Lightlike shell solitons of extremal space-time film

New exact solution class of Born -- Infeld type nonlinear scalar field model is obtained. The variational principle of this model has a specific form which is characteristic for extremal four-dimensional hypersurface or hyper-film in five-dimensional space-time. Obtained solutions are singular solitons propagating with speed of light and having energy, momentum, and angular momentum which can be calculated for explicit conditions. Such solitons will be called the lightlike ones. The soliton singularity has a form of moving two-dimensional surface or shell. The lightlike soliton can have a set of tubelike singular shells with the appropriate cavities. A twisted lightlike soliton is considered. It is notable that its energy is proportional to its angular momentum in high-frequency approximation. A case with one tubelike cavity is considered. In this case the soliton shell is diffeomorphic to a cylindrical surface with threads by multifilar helix. The shell transverse size of the appropriate finite energy soliton can be converging to zero at infinity. The ideal gas of such lightlike solitons with minimal twist parameter is considered in a finite volume. Explicit conditions provide that the angular momentum of each soliton in the volume equals Planck constant. The equilibrium energy spectral density for the solitons is obtained. It has the form of Planck distribution in some approximation. A beam of the twisted lightlike solitons is considered. The representation of arbitrary polarization for the beam with the twisted lightlike solitons is discussed. It is shown that the effect of mechanical angular momentum transfer to absorbent by the circularly polarized beam can be provided. This effect is well known for photon beam. Thus the soliton solution which have determinate likeness with photon is obtained in particular.


Introduction
A nonlinear space-time scalar field model considered here is known for a long time sufficiently. This model is related to well known Born -Infeld nonlinear electrodynamics [1], and it is sometimes called Born -Infeld type scalar field model.
This model is attractive because it has relatively simple and geometrically clear form. It can be considered as a relativistic generalization of the minimal surface or minimal thin film model in three-dimensional space.
In this generalization we have an extremal four-dimensional film in fivedimensional space-time. But the model equation appears as differential one for scalar field in four-dimensional space-time.
On the other hand, this model can provide the necessary effects which are required for a realistic filed model.
In particular, the model under consideration has a static spherically symmetric solution, which is identical to zero four-vector component of electromagnetic potential for dyon solution of Born -Infeld electrodynamics [2]. This static solution of the scalar model gives the appropriate moving soliton solution with the aid of Lorentz transform.
As it was shown in the cited work [2], in the case of nonlinear electrodynamics there are the conformity between long-range interaction of solitons and two known long-range interactions of physical particles, that is electromagnetic and gravitational ones. But the methods which was used for the investigation of soliton long-range interaction are independent of the field model. The appropriate instruments are integral conservation laws and characteristic equation.
These methods applying to the scalar model under consideration give the results, which are similar to ones for nonlinear electrodynamics. These results in detail must be matter for another article. Here we briefly discuss the obtaining of Lorentz force for interacting scalar solitons in the next section.
An essential difference of the scalar field model from the nonlinear electrodynamics is obviously caused by the different tensor character of the fields. In particular, a weak single-component scalar wave can not provide the transverse polarization of electromagnetic wave.
But at the present time we consider the light as photon beam but not a weak electromagnetic wave with constant amplitude. The photon beam could be represented by an appropriate scalar soliton beam. In this case an essential space-time nonhomogeneous of soliton solution may provide the necessary symmetry properties for the beam.
Thus at the present work we consider the model of extremal space-time film. We obtain its singular exact soliton solutions propagating with the speed of light and having energy, momentum, and angular momentum which can be calculated for explicit conditions. Such solitons will be called the lightlike ones.
Then we investigate in detail a lightlike soliton solution having a rotation about the direction of propagation that is twisted lightlike soliton.
We consider the ideal gas of such twisted lightlike solitons. Using explicit assumptions we obtain Planck distribution formula in some approximation.
At last we consider a beam with the twisted lightlike solitons. We show that this beam can represent photon one. In this case we have, in particular, the polarization property and the effect of mechanical angular momentum transfer to absorbent by circularly polarized beam.

