Dubna, Russia

A kinetic theory of vacuum particle creation under the action of an inertial mechanism is constructed within a nonpertrubative dynamical approach. At the semi-phenomenological level, the inertial mechanism corresponds to quantum field theory with a time-dependent mass. At the microscopic level, such a dependence may be caused by different reasons: The non-stationary Higgs mechanism, the influence of a mean field or condensate, the presence of the conformal multiplier in the scalar-tensor gravitation theory etc. In what follows, a kinetic theory in the collisionless approximation is developed for scalar, spinor and massive vector fields in the framework of the oscillator representation, which is an effective tool for transition to the quasiparticle description and for derivation of non-Markovian kinetic equations. Properties of these equations and relevant observables (particle number and energy densities, pressure) are studied. The developed theory is applied here to describe the vacuum matter creation in conformal cosmological models and discuss the problem of the observed number density of photons in the cosmic microwave background radiation. As other example, the self-consistent evolution of scalar fields with non-monotonic self-interaction potentials (the W - potential and Witten-Di Vecchia - Veneziano model) is considered. In particular, conditions for appearance of tachyonic modes and a problem of the relevant definition of a vacuum state are considered.

PACS: 13.87.Ce  05.20.Dd  11.15.Tk  98.70.Vc  11.10.Lm

1 Introduction

The present work is devoted to the construction of a kinetic theory of vacuum creation of particles with time-dependent masses. For brevity, this mechanism will be referred to as inertial one. Microscopic foundations of a mass change may be different. The Higgs mechanism leads to the most popular models of such a class, when the corresponding mean fields are time-dependent. General quantum field models with nonpolynomial interactions may also be considered, where the separation of nonstationary mean fields results in a time-dependent mass [1]. A well-known example of this kind is the Witten – Di Vecchia – Veneziano model [2, 3] in the framework of which the mean-field concept was analyzed in [4]. The Nambu-Jona-Lasinio [5] and [6] models are other examples, where the meson masses are defined by evolution of a quark condensate to be described at the hydrodynamic [7] or kinetic [8] level. The particle mass may depend on many-particle interactions in hot and dense non-stationary matter [9, 10, 11]. A general basis for a rather slowly-varying time dependence of the effective mass can be obtained within the Green function method [12, 13]. The field dependence of the mass is a general factor determining the time evolution in all these cases (F-class models). The conformal invariance of the scalar-tensor gravitational theory provides a time dependence of the particle mass by means of the conformal multiplier [14, 15, 16, 17]. The mass can be changed also due to the parametrization stipulated by additional space dimensions [18]. Such theories should be referred to as the other class (C-class). In the F-class theories, the vacuum particle creation admits a well-known interpretation based on the simplified vacuum tunnelling model in an external field [19, 20, 21]. A similar interpretation of the C-class models is difficult. At the phenomenological level, however, both classes have a uniform mathematical description as it will be shown in Sects. 2, 3 and 4, respectively, for the scalar, fermion and massive vector boson in quantum field theories (QFT’s).

The first consideration of the vacuum creation of particles with the variable mass was proposed apparently in [22] as a possible variant for describing a quantum system response to the time variation of system parameters [23]. Using the Bogoliubov transformation method, residual momentum distributions for fermions, and pair correlators were found for the cases of step-like and smooth variations of the fermion mass (Sect. 3.4).

In the present work, the kinetic theory will be based on the oscillator representation (OR) [24, 25], which is the most economical method for a nonpertrubative description (as compared to the Bogoliubov method of canonical transformation [26] or other accurate approaches to the problem [15, 27, 28]) of the vacuum particle creation under the action of time-dependent strong fields. This approach leads directly to the quasiparticle representation (QPR) with diagonal operator forms in the momentum space for the set of dynamical variables. It allows one to get easily the Heisenberg-type equations of motion for creation and annihilation operators. An important feature of the time-dependent Fock representation is the necessary consistence of commutation (anti-commutation) relations with the equations of motion. Otherwise, this circumstance can bring to the non-canonical quantization rules (an example will be considered in Sect. 4.1).

