On nonlinear electrodynamics with Yang-Mills equations Текст научной статьи по специальности «Физика»

Частицы и поля

УДК 541.532.78

On Nonlinear Electrodynamics with Yang-Mills

Equations

A. S. Rabinowitch

Department of Applied Mathematics and informatics, Moscow State Academy of Instrument-Making and Informatics, 20, Stromynka str., Moscow, 107996, Russia

A nonlinear generalization of the Maxwell electrodynamics is proposed which is based on the Yang-Mills equations. This nonlinear theory is intended to describe strong electromagnetic fields generated by sufficiently powerful sources. In the case of stationary spherically symmetric sources an exact solution to the examined nonlinear field equations is found. Its applications to several unsolved problems of the Maxwell electrodynamics are considered.

1. Introduction

As is well known, the classical Maxwell electrodynamics adequately describe a great diversity of electromagnetic processes. At the same time, there exist a number of natural phenomena which have not got satisfactory explanations within the framework of the Maxwell theory, despite many attempts to do it, and that is why still remain puzzling. To them one can attribute enigmatic properties of the ball lightning [1], of the Earth magnetic field, in particular, its paleomagnetic inversion [2], and a number of other phenomena. However, in our opinion, the most intriguing of them is the following puzzle of star electric fields.

As is known, many stars must have lost several percent of their protons because of explosions and constant small escape of them. For example, for the period of 5 billion years the Sun has lost about 1% of its protons [3]. At the same time, the loss of star electrons, which are much lighter than protons, should be much larger than that of protons. That is why many stars could lose substantial quantities of their electrons and could have considerable positive charges. However, according to the classical theory of electricity, such stars cannot exist, since the electric repulsion of protons is enormously larger than their gravitational attraction.

This paradox and also unsolved problems mentioned above show that the linear Maxwell theory of electromagnetic fields may break down and have to be replaced by some nonlinear theory in the case of very big charges.

It should be noted that there is one more argument that demonstrates the necessity of finding a nonlinear generalization of the Maxwell equations.

Namely, when sources of electromagnetic fields are very powerful, they may generate not only virtual photons but also virtual Z° and W± bosons. In such cases the Maxwell equations may be incorrect, since they are applicable to the fields for which only virtual photons are their carriers.

On the other hand, there are the well-known Yang-Mills equations with SO{3) symmetry generalizing the Maxwell equations and playing a leading role in various models of electroweak interactions caused by photons and Z° and W± bosons [4-8].

That is why let us apply the Yang-Mills equations with 50(3) symmetry to the case of sufficiently strong fields, when virtual Z° and W± bosons may be generated, besides virtual photons. These equations can be represented as

d^" + gsklmFl^A™ = (4tx/c)Jk'v, (1)

Fk,v* = d»Ak,v __ d»Ak,„ _ g£klmAi,i*Am,v^ (2)

where Ak'u and Fk^v are the Yang-Mills field potentials and intensities, respectively, Jk'v are field sources, yu, v = 0, 1, 2, 3, k, I, m - 1, 2, 3, £kim is the antisymmetric tensor, e123 — 1 and <j is the coupling constant of electroweak interactions. Eqs. (1) can be rewritten as

DtlFk^ = (4ir/c)Jk'lJ, (3)

where Dfl is the Yang-Mills covariant derivative which is defined for an arbitrary three-dimensional vector Uk as [4,5]

DllVk = dllUk + geklmVlA™. (4)

As is well known [8], the sources Jfe't/ of a Yang-Mills field described by Eqs. (1) and (2) satisfy the equalities

DuJk,lJ = 0. (5)

Using (4), we can represent these equalities as

d^» +geklmJUuA™ = 0. (6)

Consider the field sources of the following form

J2-" = J3-" = 0, (7)

where Ju are the classical four-dimensional current densities.

Then substituting (7) into the first equation of (6)(fc = 1), we find

(9„,/1,1/ — 0, = J2'v = 0, J3'" = 0. (8)

Therefore, from the Yang-Mills equations (l)-(2) with the classical sources of form (7) we get the classical differential equation of charge conservation du Jv ~ 0, as well as from the Maxwell equations.

Besides, the Yang-Mills equations (l)-(2) with the sources of form (7) have a trivial particular solution corresponding to the Maxwell equations:

fyF1'"" = (4tr/c) J1'" , F1'^ = d^A1'" - duA1^,

A2'u = A3'" — F2'Mt/ = — 0 J2,u — J3'" = 0. ^

Thus, the Yang-Mills equations with the classical sources of form (7) present quite a reasonable generalization of the Maxwell equations.

