december 13, 2017particular, a weak single-component scalar wave can not provide the transverse...
TRANSCRIPT
LIGHTLIKE SHELL SOLITONS
OF EXTREMAL SPACE-TIME FILM
Alexander A. Chernitskii
A. Friedmann Laboratory for Theoretical Physics
Moika 48, St.-Petersburg, Russia.
October 17, 2019
Abstract
New exact solution class of Born β Infeld type nonlinear scalar fieldmodel is obtained. The variational principle of this model has a specificform which is characteristic for extremal four-dimensional hypersurface orhyper film in five-dimensional space-time. Obtained solutions are singularsolitons propagating with speed of light and having energy, momentum,and angular momentum which can be calculated for explicit conditions.The soliton singularity here is a moving two-dimensional surface or shell,where the model action density becomes zero. The lightlike soliton canhave a set of tubelike shells with the appropriate cavities. A twisted light-like soliton is considered. It is notable that its energy is proportional toits angular momentum in high-frequency approximation. A case with onetubelike cavity is considered. In this case the soliton shell is diffeomorphicto cylindrical surface with cuts by multifilar helix. The shell transversesize of the appropriate finite energy soliton can be converging to zero atinfinity. The ideal gas of such lightlike solitons with minimal twist param-eter is considered in a finite volume. Explicit conditions provide that theangular momentum of each soliton in the volume equals Planck constant.The equilibrium energy spectral density for the solitons is obtained. It hasthe form of Planck distribution in some approximation. A beam of twistedlightlike solitons is considered. The representation of arbitrary polariza-tion for beam with twisted lightlike solitons is discussed. It is shown thatthis beam provides the effect of mechanical angular momentum transferto absorbent by circularly polarized beam. This effect well known forphoton beam. Thus the soliton solution which have determinate likenesswith photon is obtained in particular.
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Contents
1 Introduction 2
2 Extremal space-time film 3
3 Energy-momentum and angular momentum 6
4 General lightlike soliton 7
5 Twisted lightlike soliton 13
6 Relation to photons 23
7 Conclusions 32
1 Introduction
A nonlinear space-time scalar field model considered here is known for a longtime sufficiently. This model is related to well known Born β Infeld nonlinearelectrodynamics [1], and it is sometimes called Born β Infeld type scalar fieldmodel.
This model is attractive because it has relatively simple and geometricallyclear form. It can be considered as a relativistic generalization of the minimalsurface or minimal thin film model in three-dimensional space.
In this generalization we have an extremal four-dimensional film in five-dimensional space-time. But the model equation appears as differential one forscalar field in four-dimensional space-time.
On the other hand, this model can provide the necessary effects which arerequired for a realistic filed model.
In particular, the model under consideration has a static spherically symmet-ric solution, which is identical to zero four-vector component of electromagneticpotential for dyon solution of Born β Infeld electrodynamics [2]. This staticsolution of the scalar model gives the appropriate moving soliton solution withthe aid of Lorentz transform.
As it was shown in the cited work, in the case of nonlinear electrodynam-ics there are the conformity between long-range interaction of solitons and twoknown long-range interactions of physical particles, that is electromagnetic andgravitational ones. But the methods which was used for the investigation of soli-ton long-range interaction are independent of the field model. The appropriateinstruments are integral conservation laws and characteristic equation.
2
These methods applying to the scalar model under consideration give theresults, which are similar to ones for nonlinear electrodynamics. These results indetail must be matter for another article. Here we briefly discuss the obtainingof Lorentz force for interacting scalar solitons in the next section.
An essential difference of the scalar field model from the nonlinear electro-dynamics is obviously caused by the different tensor character of the fields. Inparticular, a weak single-component scalar wave can not provide the transversepolarization of electromagnetic wave.
But at the present time we consider the light as photon beam but not aweak electromagnetic wave with constant amplitude. The photon beam couldbe represented by an appropriate scalar soliton beam. In this case an essen-tial space-time nonhomogeneous of soliton solution may provide the necessarysymmetry properties for the beam.
Thus at the present work we consider the model of extremal space-time film.We obtain its exact soliton solutions propagating with the speed of light that islightlike solitons.
Then we investigate in detail a lightlike soliton solution having a rotationabout the direction of propagation that is twisted lightlike soliton.
We consider the ideal gas of such twisted lightlike solitons. Using explicitassumptions we obtain Planck distribution formula in some approximation.
At last we consider a beam with the twisted lightlike solitons. We show thatthis beam can represent photon one. In this case we have, in particular, thepolarization property and the effect of mechanical angular momentum transferto absorbent by circularly polarized beam.
2 Extremal space-time film
Let us consider the following action which has the world volume form:
π =
β«π
β|M| (dπ₯)
4, (2.1a)
where M + det(Mππ), (dπ₯)4 + dπ₯0dπ₯1dπ₯2dπ₯3, π is space-time volume,
Mππ = mππ + π2 πΞ¦
ππ₯π
πΞ¦
ππ₯π, (2.1b)
mππ are components of metric tensor for flat four-dimensional space-time, Ξ¦ isscalar real field function, π is dimensional constant. The Greek indices takevalues {0, 1, 2, 3}.
The variational principle πΏπ = 0 with action (2.1) corresponds to extremalfour-dimensional film Ξ¦({π₯π}) in five-dimensional space-time {Ξ¦, π₯0, π₯1, π₯2, π₯3}.
Determinant M in (2.1) can be represented in the form
M = m
(1 + π2 mππ πΞ¦
ππ₯π
πΞ¦
ππ₯π
), (2.2)
3
where m + det(mππ).Taking into account (2.2) we can write model action (2.1) in the form
π =
β«π
β dπ , (2.3a)
where dπ +β|m| (dπ₯)
4is four-dimensional volume element,
β +
β1 + π2 mππ
πΞ¦
ππ₯π
πΞ¦
ππ₯π
. (2.3b)
Variational principle with action (2.3) gives the following model equation:
1β|m|
π
ππ₯π
β|m|Ξ₯π = 0 , (2.4a)
where
Ξ₯π +Ξ¦π
β, Ξ¦π = mππ Ξ¦π , (2.4b)
Ξ¦π +πΞ¦
ππ₯π. (2.4c)
We have the following evident relations from (2.4c):
πΞ¦π
ππ₯πβ πΞ¦π
ππ₯π= 0 . (2.5)
Inversion for relations (2.4b) gives
Ξ¦π =Ξ₯π
β, (2.6a)
where
β +β|1 β π2 mππ Ξ₯π Ξ₯π | . (2.6b)
For the case when the field invariant Ξ¦π Ξ¦π relatively small (π2 |Ξ¦π Ξ¦π| βͺ 1)we can represent the action density β with two first terms in formal power seriesof π:
β = 1 +π2
2mππ πΞ¦
ππ₯π
πΞ¦
ππ₯π+ πͺ
(π4)πβ0
. (2.7)
The appropriate linearized equation has the form
1β|m|
π
ππ₯π
(β|m|mππ πΞ¦
ππ₯π
)+ πͺ
(π2)πβ0
= 0 . (2.8)
Also let us write the linearized relation (2.4b):
Ξ₯π = Ξ¦π + πͺ(π2)πβ0
. (2.9)
4
Nonlinear differential equation of second order (2.4) for function Ξ¦ can berepresented in the form of the first order differential equation system for four-vectors Ξ¦π or Ξ₯π. 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 met-ric signature. Here we will use the signature {+,β,β,β}. We introduce thefollowing designation for Minkowski metric:
βm00 = 1 , βm
0π = 0 , βmππ = βπΏππ (2.10)
where πΏππ is Kronecker symbol. The Latin indices take values {1, 2, 3}.The signature metric in use (2.10) allows the same spherically symmetric
solution of the model that was obtained by M. Born and L. Infeld in theirclassical work for nonlinear electrodynamics:
Ξ₯π =π
π2,
πΞ¦
ππ=
πβπ4 + π4
, (2.11)
where π is constant, π is radial spherical coordinate, π +β|π π|.