Extremal space-time film
Let us consider the following action which has the world volume form: where M det(M µν ), (dx) where m det(m µν ). Taking into account (2.2) we can write model action (2.1) in the form where dV |m| (dx) 4 is four-dimensional volume element, Variational principle with action (2.3) gives the following model equation: We have the following evident relations from (2.4c): Inversion for relations (2.4b) gives Let us write also the following useful relation, which is obtained from (2.3b) and (2.6): For the case when the field invariant Φ ρ Φ ρ relatively small (χ 2 |Φ ρ Φ ρ | 1) we can represent the action density L with two first terms in formal power series of χ: The appropriate linearized equation has the form Also let us write the linearized relation (2.4b): (2.10) Nonlinear differential equation of second order (2.4) for function Φ can be represented in the form of the first order differential equation system for fourvectors Φ µ or Υ µ . In this case we have differential field equations (2.4a) and (2.5). In addition we must consider algebraical relations (2.4b) or (2.6).
As can be seen, the model action (2.3) is susceptible to the choice of metric signature. Here we will consider both the signatures {+, −, −, −} and {−, +, +, +}. Thus we use the following designations for Minkowski metric: where δ ij is Kronecker symbol. The Latin indices take values {1, 2, 3}. The signature of metric (2.11a) allows the same spherically symmetric solution of the model that was obtained by M. Born and L. Infeld in their classical work for nonlinear electrodynamics: whereq is constant, r is radial spherical coordinate,r |q χ|. It is evident that solution (2.12) give birth to the class of soliton solutions with Lorentz transformations. Such solutions in this model also can be considered as point charged particles because their long-range interactions have electromagnetic character.
Indeed to investigate the interactions we can use the method based on integral conservation law of momentum (for Born -Infeld nonlinear electrodynamics see [2]). Let us consider the long-range interaction of an appropriate to (2.12) moving soliton-particle with the rest one (2.12). In this case the method gives pure electrical interaction between the particles. Then we can transform the obtained law of particle movement with electrical force to another moving reference frame. In this case Lorentz transform of the force gives its magnetic component. In this connection it should be noted that the Lorentz transform of the force was presented by A. Einstein in last section of his classical work on special relativity [3].
It should be mentioned that using the another metric signature (2.11b) for action (2.3) leads to the following spherically symmetric solution instead of (2.12a): Here we must consider the area r r. In this case we have infinity values for Φ r and L on the sphere r =r. But, as can be shown, the field function Φ is finite on this sphere.  (2.11). This representation can be preferable for some problems.
The appropriate solution of equations (2.4b) gives the following relations: Here and below in this section top and bottom signs are appropriate to metrics (2.11a) and (2.11b) accordingly. Comparison relations (2.13) and (2.4b) gives the following expressions for the action density L in Cartesian coordinates: (2.14) Now let us write the field equation (2.4) in Cartesian coordinates with metric (2.11). After differentiation Υ µ (2.4b) in (2.4a) and multiplication the equation As we see, obtained equation does not include radicals. It is evident that the model under consideration keep invariance for spacetime rotation and scale transformation. Thus any solution give birth to the appropriate class of solutions with the following transform: where L µ .ν are components of space-time rotation matrix, a is scale parameter.

Energy-momentum and angular momentum
Customary method gives the following canonical energy-momentum density tensor of the model in Cartesian coordinates As we see, the canonical tensor is symmetrical.
To use finite integral characteristics of solutions in infinite space-time we introduce regularized energy-momentum density tensor with the following formula: where ∞ → T µν is regularizing symmetrical energy-momentum density tensor which can be defined depending on class of solutions under consideration. Here we will use constant regularizing tensor We have conservation law for regularized energy-momentum density tensor in Cartesian coordinates Let us define angular momentum density tensor by customary way. We have the following appropriate conservation law: We introduce the following special designations for energy, momentum vector, and angular momentum vector densities: E, P, J . Let us write the appropriate expressions taking into account relations (2.3b), (2.4b), and (2.14): where ijk is Levi-Civita symbol ( 123 = 1). Let us define energy, momentum, and angular momentum of field in a threedimensional volume V : (3.8)

General lightlike soliton
Let us consider solutions in a form of wave propagating along x 3 axis of Cartesian coordinate system with the speed of light. Let this solution be have some transverse and longitudinal field distributions. Thus we can write Substitution (4.1) to field equation (2.15) gives the following equation: where top and bottom signs are appropriate to Minkowski metrics (2.11a) and (2.11b) accordingly.
As we see the obtained equation (4.2) does not include derivatives on phase of wave θ (4.1b).
Equation (4.2) is elliptical with the following condition: The similar in form (4.2) equations were considered. About this topic see the paper by R. Ferraro [4] and references therein.
