Applied Mathematics, 2010, 1, 1-7
doi:10.4236/am.2010.11001 Published Online May 2010 (http://www.SciRP.org/journal/am)
On New Solutions of Classical Yang-Mills Equations with
Cylindrical Sources
Alexander S. Rabinowitch
Department of Applied Mathematics, Moscow State University of Instrument Construction and Information Sciences,
Moscow, Russia
Received April 27, 2010; revised May 14, 2010; accepted May 17, 2010
Abstract
Strong fields generated by big electric currents are examined within the framework of the Yang-Mills nonli-
near generalization of the Maxwell electrodynamics proposed in our earlier papers. First we consider the
case of stationary currents and find a new exact solution to the Yang-Mills equations. Then we study a
Yang-Mills field inside a thin circular cylinder with nonstationary plasma and find expressions for field
strengths in it. Obtained results are applied to interpret several puzzling natural phenomena.
Keywords: Yang-Mills Equations, Su(2) Symmetry, Source Currents, Field Strengths, Lightning, Exploding
Wires
1. Introduction
As is well known, the Yang-Mills field theory proposed
in 1954 is one of the greatest achievements of the XX
century, which plays a leading role in modern quantum
physics [1-3]. At the same time, the whole area of its
applications can concern not only quantum physics but
also classical physics [4-7]. To explain this point of view,
let us examine powerful fields generated by sources with
very big electric charges and currents. Then the follow-
ing question should be raised. Are the classical Maxwell
equations always applicable to such fields?
It is beyond doubt that the Maxwell equations ade-
quately describe a great diversity of electromagnetic
fields for which photons are their carriers. At the same
time, powerful sources with very big charges and cur-
rents may generate not only photons but also 0
Zand
W bosons. In such cases, the Maxwell equations may
be incorrect since they are applicable to fields for which
only photons are the carriers. On the other hand, there
are the well-known Yang-Mills equations with SU(2)
symmetry which are a nonlinear generalization of the
linear Maxwell equations playing a leading role in vari-
ous models of electroweak interactions caused by pho-
tons and 0 and
Z
W bosons. For this reason, in [4-7]
the classical Yang-Mills equations with SU(2) symmetry
are applied in the case of powerful field sources with
very big electric charges and currents when 0 ZWand

bosons may be generated, along with photons. These
equations can be represented in the form [1-3]
,, ,,
(4/ )
kklm k
klm
DFFgF Acj 

 
 
 , (1)
,,, ,,kkk lm
klm
F
AAgAA  

 
 , (2)
where ,,
,0, 1, 2, 3; ,,1, 2, 3, and
kk
klmA F



are potentials and strengths of a Yang-Mills field, re-
spectively, ,k
j

are three 4-vectors of source current
densities, klm

is the antisymmetric tensor, 123 1

,
D

is the Yang-Mills covariant derivative, g is the
constant of electroweak interaction, and /
x



,
where

x are orthogonal space-time coordinates of the
Minkowski geometry.
It is worth noting that Equations (1-2) have the fol-
lowing well-known consequences [1-3]:
,0 k
DDF 
 , (3)
,, ,
0
kk lm
klm
Djjgj A
 
 

 . (4)
Further we will consider the field sources
 ,k
j of
the form
1,2, 3,
, 0jjjj


 
, (5)
where j

is a classical 4-vector of current densities.
Then when the potentials2,3, 0AA

, the Yang-
Mills Equations (1-2) become coinciding with the Max-
well equations for the potentials1,
A. Moreover, from (4)

A. S. RABINOWITCH
2
with k1 and from (5) we obtain the differential
charge conservation equation
1, 0j 

. (6)
That is why Equations (1-2) with field sources of
Formula (5) can be regarded as a reasonable nonlinear
generalization of the classical Maxwell equations. This
nonlinear generalization was studied in [4-7], where new
classes of spherically symmetric and wave solutions to
the considered Yang-Mills equations were obtained.
These solutions were applied to interpret puzzling prop-
erties of atmospheric electricity, the phenomenon of ball
lightning, and some other natural phenomena unex-
plained within the framework of the linear Maxwell
theory [4-7].
It should be noted that from (1)-(6) we come to the
identity
,,
[(4/)]0 for 1
kk
DDF cjk
 