It is evident that solution (2.11) give birth to the class of soliton solutionswith Lorentz transformations. Such solutions in this model also can be con-sidered as point charged particles because their long-range interactions haveelectromagnetic character.
Indeed to investigate the interactions we can use the method based on inte-gral conservation law of momentum (for Born β Infeld nonlinear electrodynamicssee [2]). Let us consider the long-range interaction of an appropriate to (2.11)moving soliton-particle with the rest one (2.11). In this case the method givespure electrical interaction between the particles. Then we can transform theobtained law of particle movement with electrical force to another moving ref-erence frame. In this case Lorentz transform of the force gives its magneticcomponent. The Lorentz transform of the force was presented by A. Einsteinin last section of his classical work on special relativity [3].
It should be noted that using the metric signature {β,+,+,+} for action(2.3) leads to minus sign before the term π4 in (2.11). This gives infinity value ofΞ¦π on the sphere. If we want to consider the solution (2.11) with this signature,we must change the sign before π2 in (2.1b) and (2.3b).
Instead relations between four-vectors {Ξ¦π} and {Ξ₯π} (2.4b) and (2.6) wecan consider relation between quadruples of components {Ξ¦0,Ξ₯1,Ξ₯2,Ξ₯3} and{Ξ₯0,Ξ¦1,Ξ¦2,Ξ¦3} in Minkowski metric (2.10). This representation can be prefer-able for some problems.
The appropriate solution of equations (2.4b) gives the following relations:
Ξ₯0 =
β1 + π2 Ξ₯π Ξ₯πβ
1 + π2 Ξ¦20
Ξ¦0 , Ξ¦π =
β1 + π2 Ξ¦2
0β1 + π2 Ξ₯π Ξ₯π
Ξ₯π . (2.12)
Comparison relations (2.12) and (2.4b) gives the following expression for the
5
action density β in Cartesian coordinates:
β =
β1 + π2 Ξ¦2
0β1 + π2 Ξ₯π Ξ₯π
. (2.13)
Now let us write the field equation (2.4) in Cartesian coordinates with metric(2.10). After differentiation Ξ₯π (2.4b) in (2.4a) and multiplication the equationby β3 we obtain (
βmππ β2 β π2 Ξ¦π Ξ¦π
) π2 Ξ¦
ππ₯π ππ₯π= 0 . (2.14)
As we see, obtained equation does not include radicals.It is evident that the model under consideration keep invariance for space-
time rotation and scale transformation. Thus any solution give birth to theappropriate class of solutions with the following transform:
Ξ¦({π₯π}) β πΞ¦({πΏπ.π π₯
π/π}) , (2.15)
where πΏπ.π are components of space-time rotation matrix, π is scale parameter.
3 Energy-momentum and angular momentum
Customary method gives the following canonical energy-momentum density ten-sor of the model in Cartesian coordinates
ββπππ =
1
4π
(Ξ¦π Ξ¦π
ββ βm
ππ
π2β)
. (3.1)
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 for-mula:
βπππ =ββπππ β
ββπππ . (3.2)
whereββπππ is regularizing symmetrical energy-momentum density tensor which
can be defined depending on class of solutions under consideration. Here wewill use constant regularizing tensor
ββπππ = β 1
4π π2 βmππ . (3.3)
We have conservation law for regularized energy-momentum density tensorin Cartesian coordinates
πβπππ
ππ₯π= 0 . (3.4)
Let us define angular momentum density tensor by customary way. We havethe following appropriate conservation law:
π βππππ
ππ₯π= 0 , (3.5)
6
whereβππππ + π₯πβπ ππ β π₯π βπππ . (3.6)
We introduce the following special designations for energy, momentum vec-tor, and angular momentum vector densities: β° , π« , π₯ . Let us write the appro-priate expressions taking into account relations (2.3b), (2.4b), and (2.13):
β° +βπ 00 =1
4π
(Ξ¦0 Ξ¦0
ββ ββ 1
π2
)(3.7a)
=1
4π π2
(π2 (Ξ¦π Ξ¦π) β 1
β+ 1
)(3.7b)
=1
4π π2
(π4 (Ξ₯π Ξ₯π) Ξ¦2
0 β 1β1 + π2 (Ξ₯π Ξ₯π)
β1 + π2 Ξ¦2
0
+ 1
), (3.7c)
π«π +βπ 0π = βπ π0 =1
4π
Ξ¦0 Ξ¦π
β=
1
4πΞ¦0 Ξ₯π =
1
4πΞ¦π Ξ₯0 , (3.7d)
π₯π + ππππ π₯π π«π , (3.7e)
where ππππ is Levi-Civita symbol (π123 = 1).Let us define energy, momentum, and angular momentum of field in a three-
dimensional volume π :
Eπ +β«π
β° dπ , PPPπ +β«π
π« dπ , JJJπ +β«π
π₯ dπ . (3.8)
4 General lightlike soliton
Let us consider solutions in a form of wave propagating along π₯3 axis of Cartesiancoordinate system with the speed of light. Let this solution be have sometransverse and longitudinal field distributions. Thus we can write
Ξ¦ = Ξ¦(π, π₯1, π₯2
), (4.1a)
π = π π₯0 β π3 π₯3 , π3 = Β±π , π > 0 . (4.1b)
Substitution (4.1) to field equation (2.14) gives the following equation:(1 β π2
(πΞ¦
ππ₯2
)2)π2Ξ¦
(ππ₯1)2+ 2π2 πΞ¦
ππ₯1
πΞ¦
ππ₯2
πΞ¦2
ππ₯1ππ₯2
+
(1 β π2
(πΞ¦
ππ₯1
)2)π2Ξ¦
(ππ₯2)2= 0 . (4.2)
As we see this equation does not include derivatives on phase of wave π (4.1b).Equation (4.2) is elliptical for the following condition:
1 β π2(Ξ¦2
1 + Ξ¦22
)> 0 . (4.3)
7
The similar in form (4.2) equations were considered. About this topic seethe paper by R. Ferraro [4] and references therein.
In particular, the similar in form but different in type equation was consid-ered by Barbashov and Chernikov [5]. The monograph by Whitham [6] containsrelatively simple way for obtaining the Barbashov β Chernikov solution with thehelp of hodograph transformation (see, for example, [7]).
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 ellipticequations causes the appropriate difference in the solution way.
Let us introduce new independent variables
π = π₯1 + π€ π₯2 , *π = π₯1 β π€ π₯2 , (4.4)
where π€2 = β1.Also we will use cylindrical coordinates {π, π, π₯3}. We have the following
evident relations:
π = π eπ€ π , *π = π eβπ€ π , (4.5a)
π =β
π *π , π = βπ€ ln(πβ§Έβ
π *π). (4.5b)
Using new variables (4.4) we obtain from (4.2) the following equation:(1 β 2π2 πΞ¦
ππ
πΞ¦
π*π
)π2Ξ¦
ππ π*π
+ π2
(πΞ¦
ππ
)2π2Ξ¦
(π*π)2+ π2
(πΞ¦
π*π
)2π2Ξ¦
(ππ)2= 0 . (4.6)
Equation (4.6) is hyperbolic for the following condition:
1 β 4π2(Ξ¦2
π + Ξ¦2*π
)> 0 . (4.7)
As noted in section 2 the field model under consideration is invariant byspace-time rotation and scale transformation. But equation (4.2) does not con-tain derivatives with respect to coordinates {π₯0, π₯3}. Because this here we havespace-time rotation and scale invariance in the planes {π₯1, π₯2} and {π₯0, π₯3} withmutually independent parameters. Thus equation (4.2) is invariant with respectto rotation about π₯3 axis and scale transformation in {π₯1, π₯2} plane.