As will be shown below, this generalization is nontrivial: Aside from the trivial particular solution satisfying Eqs. (9), the Yang-Mills equations (l)-(2) with the sources of form (7) have also a class of nontrivial solutions.

In section 2 we will examine the Yang-Mills equations (l)-(2) with the stationary spherically symmetric sources of form (7) and will find their nontrivial exact solutions.

In section 3, using these solutions to the Yang-Mills equations, we will propose a nonlinear theory of strong electric fields generalizing the classical theory in the case of big electric charges.

In section ?? this nonlinear theory will be applied to the unsolved problems of the origin of the Earth magnetic field [2] and of the nature of ball lightning [1].

In section 5 we will discuss obtained results and their applications.

2. Exact solutions to the Yang-Mills equations with

spherically symmetric classical sources

Consider the Yang-Mills equations (l)-(2) when the field sources Jkhave form (7) and correspond to a spherically symmetric body with stationary charges:

J1-0 = c0(r), Ju = 0, 1 = 1,2,3,

,- (10)

J2-" = J3'" = 0, r = -^(x1)2 + (x2)2 + (x3)2,

where 9(r) is the charge density of the spherically symmetric body, r is the distance between a body point and its center, x1, x2, x3 are rectangular spatial coordinates.

Let us start with equalities (6). Substituting (10) into them, we easily find that in the region where the charge density 6 0

^2,0 = ¿3,0 = a (11)

Since the sources of form (10) are invariant under their gauge rotations about the first axis, we can choose such a gauge transformation so as to have .41,0 = 0. That is why and taking into account (11), we will examine potentials Ak,v that are zero when v = 0:

= fc-1,2,3. (12)

Let us turn to the components Ak'1, I = 1, 2, 3. We will seek potentials Ak-1 that satisfy the Yang-Mills equations (l)-(2) with the spherically symmetric sources (10) in the following form:

^•WMrV + uoir)], A2<1 =xl[v(r)x° + v0(r)}, A3'1 =xl[w(r)x° + w0 (r)], / = 1,2,3, x° = ct,

where uo, u, vq, v, wo, w are some functions of r and t is time.

Then, substituting expressions (12)-(13) for the field potentials Ak-y into formulae (2), we obtain the following stationary and spherically symmetric expressions for the field intensities Fk'fll/:

Ffc-m' = 0, k, rn, I = 1, 2, 3,

F1'01 — xlu{r), F2,01 = xlv(r), F3'0' = xlw(r), pk^=-Fkfil. (H)

Let us now substitute expressions (10) and (12)—(14) for Jfc'", Ak'v and Fk'^ into the Yang-Mills equations (1). Then we come to the following system of equations:

ru + 3u + gr2(wvo — vwo) — -4n6, (15)

rv' + 3f + gr2{uwo - wuq) = 0, (16)

rw' + 3w + gr'2(vuo - uvo) = 0. (17)

From Eqs. (16) and (17) we easily get

wo — (9'"2w)~1(.gr2u>uo - rv' - 3u), (18)

vQ — (gr2u) ~1 (gr2vu0 + rw' + 3w). (19)

Eqs. (18) and (19) give

wv0 - vwQ = {2gr2u)~i [r(v2 + w2)' + 6(<;2 + w2)}. (20)

Substituting (20) into Eq. (15) and multiplying this equation by 2u, we easily find r(u2 + v2 + w2)' + 6 (u2 + v2 + w2) = -87rdu, (21)

which is the only differential equation for the three functions u(r), v(r) and w(r).

Let us represent the functions u(r), v(r) and w(r) in the form

u = -i?cos£/r3 , v = -Rsin £ cos T)/r3 , w = —Rsin£sinr;/r3, R = R(r), f = £(r), v = v{r), u2 + v2 + w2 = R2/r6. (22)

Then from Eq. (21) we get

R'(r) = 47rr20(r) cos £(r), /2(0) = 0. (23)

Thus, we have found some exact spherically symmetric solutions to the Yang-Mills equations (l)-(2) with the sources of form (10) that are described by formulae (12)-(14).

In these formulae the functions wo(^) and wo(r) are given by expressions (18) and (19), the functions u(r), v(r) and w(r) are determined by expressions (22) and (23) and the three functions uo(r), £(r) and r/(r) are arbitrary.