In particular, the similar in form but different in type equation was considered by B.M. Barbashov and N.A. Chernikov [5]. A Lax representation (see, for example, [6]) for this equation was presented in article by J.C. Brunelli and A. Das [7]. 1 The monograph by G.B. Whitham [6] contains relatively simple way for obtaining the Barbashov -Chernikov solution with the help of hodograph transformation (see, for example, [8]).
Here we use in outline the Whitham method but for the elliptic (for condition (4.3)) equation (4.2). The qualitative difference between hyperbolic and elliptic equations causes the appropriate difference in the solution way.
Let us introduce new independent variables where ı 2 = −1. Also we will use cylindrical coordinates {ρ, ϕ, x 3 }. We have the following evident relations: Using new variables (4.4) we obtain from (4.2) the following equation: The possible application of Lax representation to the model under discussion can be considered in future investigation. Equation (4.6) is hyperbolic with the following condition: As noted in section 2 the field model under consideration is invariant by space-time rotation and scale transformation. But equation (4.2) does not contain derivatives with respect to coordinates {x 0 , x 3 }. Because this here we have space-time rotation and scale invariance in the planes {x 1 , x 2 } and {x 0 , x 3 } with mutually independent parameters. Thus equation (4.2) is invariant with respect to rotation about x 3 axis and scale transformation in {x 1 , x 2 } plane.
As applied to equation (4.6), taking into account relations (4.5) and (2.16), these two types of invariance are provided by the following general substitution: where ρ and ϕ are real constants with respect to coordinates {x 1 , x 2 }. But in general case these constants can be depend on phase of soliton θ (4.1b): Because this we will call o(θ) the scale-rotation function of the soliton. It is evident that the function ρ(θ) defines the phase dependence of transversal scale and the function ϕ(θ) defines the phase dependence of rotation about x 3 axis. Thus if we have a solution Φ(ξ, * ξ) to equation (4.6) then by means of invariant substitution (4.8) we obtain wave propagating along x 3 axis and preserving its transversal form. Longitudinal form of the wave defined by scale-rotation phase function o(θ) is also preserved.
As result we have in (4.8) a wave packet propagating with speed of light and preserving its shape. It will be called the lightlike soliton.
Hereafter up to formulas (4.23) we consider the equation (4.6) for the case of top signs that is the metric (2.11a). But the obtained solution will be represented in general form which is appropriate to both metrics (2.11).
Thus equation (4.6) with top signs is equivalent to the following first order system: By interchanging the roles of the dependent and independent variables in (4.9) we obtain the linear system which is equivalent to single equation Let us introduce new independent variables Inversion for relations (4.13) gives .
It can be checked also that equation (4.6) is hyperbolic with the solution (4.23) for condition (4.24a).
One could say that relations (4.23b) define transformation of independent variables {ξ, * ξ} to {ξ 1 ,ξ 2 } for equation (4.6). But the definition of transformation with differential relations is not complete. Direct connection between the variables can be obtained by path integration of relations (4.23b) in nonsingular area, that is for condition (4.24a). At the same time we must define an initial correspondence between the variables {ξ, * ξ} and {ξ 1 ,ξ 2 }.
We can consider the simplest case by taking χ = 0 in (4.23b). In this case we can put ξ 1 and expression (4.23a) is evident solution of appropriate to (4.6) linear equation when χ = 0. In general case let us consider relation (4.25) as asymptotic for ρ → ∞. Then we can designateξ It is useful to introduce polar coordinates {ρ,φ} for variables {ξ, * ξ } by analogy with (4.5) for variables {ξ, * ξ}: General solution in the form of lightlike soliton depending on phase θ (4.1b) can be obtained from (4.23) by invariant substitution (4.8).
It is notable that the action density for obtained solution does not contain radical. Substitution solution (4.8) with (4.23a) into (2.3b) with (4.4) and (4.24b) gives expression As we see, explicit dependence on phase θ, which we have in (4.8), here is absent. As a consequence of probes of various arbitrary functions Ξ 1 (ξ 1 ) and Ξ 2 (ξ 2 ) we have obtained that the solitons under consideration have singular tubelike shells, where the condition (4.24a) is broken. Note that one soliton can have a set of such singular shells. More detail consideration is presented in the next section for simplest arbitrary functions Ξ 1 and Ξ 2 .
Thus the obtained solution can be called the shell lightlike soliton. Now let us obtain the energy, momentum, and angular momentum densities for a lightlike soliton. For this purpose we substitute solution (4.23a) with scale-rotation transformation (4.8) to formulas (3.7).