As applied to equation (4.6), taking into account relations (4.5) and (2.15),these two types of invariance are provided by the following general substitution:
Ξ¦(π, *π) ββπ *π Ξ¦ (π/π, *π/*π) , (4.8a)
where π is arbitrary complex constant with respect to coordinates {π, *π}, *π iscomplex conjugate to π quantity. The constant π will be called the scale-rotationparameter of solution in the plane {π₯1, π₯2}.
The complex constant π can be written in the form
π = π eπ€ π , (4.8b)
8
where π and π are real constants with respect to coordinates {π₯1, π₯2}. But ingeneral case these constants can be depend on phase of soliton π (4.1b):
π = π(π) , π = π(π) . (4.8c)
Because this we will call π(π) the scale-rotation function of the soliton. It isevident that the function π(π) defines the phase dependence of transversal scaleand the function π(π) defines the phase dependence of rotation about π₯3 axis.
Thus if we have a solution Ξ¦(π, *π) to equation (4.6) then by means of invari-ant substitution (4.8) we obtain wave propagating along π₯3 axis and preservingits transversal form. Longitudinal form of the wave defined by scale-rotationphase function π(π) is also preserved.
As result we have wave packet propagating with speed of lite and preservingits shape. It can be called the lightlike soliton.
Equation (4.6) is equivalent to the following first order system:
πΞ¦π
π*πβ πΞ¦*π
ππ= 0 , (4.9a)(
1 β 2π2 Ξ¦π Ξ¦*π
) πΞ¦π
π*π+ π2 Ξ¦2
π
πΞ¦*π
π*π+ π2 Ξ¦2
*π
πΞ¦π
ππ= 0 , (4.9b)
where
Ξ¦π +πΞ¦
ππ, Ξ¦*π +
πΞ¦
π*π. (4.10)
By interchanging the roles of the dependent and independent variables in(4.9) we obtain the linear system
ππ
πΞ¦*πβ π*π
πΞ¦π= 0 , (4.11a)
(1 β 2π2 Ξ¦π Ξ¦*π
) ππ
πΞ¦*πβ π2 Ξ¦2
π
ππ
πΞ¦πβ π2 Ξ¦2
*π
π*π
πΞ¦*π= 0 , (4.11b)
which is equivalent to single equation
(1 β 2π2 Ξ¦π Ξ¦*π
) π2π
πΞ¦ππΞ¦*πβ π2 Ξ¦2
π
π2π
(πΞ¦π)2β π2 Ξ¦2
*π
π2π
(πΞ¦*π)2
β 2π2 Ξ¦πππ
πΞ¦πβ 2π2 Ξ¦*π
ππ
πΞ¦*π= 0 . (4.12)
Let us introduce new independent variables
π = π€
β1 β 4π2 Ξ¦π Ξ¦*π β 1
2πΞ¦*π, (4.13a)
*π = βπ€
β1 β 4π2 Ξ¦π Ξ¦*π β 1
2πΞ¦π. (4.13b)
9
Inversion for relations (4.13) gives
Ξ¦π =π€ π
π (1 + π*π), (4.14a)
Ξ¦*π = β π€ *π
π (1 + π*π). (4.14b)
Substituting (4.13) and (4.14) into (4.11), we obtain
π2ππ
ππ+
π*π
ππ= 0 , (4.15a)
ππ
π*π+ *π2
π*π
π*π= 0 . (4.15b)
Sequential elimination each of the dependent variables *π(π, *π) and π(π, *π)from system (4.15) gives two simple equations
π2π
ππ π*π= 0 ,
π2*π
ππ π*π= 0 , (4.16)
solutions of which have the form
π = π1(π) + π3(*π) , *π = π2(*π) + π4(π) , (4.17)
where π1(π), π2(*π), π3(*π), π4(π) are arbitrary functions.Substitution (4.17) to (4.15) gives
π2 πβ²1 + πβ²4 = 0 , πβ²3 + *π2 πβ²2 = 0 . (4.18)
Taking into consideration (4.17) and 4.18), we can write the following rela-tions for general solution of the system (4.15):
dπ = dπ1 β *π2 πβ²2 d*π = dπ1 β *π2 dπ2 , (4.19a)
d*π = dπ2 β π2 πβ²1 dπ = dπ2 β π2 dπ1 . (4.19b)
Thus the general solution of the system (4.15) contains only two arbitrary func-tions π1(π) and π2(*π).
Using (4.14) and (4.19), we obtain
πΞ¦
ππ= Ξ¦π
ππ
ππ+ Ξ¦*π
π*π
ππ=
π€
ππ πβ²1 , (4.20a)
πΞ¦
π*π= Ξ¦π
ππ
π*π+ Ξ¦*π
π*π
π*π= β π€
π*π πβ²2 . (4.20b)
From (4.20) we have
dΞ¦ =π€
π
(π πβ²1 dπ β *π πβ²2 d*π
)=
π€
π
(π dπ1 β *π dπ2
). (4.21)
10
Here the variables π and *π in last expression must be considered as inversefunctions for π1(π) and π2(*π) that are π = π(π1) and *π = *π(π2).
Let us introduce the designations
π€
ππ(π1)+
dΞ1
dπ1, β π€
π*π(π2)+
dΞ2
dπ2, (4.22)
where functions Ξ1(π1) and Ξ2(π2) are arbitrary because of arbitrariness of thefunctions π1(π) and π2(*π).
Then, using (4.21) and (4.22), we have the general solution of equation (4.6)in the form
Ξ¦ = Ξ1(π1) + Ξ2(π2) . (4.23a)
Here arbitrariness of functions Ξ1(π1) and Ξ2(π2) is restricted by reality of fieldfunction Ξ¦. The connection between variables {π1, π2} and {π, *π} is defined byrelations (
dπd*π
)=
(1 π2
(Ξβ²2
)2π2(Ξβ²1
)21
)(dπ1dπ2
), (4.23b)
which are obtained from (4.19) with (4.22).Relations (4.23b) can be inverted on the assumption of nonsingularity of the
transition matrix:
1 β π4(Ξβ²1
)2 (Ξβ²2
)2 = 0 , (4.24a)(dπ1dπ2
)=
1
1 β π4(Ξβ²1
)2 (Ξβ²2
)2(
1 βπ2(Ξβ²2
)2βπ2
(Ξβ²1
)21
)(dπd*π
). (4.24b)
Obtained solution (4.23) can be checked directly. Substitution (4.23a) toequation (4.6) and using (4.24) reduce to identity.
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 independentvariables {π, *π} to {π1, π2} for equation (4.6). But the definition of transforma-tion with differential relations is not complete. Direct connection between thevariables can be obtained by path integration of relations (4.23b) in nonsingulararea, that is for condition (4.24a). At the same time we must define an initialcorrespondence between the variables {π, *π} and {π1, π2}.
We can consider the simplest case by taking π = 0 in (4.23b). In this casewe can put (
π1π2
)=
(π*π
), (4.25)
and expression (4.23a) is evident solution of appropriate to (4.6) linear equationwhen π = 0.