In terms of the OR it is possible to immediately derive the corresponding kinetic equations (KE) by the well known method [29]. Some particular results are published in [1, 30]. The kinetic theory for scalar, spinor, and massive vector fields is constructed in Sects. 2, 3 and 4, respectively. The main attention is paid to the particle creation in conformal cosmological models [31, 32, 33] (Sec. 5). It is shown that the choice of the equation of state (EoS) of the Universe allows one to obtain, in principle, the observed number density of matter participants and photons and, possibly, dark matter. The basic problem here is the description of vacuum particle creation which should be consistent with EoS but it is beyond the present article.

Finally, in the Sect. 6 other class of scalar QFT systems is considered with non-monotonic self-interaction potentials to apply the decomposition of the field amplitude into the quasiclassical space-homogeneous time-dependent background field and the fluctuation part. In this case, the particle mass is defined by intensity of a quasiclassical field. As an example, self-interaction potentials of the simplest polynomial type and that for a nontrivial case (the pseudoscalar sector of the Witten – Di Vecchia – Veneziano model) are analyzed. It is shown, that the relevant definition of vacuum states allows one to avoid of the tachyonic mode beginning. The main purpose of this review is to summarize all known relevant results on the inertial mechanism of the vacuum particle production and to call attention to unsolved problems which are shortly listed in Sect. 7.

We use the metric and natural units .

2 Scalar field

2.1 Oscillator and quasiparticle representations

Let us start our consideration with the simplest case of the real scalar field with the time-dependent mass , whose equation of motion is


The corresponding Lagrange function is given as


In the considered case, the system is space-homogeneous and nonstationary. Therefore, the transition to the Fock space can be realized on the basis functions , and creation and annihilation operators become time-dependent. The assumption about the space homogeneity allows one to look for solution of Eq. (1) in discrete momentum space in the following form:


where and at that the integers run from to . The thermodynamic limit can be covered in the resulting equations. Then the oscillator-type equation of motion follows from Eqs. (1) and decomposition (3) as




The symbols correspond to the positive and negative frequency solutions of Eq. (4) defined by its free asymptotics in the infinite past (future) [26],


where are defined by asymptotics of the mass


The asymptotics (6) corresponds to the in(out)-states and is necessary for definition of in(out)-vacuum. This requirement, however, can be broken in cosmology [34]. We suppose here that such asymptotics exists and the relevant vacuum states will be denoted by without indices ”in”  or ”out” , that is evident from the context. In the considered class of problems, the classification of states in the frequency sign turns out to be impossible for an arbitrary time moment. According to a general analysis [35], this leads to instability of a vacuum state during the action period of the external fields and to the vacuum particle creation. In this case, it is possible to consider quasiparticle excitations during the system evolution () and describe their creation and annihilation in the vacuum state . When the external field action is completed, residual particles of some finite density remain in the out-state. However, it is necessary to emphasize that these particles are defined in respect of the in-vacuum state [26].

In the general case, S-matrix does not exist in the considered formalism. Its role in description of the system with unstable vacuum is performed by other mathematical objects: The operator of the canonical Bogoliubov transformation [26], distribution functions in the kinetic approach [24, 29, 30, 36, 37, 38] or some set of correlation functions [39, 40] etc.

The conception of ”quasiparticle” plays the central role in the QFT with strong time-dependent quasiclassical external fields [26, 35]. Under the considered conditions, this approach is the shortest realization of the quasiparticle concept within the standard Fock representation of the QFT, where the external field can be taken into account nonperturbatively. Thus, the QPR corresponds to a possibility of writing down the set of commutative operators of physical (observable) quantities (the complete QPR) in the diagonal form in an arbitrary time. It is naturally related to the question whether operators have the quadratic form in the Fock representation. Hence, the interaction between the field constituents and self-interaction is not taken into account. It corresponds to a nondissipative approximation in the kinetic theory [41]. An alternative definition of the quasiparticle was given in [25] for constrained systems.

The transition to the QPR can be realized in different ways. The traditional method is based on the time-dependent canonical Bogoliubov transformation [26]. The alternative approach uses the oscillator (”holomorphic”) representation, which leads directly to the QRP [24]. In the considered case the transition to the OR is made by substituting into the dispersion law for the free field and postulating a following decompositions:


where is the generalized momentum, and are the creation and annihilation operators of particles with the momentum at the time moment . The in-vacuum state is defined as


The canonical commutation relation


together with the decomposition (2.1) provides the standard commutation relation for time-dependent creation and annihilation operators


The substitution of the decompositions (2.1) into the Hamiltonian


leads immediately to a diagonal form which corresponds to the QPR


In the considered case the vacuum energy of zero oscillations (”Zitterbewegung” ) depends on time.