Using relations (4.4) and (4.24b), we obtain the expressions for energy, momentum, and angular momentum densities with some common functions, which will be designate as f E i . Then we have where k 2 3 = ω 2 according to (4.1b). Here we write explicitly only two functions f E 0 and f E 1 : (4.30) These functions play main role in the area, where the scale function ρ(θ) is almost constant: ρ → 0. We have from (4.29) the following notable relation for the case ρ → 0: The arbitrary functions ρ(θ) and ϕ(θ) (4.8c) define scale and rotation in the plane {x 1 , x 2 } accordingly. Using (5.2), (4.8), and (4.1b), we can show that the case ϕ > 0 corresponds to positive rotation by angle ϕ in time x 0 and in x 3 axis for k 3 > 0.
Thus for right-handed coordinate system {x 1 , x 2 , x 3 }, the cases ϕ > 0 and ϕ < 0 correspond to right and left local twist of the soliton accordingly.
It is interesting to consider the solitons with constant twist: Such solitons can be called the uniformly twisted ones. For conciseness we will call them the twisted solitons.
As we see in (4.31), for the case (4.32) the soliton energy density E is proportional to its angular momentum density J in high-frequency approximation, The appropriate proportionality relation between soliton energy and its angular momentum is notable property of the twisted lightlike soliton.
To obtain integral characteristics of the soliton it is necessary to integrate the functions {f E 0 , ..., f E 6 } in the plane {x 1 , x 2 }. Considering (4.23b) and (4.27), we can see that the appropriate integrands have notable simple form in the variables {ρ,φ} (4.27).

Twisted lightlike soliton
For further calculations, we define the arbitrary functions Ξ 1 and Ξ 2 . Let us take power function with integer negative exponent. Introducing necessary multiplicative constants for concordance of physical dimension and for simplification of resulting formulas, we have where m is natural number, real constant − ρ has a physical dimension of length.
Then formula (4.23a) representing the solution of equation (4.6) has the form Because ξ ∼ξ and * ξ ∼ * ξ at ρ → ∞, we have from (5.2) with the scalerotation transformation (4.8) the following asymptotic solution: In view of dependence on phase ρ(θ) and ϕ(θ), the formula (5.3) describes the propagating wave along the x 3 axis. The dependence ϕ(θ) in (5.3) describes also the twist of this wave about the propagation direction.
Let us consider the twisted lightlike soliton with condition (4.32). We put for this case ϕ = ± θ m , (5.4) where the signs '+' and '−' correspond to right and left twisted soliton accordingly.
In addition, let us consider that the scale function ρ(θ) is almost constant: ρ ∼ 0. As we can see in (5.3) with (5.4), in this case ω in (4.1b) is radian frequency of the soliton wave and 2π/ |k 3 | is the appropriate wave length.
Thus Thus the solution under consideration in this section has identical form for metrics with both signatures (2.11a) and (2.11b). Then to obtain the field configuration of the solution we can consider only one from the two cases. Let us consider the case of top signs, which is appropriate to the metric (2.11a).
Expression (5.8) represents epicycloid with 2 m cusps. For m = 1 this line is shown on Fig. 5.1 and for m = 2 it is shown on Fig. 5.2. These figures was obtained also by R. Ferraro [9,10] for mathematically similar but another problem.
In the present investigation, the system (5.5) with − ρ = 1 for given values of parameter m and variables {ξ, * ξ} is solved numerically with respect to variables {ξ, * ξ } in all characteristic areas of the plane {x 1 , x 2 }.
In the area of the plane {x 1 , x 2 } outside of the singular line (5.8) we have one-to-one mapping (5.6) with the condition (4.25) at infinity ρ → ∞.
This mapping can keep continuity for transition through the singular line, if we resign the condition of mutual complex conjugation for variablesξ 1 and ξ 2 (4.26). But in this case the field function Φ becomes complex-valued nearly everywhere in the inner area of the singular line (5.8), excepting some radial But such transition through the singular line is forbidden in the framework of the obtained solution. Thus we can find a solution in the inner area of the singular line in the plane {x 1 , x 2 } independently of the solution in the outside area. Then we could try to satisfy any conditions for the field function and its derivatives on the singular line. But here such conditions could appear to be forcible.
A radical solution of the inner area problem is exclusion of this area form the space. In this case we have a soliton with the appropriate cavity. Here we follow this way.