In general case let us consider relation (4.25) as asymptotic for π β β.Then we can designate
π1 = π , π2 = *π . (4.26)
11
It is useful to introduce polar coordinates {π, π} for variables {π, *π} byanalogy with (4.5) for variables {π, *π}:
π = π eπ€ π , *π = π eβπ€ π , (4.27a)
π =
βπ *π , π = βπ€ ln
(π
β§Έβπ *π
). (4.27b)
General solution in the form of lightlike soliton depending on phase π (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 con-tain radical. Substitution solution (4.8) with (4.23a) into (2.3b) with (4.4) and(4.24b) gives expression
β =
1 β π2 Ξβ²
1 Ξβ²2
1 + π2 Ξβ²1 Ξβ²
2
. (4.28)
As we see, explicit dependence on phase π, which we have in (4.8), here is absent.Let us obtain the energy, momentum, and angular momentum densities for
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, mo-mentum, and angular momentum densities with some common functions, whichwill be designate as πβ°
π . Then we have
β° = πβ°0 + π2
((πβ²)2 πβ°
1 +(πβ²)2 (
πβ°2 /π
2 + πβ°3 /π + πβ°
4
)+ πβ² πβ² (πβ°
5 /π + πβ°6
) ), (4.29a)
π«3 = π π((
πβ²)2 πβ°1 +
(πβ²)2 (
πβ°2 /π
2 + πβ°3 /π + πβ°
4
)+ πβ² πβ² (πβ°
5 /π + πβ°6
) ), (4.29b)
π₯3 = π(πβ² πβ°
1 + πβ²(πβ°5 /π + πβ°
6
)/2)
, (4.29c)
where π = Β±π according to (4.1b).Here we write explicitly only two functions πβ°
0 and πβ°1 :
πβ°0 +
1
2π
Ξβ²1 Ξβ²
2
1 + π2 Ξβ²1 Ξβ²
2
, πβ°1 + β 1
4π
(π eβπ€ π Ξβ²
1 β *π eπ€ π Ξβ²2
)21 β π4
(Ξβ²1
)2(Ξβ²2
)2 . (4.30)
These functions play main role in the area, where the scale function π(π) isalmost constant: πβ² β 0.
We have from (4.29) the following notable relation for the case πβ² β 0:
β° β πβ°0 = |π«3| = π |πβ²π₯3| . (4.31)
The arbitrary functions π(π) and π(π) (4.8c) define scale and rotation in theplane {π₯1, π₯2} accordingly. Using (5.2), (4.8), and (4.1b), we can show that the
12
case πβ² > 0 corresponds to positive rotation by angle π in time π₯0 and in π₯3
axis for π3 > 0.Thus for right-handed coordinate system {π₯1, π₯2, π₯3}, the cases πβ² > 0 and
πβ² < 0 correspond to right and left local twist of the soliton accordingly.It is interesting to consider the solitons with constant twist:
πβ² = const = 0 . (4.32)
Such solitons can be called uniformly twisted ones. For conciseness we will callthem the twisted solitons.
As we see in (4.31), for the case (4.32) the soliton energy density β° is pro-portional to its angular momentum density π₯ in high-frequency approximation,that is for π |πβ²π₯3| β«
πβ°0
. The appropriate proportionality relation between
soliton energy and its angular momentum is notable property of the twistedlightlike soliton.
To obtain integral characteristics of the soliton it is necessary to integratethe functions {πβ°
0 , ..., πβ°6 } in the plane {π₯1, π₯2}. Considering (4.23b) and (4.27),
we can see that the appropriate integrands have notable simple form in thevariables {π, π} (4.27).
We must take into consideration that the functions {π, *π} and, accordingly,{Ξ, *Ξ} depended on arguments {π/π, *π/*π} after scale-rotation transformation(4.8). Thus making additional substitution {π/π, *π/*π} β {π, *π}, we have thefollowing integrands:
πβ°0 π2 π dπdπ =
π2
2πΞβ²1 Ξβ²
2
(1 β π2 Ξβ²
1 Ξβ²2
)πdπdπ , (4.33a)
πβ°1 π2 π dπdπ = β π2
4π
(π π eβπ€ π Ξβ²
1 β *π *π eπ€ π Ξβ²2
)2π dπdπ , (4.33b)
πβ°2 π2 π dπdπ =
π2
4π
(π π eβπ€ π Ξβ²
1 + *π *π eπ€ π Ξβ²2
)2πdπdπ , (4.33c)
πβ°3 π2 π dπdπ = β π2
2π(Ξ1 + Ξ2)
(π π eβπ€ π Ξβ²
1 + *π *π eπ€ π Ξβ²2
)Γ(1 + π2 Ξβ²
1 Ξβ²2
)πdπdπ , (4.33d)
πβ°4 π2 π dπdπ =
π2
4π(Ξ1 + Ξ2)
2 (1 + π2 Ξβ²
1 Ξβ²2
)2πdπdπ , (4.33e)
πβ°5 π2 π dπdπ =
π2 π€
2π
(π2 π2 eβπ€ 2π (Ξβ²
1)2 β *π2 *π2 eπ€ 2π (Ξβ²2)2)πdπdπ , (4.33f)
πβ°6 π2 π dπdπ = βπ2 π€
2π(Ξ1 + Ξ2)
(π π eβπ€ π Ξβ²
1 β *π *π eπ€ π Ξβ²2
)Γ(1 + π2 Ξβ²
1 Ξβ²2
)πdπdπ . (4.33g)
5 Twisted lightlike soliton
For further calculations, we define the arbitrary functions Ξ1 and Ξ2. Let ustake power function with integer negative exponent. Introducing necessary mul-
13
tiplicative constants for concordance of physical dimension and for simplificationof resulting formulas, we have
Ξ1 =βππ+1
πππβπ , Ξ2 =
βππ+1
ππ*πβπ , (5.1a)
Ξβ²1 = β
βππ+1
ππβ(π+1) , Ξβ²
2 = ββππ+1
π*πβ(π+1) , (5.1b)
where π is natural number, constant βπ has a physical dimension of length.Then formula (4.23a) representing the solution of equation (4.6) has the form
Ξ¦ =βππ+1
ππ
(πβπ + *πβπ
). (5.2)
Because π βΌ π and *π βΌ *π at π β β, we have from (5.2) with the scale-rotation transformation (4.8) the following asymptotic solution:
Ξ¦ βΌ 2 (πβπ)π+1
ππππcos(π (πβ π)
)at π β β . (5.3)
In view of dependence on phase π(π) and π(π), the formula (5.3) describesthe propagating wave along the π₯3 axis. The dependence π(π) in (5.3) describesalso the twist of this wave about the propagation direction.
Let us consider the twisted lightlike soliton with condition (4.32). We putfor this case
π = Β± π
π, (5.4)
where the signs β+β and βββ correspond to right and left twisted soliton accord-ingly.
In addition, let us consider that the scale function π(π) is almost constant:πβ² βΌ 0. As we can see in (5.3) with (5.4), in this case π in (4.1b) is radianfrequency of the soliton wave and 2π/ |π3| is the appropriate wave length.
To obtain the functions π(π) and *π(*π), first we must integrate relations(4.23b). It is convenient to use the coordinates {π, οΏ½οΏ½} (4.27) for integration ofright hand part of (4.23b).
Let us take the path of integration in the plane {π, οΏ½οΏ½} with sufficiently farbeginning from coordinate origin. Let the starting point be {πβ, 0}. We canintegrate by the following path: from {πβ, 0} to {π, 0} for οΏ½οΏ½ = const and from{π, 0} to {π, οΏ½οΏ½} for π = const.