Equations of motion for the operators can be obtained now from the minimal action principle [24] or from the Hamiltonian equations


Here and below we use the notation . Combining Eqs. (2.1) and (14), we get




is the factor defining the mixing of states with positive and negative energies. This equation obviously is consistent with the commutation relations (11).

Equations of motion (15) can be rewritten as the Heisenberg-type equation, e.g.,


In the instantaneous QPR these equations serve as a basis for a nonperturbative derivation of the KE describing scalar particle creation and annihilation processes within the inertial mechanism.

2.2 Kinetic equation

The key object of the kinetic theory is the quasiparticle distribution function which for the space-homogeneous case is


where is the initial () vacuum state. Differentiating the distribution function (18) with respect to time and using (15) we get


Here the auxiliary correlation function is introduced


This function provides a coherent connection between the states with positive and negative energies (so-called entangled states [28]).

The equation of motion for can be obtained by analogy with equation (19). We present it here in the integral form as


where the initial condition was used. This condition corresponds to the initial condition for the distribution function and is a direct consequence of the definition (9). Eventually, the dynamical phase in Eq.(21) is equal to


The substitution of Eqs.(21) in Eq.(19) leads us to the resulting KE written in the thermodynamical limit at the fixed particle density


The source term in the r.h.s. of Eq. (23) describes a variation of the particle number with the given momentum due to vacuum creation and annihilation processes for the inertial mechanism, is defined by Eq.(16). The non-Markovian KE (23) has the structure as that for the Schwinger mechanism of pair creation in an electric field [29]. This equation was investigated in detail for description of the pre-equilibrium evolution of quark-gluon plasma created in collisions of ultrarelativistic heavy ions [36, 37]. The case of the scalar QED was considered in [24] for an electric field of arbitrary polarization.

As it follows from Eq.(16), in the framework of the inertial mechanism the particle production rate is defined by the rate of the mass change


where is the characteristic mass to fix the time scale (e.g., ).

Let us note that the KE (23) is valid under two basic assumptions: a) there are no particles (or antiparticles) in the in-state; b) a collisionless approximation is applicable (i.e. the corresponding dissipative processes are not taken into consideration).

In the low-density approximation the KE (23) results in the following solution [36]:


The KE (23) can be transformed to linear equations of the non-Hamiltonian dynamical system with zero initial conditions


which is convenient for numerical analysis. This equation system has the first integral


according to which the phase trajectories are located on the two-cavity hyperboloid with top coordinates (physical branch) and (nonphysical one). If the function is excluded from Eqs. (26), we obtain the non-linear two-dimensional dynamical system with


The functions and have a certain physical meaning (the last function describes vacuum polarization effects, see Sect. 2.3) below and are invariants with respect to the time inversion while the auxiliary function and factor (16) change their signs. Thus, the KE (23) is invariant at the time inversion.

The presented formalism of vacuum particle creation is specific for kinetic theory and allows one natural generalization to the case of interacting fields that leads to introduction of corresponding collision integral of a non-Markovian type [42]. This approach is close to the modern method expounded in the book [26] where the time-dependent Bogoliubov transformation is used. The same method was used in pioneer works [43, 44, 45] (see also [34]). Some modification of the formalism [26] (the representation) was developed in [25] and then used widely (e.q., in [32] and references cited there). The correspondence between the representation and our approach (as well as that used in the book [26]) may be easily established.

2.3 Observable and regularization

The KE (23) describes the vacuum quasiparticle excitations rising at an external force (, in the considered case). When this action is switched off, there is still some remaining density of real (residual) particles and antiparticles. In the absence of any interaction between the system constituents, the real particles are ”on-shell” ones and have the free-particle dispersion law with the mass (7), while quasiparticles are ”off-shell” with the dispersion law (5). Within the Green function method [13], one can say that the time-dependent dispersion law like (5) corresponds to the t-parametric mass shell surface of slowly time-dependent , i.e. , where and correspond to slow and fast time scales. This case is not of interest for the considered problem. Thus, the dispersion law (5) does not belong to the mass shell surface. In the general case, the on-shell condition