Taking into account also the scale-rotation transformation (4.8), we have the following condition for the space of the solution: where dependence ξ m = ξ m ϕ(θ) corresponds to rotation of the singular contour in the plane {x 1 , x 2 } by the angle ϕ(θ) (4.8c). Thus according to (5.10) we have the soliton with an inner shell. The results of numerical calculations for the function Φ (5.2) on the singular line are shown on As we see on Figs. 5.3 and 5.4, the field function Φ on the singular line is The cusps are the derivative discontinuities for the field function along the singular line in the plane {x 1 , x 2 }. As we can see on Figs. 5.5 and 5.6, these derivative discontinuities are absent outside of the singular line. Now let us obtain the expressions of full energy, momentum, and angular momentum for the solution under consideration in bounded three-dimensional volume. For convenience we consider the tubular volume in coordinates {ρ,φ, x 3 }. Its internal radius is defined in (5.10). Let its external radius and length be designated asρ ∞ and l s accordingly. Thus in addition to condition (5.10) we haveρ ρ ∞ , − l s 2 x 3 l s 2 . Note that in this step we may obtain different results for two metric signatures (2.11a) and (2.11b), because of the appropriate difference in expressions (4.33).
Making the integration in the plane {x 1 , x 2 } we can get the rotation parameter be zero: ϕ = 0. Let us substitute (5.1) and (5.5) with (4.27) to right-hand parts of (4.33). We change the integration by variable x 3 to one by phase θ (4.1b). As result we have the following expressions for energy and absolute values of momentum and angular momentum: , where E is the part of soliton energy obtained from the part f E 0 of energy density E (4.29a), for m 2 , (5.14c) Here in (5.14) we have the different expressions for two metric signatures (2.11a) and (2.11b) (top and bottom signs accordingly).
The value of x 3 momentum projection is defined by the sign of wave vector projection k 3 (4.1b): P 3 = ±P.
In general case the x 3 angular momentum projection is defined by integral I 3 (5.13), which can be called the integral twist of the soliton with weight ρ 4 : Let us write the appropriate to (5.12) expressions for the twisted soliton x 2 x 1 Figure 5.5: The field function Φ on the plane {x 1 , x 2 } for m = 1. with constant ϕ (5.4). Using condition (5.4) and formulas (5.13), we have Using (5.15) and (5.14), we obtain from (5.12) the following expressions for the twisted soliton: , (5.16a) For the twisted soliton let us consider the case for slowly varying scale function ρ(θ), such that I 2 → 0 (5.13). Also we suppose that the frequency ω is sufficiently high, such that − ρ ω → ∞. According to expressions (5.16), in this case we have the following relations: is the angular velocity of the twisted soliton. Let us consider the twisted soliton with scale function in the form of Gaussian curve: where θ is characteristic length of the soliton measured in radians and numerically equals to a total angle of twist on the characteristic length of the soliton along x 3 axis. A twist angle 2π/m corresponds to soliton wave-length along x 3 axis. Let us consider the case of infinite space with the conditions Then the calculation of the essential integrals in (5.13) and (5.15) for the functions (5.18) gives (5.20) As we see in (5.12) and (5.14) with (5.19) and ( It is significant that the twist parameter m is a topological invariant for diffeomorphism. The shell of twisted lightlike soliton is diffeomorphic to cylin-drical surface with cuts by multifilar helix, where the number of continuous cuts is 2 m. These cuts correspond to the singular lines on the shell, which we can see on Fig. 5.7 and Fig. 5.8. The field function Φ of the Gaussian twisted soliton in the plane section {x 1 , x 3 } for x 2 = 0 is shown on Fig. 5.9 and Fig. 5.10. At last we show zero level surfaces of the field function Φ for the Gaussian twisted soliton with m = 1 (Fig. 5.11) and m = 2 (Fig. 5.12). The twist of the solitons is well seen also on these figures. We have two-sheeted helical surface with excluded cavity for m = 1 and we have four-sheeted one for m = 2.
All figures 5.7 -5.12 are appropriate to the solitons twisted on the right.
Here we have considered the simplest arbitrary functions Ξ 1 and Ξ 2 , which give the twisted shell lightlike soliton with one cavity. For more complicated cases we can have the appropriate solitons with a set of cavities. But we will have the notable asymptotic relation between energy, momentum, and angular momentum (5.17) for these cases, because of the appropriate relation for densities (4.31).

Relation to photons
Because of notable connection (5.17) between energy, momentum, and angular momentum of the twisted lightlike solitons, it is reasonable to consider their relation to photons.