Then we take that πβ β β, where relations (4.25) are satisfied. Usingvariables {π, *π} again, as result we have
π = π ββπ2(π+1)
(2π + 1) *π2π+1, *π = *π β
βπ2(π+1)
(2π + 1) π2π+1., (5.5)
Obtained relations (5.5) give the following single-valued mapping for vari-ables:
{π, *π} =β {π, *π} . (5.6a)
14
But for representation of solution (5.2) in coordinates {π₯1, π₯2}, according to(4.4), we must have inverse to (5.6a) mapping
{π, *π} =β {π, *π} . (5.6b)
As we can see, relations (5.5), considered as equations for variables {π, *π},give multi-valued mapping (5.6b). But, of curse, here we must consider onlythe realization of (5.6b) with (5.5), which leads to relations (4.25) at infinityπ β β.
Let us substitute (5.1b) to expression for the determinant of transition ma-trix in (4.23b). Then, using the coordinates {π, π} (4.27), we obtain the fol-lowing value of π for any π, which violate the reversibility condition (4.24a) fortransformation of variables (4.23b):
π = βπ . (5.7)
The action density (4.28) vanishes for the set of points appropriate to (5.7).Because this the vector components Ξ₯π (2.4b) become infinite for Ξ¦π = 0.
Thus we have the singular line for the plane {π, *π} in the form of circle withradius βπ. For the plane {π, *π}, this singular line, according to (5.5) and (5.7),is described by formula
ππ = βπ eπ€ π(
1 β eπ€ 2ππ
2π + 1
). (5.8)
Here the function ππ = ππ(π) represents the parametric expression for thesingular line in the complex plane of variable π.
Expression (5.8) represents epicycloid with 2π cusps. This line for π = 1is shown on Fig. 5.1 and it for π = 2 is shown on Fig. 5.2. These figureswas obtained also by R. Ferraro [8, 9] for mathematically similar but anotherproblem.
In the present investigation, the system (5.5) with βπ = 1 for given values ofparameter π and variables {π, *π} is solved numerically with respect to variables{π, *π} in all characteristic areas of the plane {π₯1, π₯2}.
In the area of the plane {π₯1, π₯2} outside of the singular line (5.8) we haveone-to-one mapping (5.6) with the condition (4.25) at infinity π β β.
This mapping can keep continuity for transition through the singular line,if we resign the condition of mutual complex conjugation for variables π1 andπ2 (4.26). But in this case the field function Ξ¦ becomes complex-valued nearlyeverywhere in the inner area of the singular line (5.8), excepting some radiallines, where the function Ξ¦ keeps reality. These lines in the plane {π₯1, π₯2} aretwo-dimensional surfaces in the three-dimensional space {π₯1, π₯2, π₯3}.
But such transition through the singular line is forbidden in the frameworkof the obtained solution. Thus we can find a solution in the inner area of thesingular line in the plane {π₯1, π₯2} independently of the solution in the outsidearea. Then we could try to satisfy any conditions for the field function and itsderivatives on the singular line. But here such conditions could appear to beforcible.
15
π1β¬1
π₯2
π₯1
π = 1βπ = 1, π = 1
π = 1, π = 0
Figure 5.1: Singular line on the plane {π₯1, π₯2} with parameter π = 1.
A radical solution of the inner area problem is exclusion of this area formthe space. In this case we have a soliton with the appropriate cavity. Here wefollow this way.
Taking into account also the scale-rotation transformation (4.8), we have thefollowing condition for the space of the solution:
π >ππ(π(π)
)π(π) , π > βπ , (5.9)
where dependence ππ = ππ(π(π)
)corresponds to rotation of the singular con-
tour in the plane {π₯1, π₯2} by the angle π(π) (4.8c).Thus according to (5.9) we have the soliton with an inner shell. It can be
called the shell soliton.The results of numerical calculations for the function Ξ¦ (5.2) on the singular
line are shown on Fig. 5.3 for π = 1 and on Fig. 5.4 for π = 2. The appropriateresults for the field function Ξ¦ on the plane {π₯1, π₯2} are shown on Fig. 5.5 forπ = 1 and on Fig. 5.6 for π = 2. The points {π1, β¬1} for π = 1 and{π2, β¬2, π2, π2} for π = 2 are corresponding on Figures 5.1 - 5.6.
As we see on Figs. 5.3 and 5.4, the field function Ξ¦ on the singular line isnearly triangle function by angle π. But there are slight deflection from straightlines.
The cusps are the derivative discontinuities for the field function along thesingular line in the plane {π₯1, π₯2}. As we can see on Figs. 5.5 and 5.6, thesederivative discontinuities are absent outside of the singular line.
Now let us obtain the expressions of full energy, momentum, and angular mo-mentum for the solution under consideration in bounded three-dimensional vol-
16
π2β¬2
π2
π2
π₯2
π₯1
π = 2βπ = 1, π = 1
π = 1, π = 0
Figure 5.2: Singular line on the plane {π₯1, π₯2} with parameter π = 2.
ume. For convenience we consider the tubular volume in coordinates {π, π, π₯3}.Its internal radius is defined in (5.9). Let its external radius and length be des-ignated as πβ and ππ accordingly. Thus in addition to condition (5.9) we have
π 6 πβ , β ππ 26 π₯3 6
ππ 2. (5.10)
First we calculate the integrals on right-hand parts of relations (4.33) byvariables {π, π} in area {[βπ, πβ], [βπ, π]}. That corresponds to integration onleft-hand parts of relations (4.33) by variables {π, π} in the outside area of thesingular line ππ and bounded by the line π(πβ eβπ€ π).
Making the integration in the plane {π₯1, π₯2} we can get the rotation param-eter be zero: π = 0. Let us substitute (5.1) and (5.5) with (4.27) to right-handparts of (4.33). We change the integration by variable π₯3 to one by phase π(4.1b).
As result we have the following expressions for energy and absolute valuesof momentum and angular momentum:
E = P + E , (5.11a)
E =βπ2
π π2π0 β0 = P
1
βπ2 π2
π0 β0π1 (π2 β1 + π3 β2)
, (5.11b)
P =πβπ4
π2π1 (π2 β1 + π3 β2) , (5.11c)
J =βπ4
π2π1 π2 |β3| , (5.11d)
17
Ξ¦(ππ)
π1
β¬1β¬1
π = 1βπ = 1, π = 1
π = 1, π = 0
π/π
Figure 5.3: The field function Ξ¦ on the singular line of the plane {π₯1, π₯2} forπ = 1.
where E is the part of soliton energy obtained from the part πβ°0 of energy density
β° (4.29a),
β0 +
π ππ /2β«βπ ππ /2
π2dπ , β1 +
π ππ /2β«βπ ππ /2
π4 (πβ²)2
dπ ,
β2 +
π ππ /2β«βπ ππ /2
π2 (πβ²)2
dπ , β3 +
π ππ /2β«βπ ππ /2
π4 πβ² dπ , (5.12)
π0 +π + 1
2π (2π + 1)β (βπ/πβ)2π
2π+
(βπ/πβ)2 (2π+1)
2 (2π + 1), (5.13a)
π1 + ln
(πββπ
)+
13
72β 1
6
(βπ
πβ
)4β 1
72
(βπ
πβ
)8for π = 1 , (5.13b)
π1 +1
2π2
((π + 1)2 (12π2 β 2πβ 1)
π (3π + 1) (2π + 1)2 (πβ 1)β (βπ/πβ)2 (πβ1)
πβ 1
β (βπ/πβ)4π
π (2π + 1)β (βπ/πβ)2 (3π+1)
(2π + 1)2 (3π + 1)
)for π > 2 , (5.13c)
π2 + π2 , π3 + (π + 1)2 . (5.13d)
The value of π₯3 momentum projection is defined by the sign of wave vector
18
Ξ¦(ππ)
π2 β¬2β¬2
π2 π2
π = 2βπ = 1, π = 1
π = 1, π = 0
π/π
Figure 5.4: The field function Ξ¦ on the singular line of the plane {π₯1, π₯2} forπ = 2.
projection π3 (4.1b): P3 = Β±P.In general case the π₯3 angular momentum projection is defined by integral
β3 (5.12), which can be called the integral twist of the soliton with weight π4:J3 = Β±J.