In the next section we will apply the obtained solutions of the examined Yang-Mills equations to determine electric fields generated by field sources with sufficiently big charges.

3. Nonlinear theory of strong electric fields

As stated in section I, the Yang-Mills equations with the sources of form (7) may be regarded as a reasonable generalization of the Maxwell theory which can be applied to describe strong electromagnetic fields, for which not only virtual photons but also virtual Z° and W^ bosons can be their carriers.

That is why we will apply the formulae for the Yang-Mills field potentials Ak and intensities Ffc''"/ obtained in section 2 to describe strong electric fields generated by spherically symmetric bodies with big electric charges.

It should be stressed that the found expression for the electric field intensities pi,oi given by formulae (14), (22) and (23) and generalizing the classical formula for them contains the arbitrary function £(r). This means that the generalized theory of strong electric fields based only on the Yang-Mills equations (1)~(2) cannot be complete. Therefore, in order to uniquely determine the function £(r) we should add a new equation to the Yang-Mills equations (l)-(2).

For this purpose let us consider the Yang-Mills equations (l)-(2) with the field sources Jkof the form

jk.v = (24)

where 0k can be regarded as densities of three charges corresponding to the interactions of field sources with photons, and W7± bosons and Vu is the four-dimensional vector of velocity of a source point.

Let us require that these sources should satisfy the charge conservation equations

du Jk>u = 0. (25)

Then, using (4), we find

A,J*'" - dvJk'u + gekimJ^A? - geklmJl^Aln. (26)

As is well known, the field intensities Fk->"' satisfy the identity [8]

DvDtlFk^v = 0.

(27)

Using correlations (26) and (27) and formula (24) for Jfe'", we easily get the identity

ekDv[DttFk'»lv - (4tr/c) Jfc'"] - 0. (28)

This identity means that the Yang-Mills equations with the considered sources satisfying (24) and (25) are not independent and in order to have their unique solution we need an additional equation.

That is why let us turn to its determination.

First, note that the Yang-Mills equations (1) can be represented as

= , (29)

= Jk-y _ (gc/47r)eklmFl^A^. (30)

As follows from (29), the components Ik'v satisfy the three differential charge conservation equations dwIk'v — 0. Therefore, they, as well as Jk'v, can be interpreted as some three four-dimensional vectors of current densities.

It is seen from (30) that the components Ik'l/ contain not only the source current densities Jk'u but also additional components determined by field potentials and intensities. This allows one to regard the components Ik'v as densities of full currents which also contain, besides the source current densities Jfc,t/, current densities of field virtual particles.

Let us find now a correlation between the densities Jk,u and Ik,u of field source currents and full currents, respectively. For this purpose, consider a small part of a field source and let qk and qk (k - 1, 2, 3) be its intrinsic charges and its full charges, which also include charges of virtual particles created inside it, respectively.

As is known, the classical intrinsic electric energy of a homogeneous body with charge q is proportional to the value q2. That is why, using the law of energy conservation, let us require the invariance of the value qkqk which is proportional to the full electric energy of the considered small part of a field source having the tree charges q1, q2, q3. Then, since the value qkqk should be the same in both cases: when qk — qk and when qk ^ qk, we come to the correlation

q% = qkqk- (31)

Using the components ,7fc>" and Ikiv of the densities of intrinsic currents of a field source and its full currents, respectively, this correlation for a small part of a field source can be represented as

= Jk'"Jk,u • (32)

This equality, which has been derived from the energy conservation law, just presents the sought supplementary equation that should be added to the Yang-Mills equations (l)-(2).

We will further consider the nonlinear generalization of the Maxwell equations based on the Yang-Mills equations (l)-(2) with the sources of form (10) and the additional equation (32).

As follows from (29), Eq. (32) can be rewritten as

daFk*vdPFk,eu = {4ir/c)2Jk'vJk,u . (33)

Consider now field sources of form (10) and let us turn to formulae (14), (22) and (23) for the field intensities Fk>^.

Since we consider the case J2'" — J3'" = 0, let us choose the gauge

fit/ 1 ¡III !