For this purpose first we consider an ideal gas of these solitons in bounded three-dimensional volume V . As is known, the ideal gas behaviour is characterized by zero interaction between the particles. But an interaction of the particles with the volume walls provides thermodynamic equilibrium of the ideal gas.
Let us suppose that absorptive and emissive capacities of the walls are provided by soliton-particles having the following constant absolute value of angular momentum where is Planck constant. We suppose also that each lightlike soliton can interact simultaneously with only one soliton-particle of the wall. We assume angular momentum conservation for the combination of lightlike soliton with soliton-particle of the wall in absorption or emission event.
Then, because of the angular momentum conservation, absorption or emission of a twisted lightlike soliton is possible only when the angular momentum of soliton-particle in the wall is oppositely directed to the angular momentum of the lightlike soliton. The soliton-particle angular momentum is reversed in absorption or emission event.
Thus the absolute value of angular momentum of twisted lightlike solitons in the volume V must be equal to .
The structure of twisted lightlike solitons depends on structure and states of emissive and absorbent soliton-particles. We must define the value of twist parameter m and the scale phase function ρ(θ) for the twisted lightlike solitons in the volume V .
Let us consider the case m = 1 . As we see in (5.14b), in this case the energy of the soliton is logarithmically divergent in infinite space. But here we consider the finite volume, where its energy is finite. Strictly speaking, the obtained soliton solutions must be modified for finite volume. But here we consider the integral characteristics of the solitons only. Thus we can consider the soliton solutions of infinite space for the finite volume in some approximation.
Let us suppose also that the scale phase function ρ(θ) is slow variable: Thus, taking into account (5.12) -(5.15) and (6.2), we have the following relations for the twisted lightlike solitons in the volume V : f E 0 is static part of energy density E for lightlike soliton in expression (4.29a).
As we see in (4.29a), the static part of energy E is independent explicitly of the soliton frequency ω. But the condition (6.3c) with expressions (5.16c) and (5.15) gives a dependence of the soliton transversal size characterized by the parameter − ρ from the soliton frequency ω. Thus according to (5.12b) and (5.13) the static energy E is implicitly dependent on the frequency ω. An estimation for this dependence will be made below. But at first for simplicity we consider that the static energy E is approximately constant: The finiteness of the volume under consideration confines the set of possible frequencies of the solitons. As it is known, the field in any finite volume can be represented by the appropriate mode expansion. In the case of cuboid we have the simple space-time Fourier components, which satisfies the periodic boundary conditions.
In the case of arbitrary volume with cavities, the finding of volume modes looks very complicated. Here we consider that the cavities inside the soliton shells are sufficiently small to neglect of their influence. Also we take that each soliton in the volume has one of its allowed frequencies.
Hereafter up to formulas (6.16) we obtain the equilibrium distribution function by soliton frequencies. The appropriate derivation of formulas is similar to ones represented in classical works by S. Bose [11], A. Einstein [12,13], and contained in monographs (see, for example, [14]).
As distinct from cited works, here we use the natural energy cells instead of finite phase space cells. Complete deduction is expounded to show that all assumptions are in the framework of real soliton dynamics only.
For simplicity let us consider the volume V in cubic form with side l v . Then the allowed frequencies are defined by formula where {n 1 , n 2 , n 3 } are integer numbers, excepting the case when all number are zero, i is the index for different frequencies.
According to (6.5) we have the following minimal frequency in the volume If there are N i solitons with frequency ω i in the volume V , then the full energy of solitons in it is given by formula where E i is energy of the soliton with frequency ω i , N is a total number of solitons in the volume V .
Because there is the minimal frequency ω min (6.6) for the solitons in the volume V , then according to (6.7) we have the following expression for their maximal quantity: If we suppose that a full angular momentum as well as a full momentum of the soliton gas in the volume V are zero, then the total number of solitons must be even. Thus their minimal quantity is 2 and we have from (6.7) the maximal value for frequency: Among all the possible distributions by soliton frequencies {N i } there is a piece providing an identical total energy U . According to general principles of statistical physics such distributions are considered as equally probable.
Let us introduce the size of energy cell E i , which are the quantity of solitons having the energy E i and the corresponding frequency ω i . Different states in the sell are defined with the set of numbers {n 1 , n 2 , n 3 } in (6.5) for frequency ω i and two directions of twist (right and left).
Let us count up the number of ways to provide the part of total energy U produced by the solitons with energy E i that is N i E i (6.7a). According to known representation we line up N i solitons (•) and (E i − 1) dividing walls (|) in random order: Here the dividing walls (|) separate the different soliton states ({n 1 , n 2 , n 3 } and twist direction).