Let us write the appropriate to (5.11) expressions for the twisted soliton.Using condition (5.4) and formulas (5.12), we have
|β3| =1
πβ1 , β1 =
1
π2β1 , β1 +
π ππ /2β«βπ ππ /2
π4dπ . (5.14)
Using (5.14) and (5.13), we obtain from (5.11) the following expressions forthe twisted soliton:
E = P(
1 +1
βπ2 π2
π0 β0π1 (β1 + π3 β2)
), (5.15a)
P = π J1
π
(1 +
π3 β2β1
), (5.15b)
J =βπ4
π2π π1 β1 , (5.15c)
For the twisted soliton let us consider the case for slowly varying scale func-tion π(π), such that β2 β 0 (5.12). Also we suppose that the frequency π issufficiently high, such that βππ β β. According to expressions (5.15), in this
19
Ξ¦π1
β¬1π = 1βπ = 1, π = 1
π = 1, π = 0
π₯2
π₯1
Figure 5.5: The field function Ξ¦ on the plane {π₯1, π₯2} for π = 1.
Ξ¦ π2β¬2
π2π2
π = 2βπ = 1, π = 1
π = 1, π = 0π₯2
π₯1
Figure 5.6: The field function Ξ¦ on the plane {π₯1, π₯2} for π = 2.
case we have the following relations:
E = P = οΏ½οΏ½ J , (5.16a)
whereοΏ½οΏ½ +
π
π(5.16b)
is the angular velocity of the twisted soliton.Let us consider the twisted soliton with scale function in the form of Gaussian
curve:
π = exp
(β π2
2 π2
), (5.17)
where π is characteristic length of the soliton measured in radians and numeri-cally equals to a total angle of twist on the characteristic length of the soliton
20
along π₯3 axis. A twist angle 2π/π corresponds to soliton wave-length along π₯3
axis.Let us consider the case of infinite space with the conditions
πββπ
β β , π ππ β β . (5.18)
Then the calculation of the essential integrals in (5.12) and (5.14) for the func-tions (5.17) gives
β0 = πβπ , β1 = π
βπ
2, β2 =
1
4 π
βπ
2. (5.19)
As we see in (5.11) and (5.13) with (5.18) and (5.19), for the case of infinitespace we have the finite values of energy, momentum, and angular momentumif π > 2. Using (5.15) with (5.13), (5.14), (5.18), and (5.19), let us write theappropriate expressions for π = 2:
E = P
(1 +
560β
2 π2
129 (9 + 4 π2)βπ2 π2
), (5.20a)
P = π J1
2
(1 +
9
4 π2
), (5.20b)
J =387
1400
βπ
2
βπ4 π
π2. (5.20c)
It is evident that the case π β« 1 and βπ π β« 1 for expressions (5.20) givesrelations (5.16a).
The shell of the twisted soliton with Gaussian scale phase functions is shownon Fig. 5.7 and Fig. 5.8.
BB
BBοΏ½οΏ½1οΏ½οΏ½)
ππ = 1
π = 10
Figure 5.7: The shell of Gaussian twisted soliton for π = 1.
It is significant that the twist parameter π is a topological invariant fordiffeomorphism. The shell of twisted lightlike soliton is diffeomorphic to cylin-drical surface with cuts by multifilar helix, where the number of continuous cuts
21
BBBB1)
ππ = 2
π = 10
Figure 5.8: The shell of Gaussian twisted soliton for π = 2.
is 2π. These cuts correspond to the singular lines on the shell, which we cansee on Fig. 5.7 and Fig. 5.8.
The field function Ξ¦ of the Gaussian twisted soliton in the plane section{π₯1, π₯3} for π₯2 = 0 is shown on Fig. 5.9 and Fig. 5.10.
π = 1
π = 10
Ξ¦
π₯3
π₯1
Figure 5.9: The field function Ξ¦ of Gaussian twisted soliton for π = 1 on theplane {π₯1, π₯3} for π₯2 = 0.
At last we show zero level surfaces of the field function Ξ¦ for the Gaussiantwisted soliton with π = 1 (Fig. 5.11) and π = 2 (Fig. 5.12). The twist of thesolitons is well seen also on these figures. We have two-sheeted helical surfacewith excluded cavity for π = 1 and we have four-sheeted one for π = 2.
All figures 5.7 β 5.12 are appropriate to the solitons twisted on the right.Here we have considered the simplest arbitrary functions Ξ1 and Ξ2, which
give the twisted shell lightlike soliton with one cavity. For more complicated
22
π = 2
π = 10
Ξ¦
π₯3
π₯1
Figure 5.10: The field function Ξ¦ of Gaussian twisted soliton for π = 2 on theplane {π₯1, π₯3} for π₯2 = 0.
cases we can have the appropriate solitons with a set of cavities. But we willhave the notable asymptotic relation between energy, momentum, and angularmomentum of type (5.16) for these cases, because of the appropriate relationfor densities (4.31).
6 Relation to photons
Because of notable connection (5.16) between energy, momentum, and angularmomentum of the twisted lightlike solitons, it is reasonable to consider theirrelation to photons.
For this purpose first we consider an ideal gas of these solitons in boundedthree-dimensional volume π .
As is known, the ideal gas behaviour is characterized by zero interactionbetween the particles. But an interaction of the particles with the volume wallsprovides thermodynamic equilibrium of the ideal gas.
Let us suppose that absorptive and emissive capacities of the walls are pro-vided by soliton-particles having the following constant absolute value of angularmomentum
Jπ =~2, (6.1)
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 conserva-tion for the combination of lightlike soliton with soliton-particle in the wall inabsorption or emission event.
23
π = 1
π = 10
Ξ¦ = 0
π₯3
π₯1
π₯2
Figure 5.11: Zero level surfaces of the field function Ξ¦ for the Gaussian twistedsoliton with π = 1.
Then, because of the angular momentum conservation, absorption or emis-sion of twisted lightlike soliton is possible only when the angular momentum ofsoliton-particle in the wall is oppositely directed to the angular momentum oflightlike soliton. The soliton-particle angular momentum is reversed in absorp-tion or emission event.
Thus the absolute value of angular momentum of twisted lightlike solitonsin the volume π must be equal to ~.
The structure of twisted lightlike solitons depends on structure and statesof emissive and absorbent soliton-particles. We must define the value of twistparameter π and the scale phase function π(π) for the twisted lightlike solitonsin the volume π .
Let us consider the caseπ = 1 . (6.2a)
As we see in (5.13b), in this case the energy of the soliton is logarithmicallydivergent in infinite space. But here we consider the finite volume, where itsenergy is finite.
Strictly speaking, the obtained soliton solutions must be modified for finitevolume. But here we consider the integral characteristics of the solitons only.Thus we can consider the soliton solutions of infinite space for the finite volumein some approximation.
Let us suppose also that the scale phase function π(π) is slow variable:
πβ² β 0 . (6.2b)
Thus, taking into account (5.11) β (5.14) and (6.2), we have the following
24
π = 2
π = 10
Ξ¦ = 0
π₯3 π₯1
π₯2
Figure 5.12: Zero level surfaces of the field function Ξ¦ for the Gaussian twistedsoliton with π = 2.
relations for the twisted lightlike solitons in the volume π :
E = P + E , (6.3a)
P = π ~ , (6.3b)
J = ~ , (6.3c)
where
E =
β«π
πβ°0 dπ , (6.3d)
πβ°0 is static part of energy density β° for lightlike soliton in expression (4.29a).