(34)

which implies the equivalence of the second and third axes of the gauge space and set r) — it/A in (22), corresponding to (34). Then from (14), (22) and (23) we get

Fk'ml = 0, Fifil = -Rx'cost/r3, k,m,l = 1,2,3,

F2,oi = F3,oi = _2-I/2RxI sin^ v = 7r/4) (35)

dR/dr = 47rr20cos£, R = R(r), £ = £(r), 0 = 0(r). Substituting expressions (10) and (35) for Jk'u and Ffc,/il/into Eq. (33), we easily

find

(dR/dr)2 + R2(d£/dr)2 = (4tT0)2rA. (36)

Let us introduce the variable q = q(r) through the formula

dq = An r29(r)dr, q( 0) = 0 (37)

and choose the variable q as an argument of the functions R and instead of the variable r. Then from (23), (36) and (37) we obtain

dR/dq = cost, R2(dt/dq)2 = sin2£, R — R(q), £ = £(q). (38)

As follows from (37), the value q presents the charge of the spherical region bounded by the radius r.

Let us now require that formulae (35) for the field intensities Fx'flu should coincide with the classical formulae of the Maxwell theory when the charge q is small. Then from (23), (35) and (37) we find

£(0) — 0, R(q)=q + o(q), q - 0 . (39)

Therefore, when q —> 0, d^/dq — £/q = sin £/R and hence from (38) and (39) we

get

dR/dq = cos£, d£/dq = sin£/R, R( 0) = 0, ^(0) = 0. (40)

Equating the ratio of the left-hand sides and that of the right-hand sides of the two differential equations in (40), we derive

dR/R = cos£é/£/ sin£. (41)

Integrating this equality and taking into account (39), we easily get

R = K() sin £ , A'o = const. (42)

From (40) and (42) we find

dR/dq — cosí,, dR/dq — K0 cos ¡¡.d^/dq. (43)

Hence, d^/dq = 1/Kq =const and using (39) and (42), we have

Z = q/K0, R = K0sm{q/KQ), K0 = const. (44)

Substituting expressions (44) into formulae (35), we get

F1'10 =Ksm{q/K)xl/r3 , K = K0/2 , q = q(r),

f2,io = p3,io = 2-1/2_ (45)

Fk>ml = 0, k,m,l = 1,2,3. dq = Anr29(r)dr, g(0) = 0.

As follows from (45), we can represent the found expression for the electric field intensities F1,10 generated by a spherically symmetric source with charge q in the form

F1'10 — <Zeff a-''/r3 , qt„ = Ksm(q/K), K = const, (46)

where qen can be regarded as an effective charge that determines the electric field outside the spherically symmetric source.

According to (46), the constant nK can be defined as a minimum positive value of the charge q for which <jre[f = 0.

We will further use the terms "real" and "effective" for the charges q and qeff = K sin(q/K), respectively.

The obtained formula (46) presents a generalization of Coulomb's law in which the real charge q is replaced with the effective charge qe^ — Ks\n(q/K).

As follows from (46), when < K, the effective charge qe[f and real charge q are practically coinciding and when \q\ ~ K, they are substantially different.

Thus, the obtained formula (46) for the electric field intensities F1'10 can be regarded as a generalization of the classical formula in the case of a source with a sufficiently big charge.

The charge K, which should be a sufficiently big constant, will be estimated below.

It should be stressed that when a body with a real charge q is not spherically symmetric, the effective charge qau allows one to describe the electric field of the body at distances well away from it.

Thus, the effective charge of a source determines the asymptotic behaviour of its electric field which should be added to the examined Yang-Mills equations as a boundary condition at infinity.

It is interesting to note that the obtained expressions (45) for the electric field intensities Fk'10 are periodic functions of the real charge q which have the period 2ixI< and are zero when q — 2-nnK, where the number n is any integer.

Therefore, the proposed nonlinear theory of strong electric fields describes the following property of them which can be called the nonlinear effect of electric saturation.

Nonlinear effect of electric saturation. According to the proposed nonlinear generalization of the Maxwell theory, a spherically symmetric body having the real charge q — 2-irnK, where n is any integer, generates no electric field intensities outside it.

It is worth noting that this nonlinear effect allows one to resolve the following paradox of star electric fields which was considered in Introduction.

As pointed out in Introduction, the constant loss in stars of charged particles with electrons predominating can lead to the accumulation of big positive charges in them. But according to the classical linear theory of electricity, such stars cannot exist, since the electric repulsion between protons is enormously bigger than their gravitational attraction.

However, the effect of electric saturation of the proposed nonlinear generalization of the Maxwell electrodynamics shows that the influence of the electric field of a star with a very big charge on its protons can be smaller than that of its gravitational field and hence such a star can exist.

In the next section we will consider applications of obtained results and will estimate the constant K.

4. Applications of the nonlinear electric field theory

First, let us consider the electromagnetic field of the Earth.