() is not connected directly with the condition of efficiency of the vacuum particle creation, , where is defined by Eq.(24). On the contrary, the presence of high frequencies in the function is necessary for vacuum creation and does not contradict the on-shell condition (29). In principle, the KEs of such a type are designed for the description of evolution of both real particles and quasiparticles. In particular, the distribution function of residual particles is . This simple formula for follows from Eq. (25) in the low-density approximation. However, the presence of the fast oscillated multiplier in the source term in the r.h.s. of the KE (23) leads to a large amount of numerical calculations which make impossible the study of the system evolution for rather large times after the switching on external forces. The corresponding large scaling methods of calculations based on the KE (23) have not been worked out at present. Some properties of a residual particle-antiparticle plasma due to a limited pulse of the external field action can be estmated by the imaginary time method [46].

The distribution function is the key quantity of the system. The density of observable variables is some integral in the momentum space containing the distribution function and auxiliary functions , which describe the effects of vacuum polarization. The simplest variable of such a type is the density of quasiparticles. In the thermodynamic limit we have


where . To proceed to the thermodynamical limit, the rule


is used here and below.

Other important characteristics of the system are the energy density and pressure , which can be obtained as the average value of the energy-momentum tensor corresponding to the Lagrangian density (2),


As the result we have [26]


The last two terms in integrand (34) represent the contribution of vacuum polarization.

Finally, the entropy density can be introduced


It is not conserved () even in the considered non-dissipative approximation because the system is open (the mass change is defined by external causes).

A direct proof of the convergence of integrals (30), (33), (34), (35) is complicated because of the absence of an explicit form for functions and . Therefore, one usually uses the method of asymptotic expansions in power series of the inverse momentum (N-wave regularization technique) [47] (another approach rests on the WKB approximation [48]). Our present consideration is based on the explicit asymptotic solutions of the system (26) for . The integral (30) is assumed to be convergent at any time moment. Then the function should decrease at and hence in this region. This inequality corresponds to the low-density approximation (25), where the KE solution can be written in the explicit form as


because in accordance with Eq. (16). These solutions are consistent with the integral of motion (27). Thus, indeed asymptotic solutions are some quickly oscillating functions (this fact was first noted in [30] for the case of massive vector bosons, see Sect. 4.3). Such behavior matches with the quasiparticle interpretation of vacuum excitations by the inertial mechanism. The real (observed) particles are the result of the evolution by the moment when and the out-vacuum state is realized. The asymptotics (36) may be influenced by other (non-inertial) mechanisms of vacuum particle creation (e.g., in the case of harmonic ”laser” electric field [49, 50, 51]).

The asymptotics of the integral (36) can be obtained by the stationary phase method [52]


if . Using Eqs. (36) and (26) we get the leading contributions


where the upper indices show the inverse momentum degree for the corresponding leading terms (we are not interested in the asymptote of the function , which plays some auxiliary role only). Relations (2.3) are identical to the results of application of the N-wave regularization method to Eqs.(26) [47].

Now one can conclude that the integral (33) is convergent but the last integral term in Eq. (34) needs a regularization. The regularizing procedure of the Pauli-Villars type is based on the subtraction of appropriate counterterms in integrals (30), (33), (34), (35). These counterterms can be obtained by the substitution into the denominator of asymptotics (2.3),


If the regularizing mass can be chosen rather large, ( is ”the computer cut-off parameter”), the influence of counterterms on the results of numerical calculations is negligible.

The numerical investigation of the KE (23) and observable densities (30), (33)-(35) will be presented in Sect. 3.4.

3 Fermion field

3.1 Quasiparticle representation

The material of this subsection is based on papers [53, 54, 55].

Equations of motion for fermion fields with the variable mass are


where . The corresponding Hamiltonian is


By analogy to the scalar case, we use the following decompositions of field functions in the discrete momentum space:


The OR is intended to derive equations of motion for creation and annihilation operators. It is based on the primary equations (3.1) and free - spinors with the substitution . Thus, the following equations for the spinors are postulated in the OR :


with . These definitions create the set of standard orthogonality conditions [56] depending on time now parameterically


Decompositions (3.1) and relations (44) lead immediately to the diagonal form of the Hamiltonian (41)


with interpretation of (and ) as the creation and annihilation operators of quasiparticles obeying the standard anti-commutation relations


We are not interested in the subsequent diagonalization of the spin operator and such QPR can be named the incomplete representation.