As we see in (4.29a), the static part of energy E is independent of the fre-quency π of the soliton. Disregarding the dependence of E from peculiarity ofthe volume π , we take
E = const . (6.4)
The finiteness of the volume under consideration confines the set of possiblefrequencies of the solitons. As it is known, the field in any finite volume can berepresented by the appropriate mode expansion. In the case of cuboid we havethe simple space-time Fourier components, which satisfies the periodic boundaryconditions.
In the case of arbitrary volume with cavities, the finding of volume modeslooks very complicated. Here we consider that the cavities inside the solitonshells are sufficiently small to neglect of their influence. Also we take that eachsoliton in the volume has one of its allowed frequencies.
Hereafter up to formulas (6.16) we obtain the equilibrium distribution func-tion by soliton frequencies. The appropriate derivation of formulas is similarto ones represented in classical works by S. Bose [10], A. Einstein [11, 12], andcontained in monographs (see, for example, [13]).
25
As distinct from cited works, here we use the natural energy cells insteadof finite phase space cells. Complete deduction is expounded to show that allassumptions are in the framework of real soliton dynamics only.
For simplicity let us consider the volume π in cubic form with side ππ£. Thenthe allowed frequencies are defined by formula
ππ =2π
ππ=
2π
ππ£ππ =
2π
ππ£
βπ21 + π2
2 + π23 , (6.5)
where {π1, π2, π3} are integer numbers, excepting the case when all number arezero, π is the index for different frequencies.
According to (6.5) we have the following minimal frequency in the volume
πmin =2π
ππ£. (6.6)
If there are ππ solitons with frequency ππ in the volume π , then the fullenergy of solitons in it is given by formula
π =
ββπ=1
ππ Eπ , (6.7a)
where Eπ is energy of the soliton with frequency ππ,
Eπ = ππ ~ + E , (6.7b)
π =
ββπ=1
ππ , (6.7c)
π is a total number of solitons in the volume π .Because there is the minimal frequency πmin (6.6) for the solitons in the
volume π , then according to (6.7) we have the following expression for theirmaximal quantity:
πmax =π
πmin ~ + E. (6.8)
If we suppose that a full angular momentum as well as a full momentum ofthe soliton gas in the volume π are zero, then the total number of solitons mustbe even. Thus their minimal quantity is 2 and we have from (6.7) the maximalvalue for frequency:
πmin = 2 , πmax =π β 2E
2 ~. (6.9)
Among all the possible distributions by soliton frequencies {ππ} there is apiece providing an identical total energy π . According to general principles ofstatistical physics such distributions are considered as equally probable.
Let us introduce the size of energy cell πΈπ, which are the quantity of solitonshaving the energy Eπ and the corresponding frequency ππ. Different states in
26
the sell are defined with the set of numbers {π1, π2, π3} in (6.5) for frequencyππ and two directions of twist (right and left).
Let us count up the number of ways to provide the part of total energyπ produced by the solitons with energy Eπ that is ππ Eπ (6.7a). According toknown representation we line up ππ solitons (β) and (πΈπ β 1) dividing walls (|)in random order:
β β | β β β | | β | β β β β| β β| Β· Β· Β· β β Β· Β· Β· | β β β | | β . (6.10a)
Here the dividing walls (|) separate the different soliton states ({π1, π2, π3} andtwist direction).
In that case the permutation number (ππ + πΈπ β 1)! is a total number ofdistributions for solitons with energy Eπ. Then we take into account that ππ!permutations of solitons and (πΈπβ1)! permutations of dividing walls correspondto one state. As result we have the sought number of ways to provide the partππ Eπ of total energy π :
ππ =(ππ + πΈπ β 1)!
ππ! (πΈπ β 1)!. (6.10b)
We obtain the total number of ways providing the energy π by multiplicationof the numbers ππ:
π =
ββπ=1
ππ =
ββπ=1
(ππ + πΈπ β 1)!
ππ! (πΈπ β 1)!. (6.11)
According to usual method, we take that the most probable distributionprovided with maximum number of the ways π corresponds to equilibrium.The total number of solitons π is not fixed here.
Let us solve the problem for maximization of number π with fixed totalenergy π . (6.7a). For this purpose the method of Lagrange multipliers is used.For convenience we maximize the natural logarithm of number π . Thus theproblem for finding of the equilibrium distribution {ππ} take the form:
π = lnπ β π― β1 π , π β max , (6.12)
where π― β1 is Lagrange multiplier, the parameter π― has a physical dimension ofenergy.
Let us consider the case when the numbers ππ and πΈπ are sufficiently great.In this case we use the Stirling formula for factorial of number. Thus for ππ β« 1and πΈπ β« 1 we have
lnπ βββπ=1
((ππ + πΈπ) ln (ππ + πΈπ) βππ lnππ β πΈπ lnπΈπ
). (6.13)
Considering the sequence of numbers ππ as quasicontinuous, we have thefollowing necessary conditions for maximum of the function π:
ππ
πππ= 0 . (6.14)
27
From (6.14) with (6.12), (6.13), and (6.7a) we have the following equilibriumdistribution:
ππ =πΈπ
eEπ/π― β 1. (6.15)
Here the constant π― can be expressed through the total energy π by using thecondition (6.7a). Thus the physical quantity π― is an energy parameter of thedistribution (6.15).
Let us use the representation of quasicontinuous soliton energy spectrum toobtain the size of energy cell πΈπ. In this case the energy cell πΈπ is characterizedby energy gap from Eπ to Eπ + βEπ.
Having in view one-to-one correspondence between the number π in (6.5),frequency, and energy (6.3), we can obtain the quantity of different soliton stateswith frequencies from ππ to ππ + βππ. A spherical layer in the space of numbers{π1, π2, π3} corresponds to the frequency interval βππ. Taking into account alsothe two directions of twist and proceeding to the limit βππ β 0, we obtain
πΈπ,Ξπ β 2 Β· 4π π2 βπ β π3π£π2
π2 βπ β πΈπ dπ =π3π£π2
π2 dπ . (6.16a)
ππ =π3π£π2
π2
eEπ/π― β 1, (6.16b)
whereEπ = ~π + E . (6.16c)
Then we integrate the expressions Eπ ππ and ππ with substitution ππ andEπ from (6.16) over frequency from π = 0 to infinity. As result we obtain thefollowing expressions for total energy and number of solitons in the volume π :
π =π3π£ π― 4 6 Li4
(eβE/π― )
π2 ~3+ π E , (6.17a)
π =π3π£ π― 3 2 Li3
(eβE/π― )
π2 ~3, (6.17b)
where Liπ (π§) is polylogarithm function.For connection between energy parameter π― of distribution {ππ} (6.15) and
absolute temperature βπ we take
π― = ππ΅ βπ , (6.18)
where ππ΅ is Boltzmann constant.The relation (6.18) can be validated by means of comparison between sta-
tistical determination for entropy π and its thermodynamic one for the case ofconstant volume (π = const):
π = ππ΅ lnπ , (6.19a)
dπ =dπβπ
. (6.19b)
28
But because the equivalence of these determinations must be postulated, it isreasonable here to postulate the relation (6.18).
Let us write the equilibrium energy spectral density for the twisted lightlikesolitons in the volume π . According to (6.16) and (6.18) we have
π’(π, βπ ) +Eπ ππ
π=
π2
π2
~π + Eexp ~π+E
ππ΅ βπ β 1. (6.20)
For the cases of negligible static soliton energy E β 0, we have from (6.20)the following known Planck formula for photons:
π’(π, βπ ) =π2
π2
~πexp ~π
ππ΅ βπ β 1. (6.21)
Thus we can consider the relation between twisted lightlike solitons andphotons.