It is known that the origin of the Earth magnetic field remains enigmatic despite many attempts to explain it [2]. For instance, it should be noted that the Earth core cannot be ferromagnetic, since its temperature is considerably higher than the Curie temperature. Besides, since it is unlikely that the Earth has a constantly

acting electromotive force, the explanations of the Earth magnetic field based on its hypothetical currents run into serious obstacles [2].

That is why let us investigate this problem within the framework of the proposed nonlinear electric field theory based on the Yang-Mills eq

As is known (see [9, p. 206]), the absolute value

charge, which is determined by measuring the external electric field of the Earth, is as follows:

4.5 x 105 coul. (47)

uations. <7® of the Earth effective

(E)

Let </(E' be the real charge of the Earth. Then from (46 ) and (47) we get

К ~ 4.5 x 105/ \sm{q(E)/K)

coul.

(48)

As follows from (46), the effective charge qsu of a big spherically symmetric body, which can be determined by measuring its external electric field, does not allow one to determine uniquely the real charge q of the body. Actually, the function <?eii = Ks\n(q/K) is periodic and there is an infinitely large number of values of q corresponding to a given value of qc\

That is why and since the absolute value comparatively small for such a big body as the Eart

of the Earth effective charge is i, let us assume that the absolute

value

,(E)

of the Earth real charge is much bigger then the value

(E)

9elf

7(E)

>

(E)

9etf

4.5 x 105 coul.

(49)

Then it becomes possible to explain the enigmatic magnetic field of the Earth. Namely, owing to (49), this magnetic field could be regarded as a result of the axial rotation of the Earth charges. It should be noted that the charge g®, which is regarded as the Earth charge within the framework of the Maxwell theory, is very small for such an explanation of the Earth magnetic field.

Thus, the proposed nonlinear generalization of the Maxwell theory allows one to interpret the observed magnitude of the Earth magnetic field.

Taking into account estimate (48) and replacing in it the function |sin(<//A")| of q with its average value 2/n, we find the following estimate for the constant K.

К ~ 7 x 105 coul.

(50)

Let us turn now to the problem of the nature of ball lightning which still has no satisfactory solution within the framework of the Maxwell electrodynamics [1].

Consider a ball lightning that is formed during a thunder-storm and could be composed of free electrons, neutral molecules of the atmosphere and negative ozone ions generated from the oxygen in it due to thunder-storm discharges [1,10].

As will be shown later on, the existence of such a stable charged ball can be explained by using formulae (46).

For this purpose let us examine a ball composed of negative ozone ions, free

electrons and neutral molecules and having a radius 7-<b) and the real charge

.(b) _

-2тг К.

Consider the ball part rf] < r < r(b) that has the real charge q — q(b) /2 = —irK. As

,(b)

follows from formula (46 ) for the field intensities Fuo, in the region rj < r < r attractive forces act on negative ions and electrons, producing pressure in the ball, whereas repulsive forces act on the negative charges surrounding the ball.

Therefore, the sphere r — r(b) can be the boundary of a stable ball lightning with the following real charge q(b):

»

2ixK ~ —4.4 x 106 coul,

(51)

where we have used estimate (50).

Let us estimate the number Nmo, of the molecules in a ball lightning with the radius r(b). Let rmo! be the average radius of atmosphere molecules. Then the number Nmol can be estimated as follows, assuming that the molecules in the ball lightning are closely packed:

AU, - (r(b)Aw)3. (52)

On the other hand, the number iVmoi of the molecules in the ball lightning can be expressed by means of the following identity:

Nmo, - qw /[noz + ne)eo], e0 - 1.602 x HTiy coul, n0i = Noz/Nmo\, ne = Ne/Nmoi,

where q(b) is the real charge of the ball lightning, Noz and Ne are the numbers of the negative ozone ions and electrons in it, respectively, and eo is the elementary charge.

From (51)-(53) we derive

r(b> ~ rmol {2TTK/[(noz + ne)eo]}1/3 . (54)

The fraction noz of the ozone in the atmosphere which could form from the oxygen due to thunder-storm discharges is ~ 1% [10] and the average radius rmo| of the atmosphere molecules is ~ 1.5 x 10~8 cm [9].

Using these estimates and the values of K and eo in (50) and (53), we have

noz ~ 10"2, rmol ~ 1.5 x 10"8 cm,

K ~ 7 x 10d coul, e0 = 1.602 x 10~19 coul.

Let us now estimate the maximum radius r^L of ball lightnings.