Now in order to get equations of motion for creation and annihilation operators in the OR, let us substitute the decomposition (3.1) in Eqs. (3.1) and use relations (44). Then, as an intermediate result, we obtain the following closed set of equations of motion which is valid in a general case


The spinor construction is introduced here as


The matrices and describe the transitions between states with positive and negative energies and different spins, while the antiunitary matrices and show the spin rotations only


Equations (3.1) are compatible with the canonical commutation relations (46).

Let us write now the - spinors in an explicit form, according to [57]:


where and . Spin rotation matrices (3.1) in this representation are equal to zero


For the remaining matrices (3.1) we have , where is the hermitian matrix


Thus, the system of equations of motion (3.1) reduces to the following one:


3.2 Kinetic equation

Equations of motion (53) do not contain the spin rotation matrices (51) and they are similar to Eqs. (15); therefore, the KE derivation meets no problem now. To be specific, let us introduce the one-particle correlation functions


The differentiation of (3.2) with respect to time leads to the following matrix equations


where the auxiliary function was introduced


Together with Eqs. (3.2), the corresponding equation of motion


forms a closed set of equations for correlation functions. With the help of Eqs. (57), one can exclude the auxiliary correlator from the system (55)


using zero initial conditions. The subsequent transformation is based on the relation


for an arbitrary second-rank matrix , which follows from Eq. (52). The function


plays a role of some analog of Eq. (16).

Using isotropy of the considered system, we will limited ourselves to the spin-averaged scalar distributions


Calculating the trace of (58) we get


In the case of the vacuum initial state, , we have


The KE’s (23) and (63) are similar but differ by statistical factors (the Bose enhancement or the Fermi suppression) and by structure of the factors (16) and (60). The corresponding linear equations for the non-Hamiltonian dynamical system become


This system possesses one first integral of motion (see [38])


This relation represents an ellipsoid in the phase space of variables. After exclusion of the function from Eq. (64), we obtain the system of non-linear equations


It can easily be proved that the KE (63) is invariant with respect to time inversion. Equations analogous to Eqs. (26) and (64) were obtained in [53] (see also [26]) for the conformal flat space-time. The KE (63) to the case of spinor QED was generalized in [54, 55]. In the general case, the spin correlation functions (3.2) and (56) can be decomposed in respect of the Pauli matrices (e.g., [58]).

3.3 Observables and regularization

The total particle number density and energy density in the considered case are distinguished from the corresponding expressions (30) and (33) for the scalar system by the spin degeneration factor (for an equal number of particles and antiparticles)


where is the zero component of the energy-momentum tensor


By definition, the entropy density of the fermion system is equal to


Finally, the pressure is


Using (69) and (3.1), this relation can be reduced to the following form:


Here the correlation function is calculated by means of relations (44)


where the last term takes into account the contribution of the vacuum polarization,


This result was obtained under additional conditions for ”observable” correlation functions


which is a consequence of space-homogeneity and isotropy (absence of the spin moment) of the system. The auxiliary correlation function (56) is not connected with the spin moment and, therefore, it remains off-diagonal with respect to spin indices.

The same regularization procedure can be realized here for the calculation of divergence integrals, as presented in Sect. 2.3 for the scalar bosons. Equation (25) for the distribution function in the low-density approximation is valid also after the replacement . Asymptotics of the factor (60) is equal to . Using Eq. (25) and the rules of Sect. 2.3, one can derive the following expressions for counterterms for the case of fermion fields:


However, these counterterms can be ignored in computer calculations.

It is known [26] that the phase density of pairs created in an electric field for the whole period of its action is related to long-time asymptotics of solutions of some oscillator equations. For the inertial mechanism, analogous derivation results in the following relation [22]


where , the indices correspond to the initial and final states. This relation is convenient for calculation of observables for long pulses , e.g. Fig. 4 where the direct solution of kinetic equation is a very robust numerical problem.