Now let us estimate the possible values of soliton parameters in the volumeπ using certain suppositions.
Taking into account (6.2b), we put πβ² = 0 and without loss of generalityπ = 1.
Let the longitudinal size of the soliton in (5.12) and the external diameterof cylindrical integration domain in (5.13) be equal to the side of the consideredcubic volume:
ππ = 2 πβ = ππ£ . (6.22)
Then, taking into account (6.2a), we have the following values contained in(5.12) β (5.14):
β0 = β1 = π ππ£ , β2 = 0 , (6.23a)
π0 =1
3+
32βπ6
3 π6π£β 2βπ2
3 π2π£, π2 = 1 ,
π1 = ln
(ππ£2βπ
)+
13
72β
βπ4
6 π6π£β
βπ8
72 π8π£. (6.23b)
Condition (6.3c) with expression (5.15c) gives relation
βπ4 π ππ£ π1 = π2 ~ . (6.24)
Thus, by virtue of fixedness of the angular momentum of the soliton (6.3c),the radius of its shell βπ depends on frequency π. But to calculate βπ we musthave the value of the constant π.
Nevertheless, to make a very rough estimate, we assume that for a visiblelight frequency π the shell radius βπ has an order of values in the range from theelectron classical radius to half of soliton wave-length.
Let
π = π βΌ 107 mβ1 , (6.25a)
βπ βΌ 3 Β·(10β15 Γ· 10β7
)m , ππ£ βΌ 0.1 m . (6.25b)
29
Expressions (6.23b) with (6.25b) give
π0 β 1
3, π1 βΌ (12 Γ· 31) , π2 = 1 . (6.25c)
Standard value of Planck constant must be multiply by the velocity of lightfor used unit of frequency (6.25a):
~ β 2 Β· 10β7 eV Β· m . (6.26)
Then relation (6.24) with (6.25) and (6.26) give
π βΌ(1 Β· 10β22 Γ· 7 Β· 10β7
)m3/2 Β· eVβ1/2 , (6.27a)
πβ2 βΌ(2 Β· 1012 Γ· 1 Β· 1044
) eV
m3βΌ(3 Β· 10β7 Γ· 2 Β· 1025
) J
m3, (6.27b)
where minimal value of π in (6.27a) and maximal value of πβ2 in (6.27b) cor-respond to minimal value of βπ in (6.25b).
According to formula (5.11b) and taking into account (6.23), (6.25), and(6.27), we have the following values for the static part of soliton energy:
E ββπ2 ππ£3π2
βΌ(6 Β· 10β3 Γ· 2 Β· 1013
)eV , (6.28)
where minimal value of E corresponds to minimal value of πβ2 in (6.27b) andmaximal value of βπ in (6.25b).
Thus to provide the condition E βͺ π ~, the diameter of soliton shell 2βπ mustbe closer to the soliton wavelength than to the electron classical diameter.
Expressing βπ from (6.24) and substituting it to formula for E in (6.28), weobtain from (6.3) the following formula for soliton energy:
E β ~π +1
3π
β~ ππ£π π1
. (6.29)
This dependence is shown on Fig. 6.1 for the explicit values of parameters. Ofcourse, it can be considered only for a qualitative analysis.
As we see on Fig. 6.1, the distinction of soliton energy function from thelinear one ~π (dashed line) can be noticeable in low-frequency region.
The question arises as to whether there is a static part of energy for realphotons. The appropriate experimental check may be possible with the helpof extrinsic photoeffect. If the photon energy not exactly equals to ~π, thenthe frequency dependence of photoelectron energy may have a weak nonlinearitynear photoemission threshold. The substances with low photoemission thresholdis preferable for such experiments.
Let us next consider all values of twist parameter π for lightlike solitons.For π = 1 we have the known expression for photon energy in the case ~π β« E.
Thus for π > 2 here we could be considered a fractional photon with thefollowing energy expression, according to (5.16):
E =~ππ
+ E . (6.30)
30
π = 7 Β· 10β7 m3/2 Β· eVβ1/2
~ = 2 Β· 10β7 eV Β· m
π = 0.1 m
π1 = 12
E, eV
π, mβ1
Figure 6.1: Dependence of soliton energy from soliton frequency.
But we must pay attention once again to the fact that the twisted lightlikesolitons with π > 2 have qualitative distinction from ones with π = 1 in thepart of energy representation.
The energy of longitudinal limited and twisted lightlike soliton with π = 1logarithmically diverges in infinite space, but for π > 2 its energy is finite. Inthis point of view the solitons with π = 1 more closely resemble the plane waveswith constant amplitude, the energy of which also diverges in infinite space.
Let us consider the representation of polarization property of light by twistedlightlike solitons.
A beam of these solitons with right or left twist has a necessary symmetry ofright or left circularly polarized light wave accordingly. This beam, in particular,provides the Sadovskii effect [14], which is a mechanical angular momentumtransfer to absorbent by circularly polarized electromagnetic wave. This effecthas the experimental verification [15, 16], including one for electromagneticcentimeter waves [17].
As it is known, the plane circularly polarized electromagnetic wave withconstant amplitude does not have angular momentum. Thus this wave does notprovide the Sadovskii effect. But the twisted lightlike solitons as well as photonshave angular momentum and provide this effect.
Elliptical polarization and, as limiting case, linear one of the soliton beamcould be provided by a coherent combining of solitons twisted to the right andto the left.
This representation for elliptical polarization conforms to one for the beam
31
of photons, which have two helicity states only.Peculiarity of the value π = 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 beamhaving a crystal like symmetry with axes of the first order. This case can beinterpreted as a linear polarization.
But for the case of solitons with higher twist parameters we have for the sameconditions the appropriate crystal like symmetry with axes of π > 2 order (seeFig. 5.6). This case can not be interpreted as a linear polarization.
Thus the lightlike solitons with twist parameter π = 1 can be considered asusual photons in some approximation. But the solitons of higher twist π > 2have qualitative differences from the solitons of the lowest twist π = 1.
7 Conclusions
Thus we have considered the field model for extremal space-time film, which issometimes called Born β Infeld type scalar field model.
We have obtained new exact solution class for this model that is lightlikesolitons. We have considered an appropriate significant subclass that is twistedlightlike solitons. It is notable that its energy is proportional to its angularmomentum in high-frequency approximation.
The soliton under consideration has a singularity which is a moving two-dimensional surface or shell, where the model action density becomes zero. Thelightlike soliton can have a set of tubelike shells with the appropriate cavities.
A relatively simple twisted lightlike soliton with one cavity has consideredin details. This soliton is characterized, in particular, by a twist parameter πwhich is a natural number. The energy of this soliton in infinite space can befinite for π > 2, but for π = 1 its energy is logarithmically divergent. For thecase π = 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 withπ = 1 to photons. The model of ideal gas of the twisted lightlike solitons in abounded volume has considered for this purpose. Planck formula for the solitonenergy spectral density in the volume has obtained with explicit assumptions insome approximation.
An experimental check for a conformity of exact formula for the twistedsoliton energy with known formula for photon is proposed.
A beam of twisted lightlike solitons have considered. We have shown that thisbeam provides the effect of mechanical angular momentum transfer to absorbentby circularly polarized beam. This effect well known for photon beam.
It has been found that a twisted lightlike soliton beam with π = 1 canprovide polarization as well as photon beam.
Thus we have a correspondence between photon and lightlike twisted solitonwith the minimal value of twist parameter.
32
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