In that case the permutation number (N i + E i − 1)! is a total number of distributions for solitons with energy E i . Then we take into account that N i ! permutations of solitons and (E i −1)! permutations of dividing walls correspond to one state. As result we have the sought number of ways to provide the part N i E i of total energy U : We obtain the total number of ways providing the energy U by multiplication of the numbers W i : According to usual method, we take that the most probable distribution provided with maximum number of the ways W corresponds to equilibrium. The total number of solitons N is not fixed here.
Let us solve the problem for maximization of number W with fixed total energy U . (6.7a). For this purpose the method of Lagrange multipliers is used. For convenience we maximize the natural logarithm of number W . Thus the problem for finding of the equilibrium distribution {N i } take the form: where T −1 is Lagrange multiplier, the parameter T has a physical dimension of energy. Let us consider the case when the numbers N i and E i are sufficiently great. In this case we use the Stirling formula for factorial of number. Thus for N i 1 and E i 1 we have Considering the sequence of numbers N i as quasicontinuous, we have the following necessary conditions for maximum of the function S: From (6.14) with (6.12), (6.13), and (6.7a) we have the following equilibrium distribution: Here the constant T can be expressed through the total energy U by using the condition (6.7a). Thus the physical quantity T is an energy parameter of the distribution (6.15).
Let us use the representation of quasicontinuous soliton energy spectrum to obtain the size of energy cell E i . In this case the energy cell E i is characterized by energy gap from E i to E i + ∆E i .
Having in view one-to-one correspondence between the number n in (6.5), frequency, and energy (6.3), we can obtain the quantity of different soliton states with frequencies from ω i to ω i + ∆ω i . A spherical layer in the space of numbers {n 1 , n 2 , n 3 } corresponds to the frequency interval ∆ω i . Taking into account also the two directions of twist and proceeding to the limit ∆ω i → 0, we obtain Then we integrate the expressions E ω N ω and N ω with substitution N ω and E ω from (6.16) over frequency from ω = 0 to infinity. As result we obtain the following expressions for total energy and number of solitons in the volume V : where Li s (z) is polylogarithm function.
For connection between energy parameter T of distribution {N i } (6.15) and absolute temperature • T we take where k B is Boltzmann constant. The relation (6.18) can be validated by means of comparison between statistical determination for entropy S and its thermodynamic one for the case of constant volume (V = const): But because the equivalence of these determinations must be postulated, it is reasonable here to postulate the relation (6.18). Let us write the equilibrium energy spectral density for the twisted lightlike solitons in the volume V . According to (6.16) and (6.18) we have For the cases of negligible static soliton energy E → 0, we have from (6.20) the following known Planck formula for photons: (6.21) Thus we can consider the relation between twisted lightlike solitons and photons. Now let us estimate the possible values of soliton parameters in the volume V using certain suppositions.
Let the longitudinal size of the soliton in (5.13) and the external diameter of cylindrical integration domain in (5.14) be equal to the side of the considered cubic volume: Then, taking into account (6.2a), we have the following values contained in (5.13) -(5.15) for the metric signature (2.11a): (6.23b) Condition (6.3c) with expression (5.16c) gives relation Thus, by virtue of fixedness of the angular momentum of the soliton (6.3c), the radius of its shell − ρ depends on frequency ω. But to calculate − ρ we must have the value of the constant χ.
Nevertheless, to make a very rough estimate, we assume that for a visible light frequency ω the shell radius − ρ has an order of values in the range from the electron classical radius to half of soliton wave-length. Let Expressions (6.23b) with (6.25b) give Standard value of Planck constant must be multiply by the velocity of light for used unit of frequency (6.25a): (6.26) Then relation (6.24) with (6.25) and (6.26) give χ ∼ 1 · 10 −22 ÷ 7 · 10 −7 m 3/2 · eV −1/2 , (6.27a) where the minimal value of χ in (6.27a) and the maximal value of χ −2 in (6.27b) correspond to the minimal value of − ρ in (6.25b). According to formula (5.12b) and taking into account (6.23), (6.25), and (6.27), we have the following values for the static part of soliton energy: where the minimal value of E corresponds to the minimal value of χ −2 in (6.27b) and the maximal value of − ρ in (6.25b). Thus to provide the condition E ω , the diameter of soliton shell 2 − ρ must be closer to the soliton wavelength than to the electron classical diameter.