3.4 Numerical results

Here numerical investigations of the KE’s are presented for bosons (23), fermions (63) and appropriate densities of observable variables are estimated (Sects. 2.3 and 3.3). As an example, two variants of time-dependent masses are considered. The first case qualitatively corresponds to a typical meson mass change under the phase transition within the NJL model [59]


with the parameters (initial mass), (final mass), and (transition time). Another variant suggested in [22, 23] allows an analytical solution of the Dirac equation

Figure 2: The pair density evolution of fermions for various initial masses
(p - proton, e - electron).
Figure 1: Time dependence of the pair density for bosons and fermions at .
Figure 2: The pair density evolution of fermions for various initial masses
(p - proton, e - electron).
Figure 3: The residual pair density versus the mass ratio.
Figure 4: The residual energy density versus the transition time , .
Figure 1: Time dependence of the pair density for bosons and fermions at .

Numerical results for solution of the KEs are presented in Figs. 4-8 for defined by Eq. (78) with and in Figs. 10,10 for the mass (79) with . Masses are specified in natural units, MeV.

At a glimpse the time dependence of quasiparticle density for bosons and fermions repeats qualitatively the curve ; however, when the densities go asymptotically to certain finite values (residual density, Fig. 4, ), which characterize the real (free) particles (at the active stage of the process, , one may talk about quasiparticles only). The dependence for fermions on the initial mass value is shown in Fig. 4 in the range from the electron mass to proton one. This dependence is nonmonotonic: With increasing the residual density reaches the maximum at MeV and then begins to decrease. This effect appreciably depends on the variant used for the mass change: For the case (79) it manifests itself much more clearly than for the model (78). The residual particle density dependence on the mass change is presented in Fig. 4.

Figure 6: The distribution function of bosons in time-momentum variables, ,  fm/c.
Figure 5: Entropy density evolution of bosons for different values of the relaxation time , .
Figure 6: The distribution function of bosons in time-momentum variables, ,  fm/c.
Figure 7: Distribution functions at the time  fm/c, .
Figure 8: Asymptotic form of distribution functions at .
Figure 5: Entropy density evolution of bosons for different values of the relaxation time , .
Figure 9: Pressure evolution corresponding to mass changing (79), ,  fm/c.
Figure 10: Asymptotic oscillations of pressure at without ‘switching off’ of vacuum polarization effects.

Qualitative behavior of the energy density is similar to that of particle density, showing a smooth asymptotic decrease (see Fig. 4). In Fig. 8 the time dependence of the boson entropy density is presented for different values of the relaxation time. Non-monotonic behavior of entropy is caused by the fact that the system is opened and treated in the non-dissipative approximation.

Momentum spectra of particles at different stages of the interaction process are shown in Figs. 8-8. The maximal number of bosons is created with zero momentum, whereas there are no fermions with . This feature differs qualitatively from the case of the Schwinger mechanism of particle creation [38].

It is important that the formation of appreciably non-monotonic distributions with the ”fast” mass changing (78), Fig. 8, assists in the development of plasma oscillations. For smoother mass changing (79) this effect becomes much less pronounced.

The most interesting features are observed in the pressure behavior, Fig. 10-10. For both variants of the mass evolution and independently of particle statistics, the pressure is negative at the beginning of process, then it changes its sign in the reflection point of and gradually decreases. However, contrary to other observables, the pressure has no constant asymptotics and at looks like almost un-damped oscillations, Fig. 10. It is distinctive for pressure of the bosonic quasiparticle system which strongly oscillates around zero, Fig. 10. The reason is that unlike the other considered quantities, the pressure is not completely determined by the quasiparticles distribution function , but it depends also on the function , which describes vacuum polarization effects. At the operator language, this means incomplete diagonalization of the energy-momentum tensor in the Fock space: Averaged over the initial vacuum, its spatial components include the contribution of anomalous correlators like .

Thus, if the process of particle creation stops when the time mass evolution is completed (), the vacuum polarization effects are not ”switched off” simultaneously but continue to influence some observables, e.g., pressure. As a consequence, in such non-dissipative nonequilibrium model it is impossible to determine unambiguously the equation of state [60].

4 Massive vector bosons

4.1 The complete QPR

The simplest version of quantum field theory of neutral massive vector bosons is given by the Lagrangian density [56]


which corresponds to the equation of motion


with the additional ”external” constraint


An alternative way is to proceed from the Wentzel Lagrangian [61]


where the strength tensor . Here the constraint (82) is a consequence of dynamical equations. Lagrangians (80) and (83) result in different energy-momentum tensors (e.g., see [61, 62]).

The transition to the QPR is carried out by the standard decomposition of free fields and momenta in the discrete momentum space with the replacement in the dispersion law (see Sect. 2.1),