As follows from (54) and (55), the maximum radius r max can be reached when ne < noz. Hence, formula (54) and estimates (55) give

~ ''"moi [27rif/(nozeo)]1/3 ~ 21 cm. (56)

It should be noted that the obtained estimate for the maximum radius r^ix of ball lightnings, appearing sometimes during thunder-storms, is in accord with observational data. Namely, as follows from them [1,10], the radius of a ball lightning, as a rule, does not exceed 30 cm.

Thus, the considered Yang-Mills generalization of the Maxwell electrodynamics allows one to explain the phenomenon of ball lightning.

5. Conclusions

We have considered the nonlinear generalization (l)-(2) and (7) of the Maxwell equations which is based on the Yang-Mills equations. This generalization is intended to describe strong fields generated by macroscopic bodies with big electric charges, when not only virtual photons but also virtual Z° and W± bosons can come into being.

In the case of stationary spherically symmetric sources the examined Yang-Mills equations were above reduced to the system of the three ordinary differential equations (15)-(17). Examining these equations, we found their exact solutions and obtained formulae for the Yang-Mills field potentials and intensities containing three arbitrary functions.

The found formula for the electric field intensities F1'10 in the spherically symmetric case under consideration presents a nontrivial generalization of the classical formula for them. This generalization contains one arbitrary function £(r).

Since the Yang-Mills equations (l)-(2) with the sources of form (7) do not allow one to uniquely determine the electric fields intensities, we proposed the additional equation (32). This equation presents the condition of conservation of the intrinsic electric energy of field sources.

Using Eq. (32), we determined the sought function £(r) and found that it is proportional to the charge q(r) of the spherical region bounded by the radius r. The obtained formulae for the electric field intensities show that they practically coincide with the classical formulae when the charge K, where К is some sufficiently

big charge, but substantially differ from them when \q\ ~ К and describe sinusoidal dependence of the electric field intensities on the charge q.

Obtained results were used to investigate the unsolved problem of the origin of the Earth magnetic field. It was shown that the estimate ~ 7x 1Q5 coul for the constant К of the proposed nonlinear theory could allow one to interpret the observed magnitudes of both the electric and magnetic fields of the Earth.

Then we turned to the problem of the nature of ball lightning and used our results to explain the existence of such objects. Applying the found formula for the electric field intensities F1,i0, we came to the conclusion that a ball lightning should have the charge = ~'2тгК. Then, using available data concerning the chemical composition of a ball lightning and the found estimate for its charge, we obtained the estimate ~ 21 cm for its maximum radius, which accords with observational data.

Литература

1. Singer S. The Nature of Ball Lightning. — New York: Plenum Press, 1971.

2. Ueda S. The New View of the Earth. — San Francisco: Freeman, 1978.

3. Klimishin I. A. Present-day Astronomy. — Moscow: Nauka, 1980.

4. Ryder L. H. Quantum Field Theory. — Cambridge: Cambridge University Press, 1985.

5. Nelipa N. F. Physics of Elementary Particles. Gauge Fields. — Moscow: Vysshaya shkola, 1985.

6. Rabinowitch A. S. Modified Yang-Mills Theory and Electroweak Interactions // Int. J. of Theor. Physics. - 2000. - Vol. 39. - P. 2447.

7. Dodd R. K., Elbeck J. C., Gibbon J. D., Morris H. C. Solitons and Nonlinear Wave Equations. — London: Academic Press, 1984.

8. Slavnov A. A., Faddeev L. D. Introduction to Quantum Theory of Gauge Fields. — Moscow: Nauka, 1988.

9. Kitaigorodsky A. I. Introduction to Physics. — Moscow: Nauka, 1973.

10. Smirnov В. M. // Russian Journal of Technical Physics. — 1977. — Vol. 47. — P. 814.

иОС 541.532.78

О нелинейной электродинамике с уравнениями Янга-Миллса

А. С. Рабинович

Кафедра прикладной математики и информатики, Московская государственная академия приборостроения и информатики, Россия, 107996, Москва, ул. Стромынка, 20

Предложено нелинейное обобщение максвелловской электродинамики, основанное на уравнениях Янга-Миллса. Данная нелинейная теория предназначена для описания сильных электромагнитных полей, создаваемых достаточно мощными источниками. В случае стационарных сферически-симметричных источников найдено точное решение исследуемых нелинейных уравнений поля. Рассмотрены его приложения к нескольким нерешенным проблемам максвелловской электродинамики.