Expressing − ρ from (6.24) and substituting it to formula for E in (6.28), we obtain from (6.3) the following formula for soliton energy: where C 1 is considered to be constant. Here we disregard the dependence C 1 (− ρ) (6.23b) what is justified for used approximation. The dependence (6.29) is shown on Fig. 6.1 for the explicit values of parameters. Of course, it can be considered only for a qualitative analysis.
As we see on Fig. 6.1, the distinction of soliton energy function from the linear one ω (dashed line) can be noticeable in low-frequency region. But in a bit core of the plot we see a fulfillment of the approximate condition (6.4).
The question arises as to whether there is a static part of energy for real photons. The appropriate experimental check may be possible with the help of extrinsic photoeffect. If the photon energy not exactly equals to ω, then the frequency dependence of photoelectron energy may have a weak nonlinearity near photoemission threshold. The substances with low photoemission threshold is preferable for such experiments.
Let us next consider all values of twist parameter m for lightlike solitons. For m = 1 we have the known expression for photon energy in the case ω E. Thus for m 2 here we could be considered a fractional photon with the following energy expression, according to (5.17): The energy of longitudinal limited and twisted lightlike soliton with m = 1 logarithmically diverges in infinite space, but for m 2 its energy is finite. In this point of view the solitons with m = 1 more closely resemble the plane waves with constant amplitude, the energy of which also diverges in infinite space.
Let us consider the representation of polarization property of light by twisted lightlike solitons.
A beam of these solitons with right or left twist has a necessary symmetry of right or left circularly polarized light wave accordingly. This beam, in particular, provides the Sadovskii effect [15], which is a mechanical angular momentum transfer to absorbent by circularly polarized electromagnetic wave. This effect has the experimental verification [16,17], including one for electromagnetic centimeter waves [18].
As it is known, the plane circularly polarized electromagnetic wave with constant amplitude does not have angular momentum [15]. Thus this wave does not provide the Sadovskii effect. But the twisted lightlike solitons as well as photons have angular momentum and provide this effect.
Elliptical polarization and, as limiting case, linear one of the soliton beam could be provided by a coherent combining of solitons twisted to the right and to the left.
This representation for elliptical polarization conforms to one for the beam of photons, which have two helicity states only.
Peculiarity of the value m = 1 for twist parameter becomes apparent here. According to solution symmetry for this case (see Fig. 5.5), the coherent com-bining of equal quantities of such right and left twisted solitons can give a beam having a crystal like symmetry with axes of the first order. This case can be interpreted as a linear polarization.
But for the case of solitons with higher twist parameters we have for the same conditions the appropriate crystal like symmetry with axes of m 2 order (see Fig. 5.6). This case can not be interpreted as a linear polarization.
Thus the lightlike solitons with twist parameter m = 1 can be considered as usual photons in some approximation. But the solitons of higher twist m 2 have qualitative differences from the solitons of the lowest twist m = 1.

Conclusions
Thus we have considered the field model for extremal space-time film, which is sometimes called Born -Infeld type scalar field model.
We have obtained new exact solution class for this model that is lightlike solitons. We have considered an appropriate significant subclass that is twisted lightlike solitons. It is notable that its energy is proportional to its angular momentum in high-frequency approximation.
The soliton under consideration has a singularity which is a moving twodimensional surface or shell. The lightlike soliton can have a set of tubelike shells with the appropriate cavities.
A relatively simple twisted lightlike soliton with one cavity has considered in details. This soliton is characterized, in particular, by a twist parameter m which is a natural number. The energy of longitudinal limited this soliton in infinite space is finite for m 2, but for m = 1 its energy is logarithmically divergent. For the case m = 1 we have the asymptotic relation between soliton energy, momentum, and angular momentum, which is characteristic for photon.
Then we have investigated relations of the twisted lightlike solitons with m = 1 to photons. The model of ideal gas of the twisted lightlike solitons in a bounded volume has considered for this purpose. Planck formula for the soliton energy spectral density in the volume has obtained with explicit assumptions in some approximation.
An experimental check for a validity of the obtained soliton energy exact formula for real photon is proposed.
A beam of twisted lightlike solitons have considered. We have shown that this beam provides the effect of mechanical angular momentum transfer to absorbent by circularly polarized beam. This effect well known for photon beam.
It has been found that a twisted lightlike soliton beam with m = 1 can provide polarization property of light as well as photon beam.
Thus we have a correspondence between photon and lightlike twisted soliton with the minimal value of twist parameter.