 (7)
This identity shows that there is a differential relation
for the Yang-Mills Equation (1) with the classical
sources of Formula (5).
Consider now the classical Yang-Mills Equations (1-2)
with cylindrical sources k,
j of the following form:
1,0 01,3
1,1 1,22,3,
(,,), (,,),
0, 0,
jjzjjz
jjj j

 

 
(8)
where
022123
, , , ,
x
xy xxyxzx  . (9)
Then let us seek the potentials ,k
A

in the form
,0 ,1
,2
,3
(,,),
(,,),
(,,),
(, ,)
kk k
kk
kk
k
A
zA
xzA
yzA
z








, (10)
where ,,
kkk

 are some functions.
Substituting expressions (10) into Formula (2) for the
strengths k,
F and taking into account the antisymmetry
of klm

, we find
,01 ,02,03
,12 ,13,23
, , ,
0, , ,
(,,), (,,), (,,),
kkk kkk
kkkkk
kkk kkk
FxuFyuFp
FFxhFyh
uuzp pzhhz

 

 
(11)
where the functions ,, and
kkk
up h are as follows:
/,
,
/.
kkk lm
klm
kkk lm
zklm
kkk lm
zklm
ug
pg
hg




 

 

 
(12)
Here /, /, /
kkkk kk
zz

 


  .
After substituting expressions (8-11) for k, k,
j,A,


k,
and F

 into the Yang-Mills Equation (1), we obtain
2
0
2()
(4/)
kkklmlm
zklm
k
uupgu p
csj




 
 , (13)
()0
kk lmlm
zklm
uhg uh



 , (14)
2
2( )(4/)
kkklm lmk
klm
phhgphcsj 

 ,
(15)
where
123
1, 0sss

. (16)
In the considered case (8), from the three Equations (4)
we obtain
00
z
jj



, (17)
02203 3
0, 0jj jj

   . (18)
From (8) we have 2,3, 0jj



. That is why we can
choose the following gauge by some rotation about the
first axis in the gauge space:
011 0jj

  . (19)
Then using (18), we obtain
0, 1, 2, 3 kk
jjk  (20)
In the second section we study Yang-Mills fields gen-
erated by stationary currents flowing in the direction of
the axis z and find a particular exact solution to the
Yang-Mills Equations (12-16). In the third section we
examine these equations in the case of no stationary
plasma flowing through a thin circular cylinder and study
the Yang-Mills field inside the cylinder. In the fourth
section we discuss obtained results and their applications
to some puzzling phenomena appearing in lightning and
exploding wires.
2. A Particular Exact Solution to the
Yang-Mills Equations in the Case of
a Cylindrical Source with Stationary
Current
Let us turn to the considered Yang-Mills Equations
(12-16) in the following stationary case:
0
00
(), (), constjj jj   , (21)
where ()j

is some function of

.
Let us seek the functions ,,and
kk k

 in (10) in the
form
0
0, ()()()
kkkk k
z 

  , (22)
where k

and k

are some functions of

.

A. S. RABINOWITCH
3
Then from Formulas (12) and (22) we find
0(), (), 0
kkkkk
uhp     . (23)
Substituting Formulas (21-23) into Equations (13-15)
and using the antisymmetry of klm

and hence the iden-
tity 0
lm
klm 
 , we come to the following system of
equations:
2
()2(4/), 1, 2, 3
kklm k
klm
gcsjk

 .
(24)
Therefore, we have got three equations for the six
functions () and ()
kk

 .
Taking into account (16), from Equation (24) we find
1123223
()2() (4/),
g
cj
 
  (25)
33221
2
21
22231
3
21
() 2,
()2
.
g
g
g
g
 

 





(26)
From (26) we derive
23223
112 23 32232
()
(){[()() ]2[() ()]}.
g  
 




(27)
Substituting (27) into Equation (25), we readily obtain
1
(2)(4/)cj


  , (28)
where
2122232
()( )()()

 . (29)
Since in the case (8) under consideration the axes with
2, 3kin the gauge space are equivalent, let us choose
the relativistic-invariant gauge condition
2, 23, 3
F
FFF
 


. (30)
Then we can take the following form for the compo-
nents k

:
1231/2
cos, 2sin



  , (31)
which satisfies (29) and (30), where ()

 and
()

 .
From (28) and (31) we find
2(4/)coscj


  . (32)
Equation (32) is the only equation for the two un-
known functions () and ()

 . Therefore, in the
case under consideration the Yang-Mills equations can-
not allow us to uniquely determine the field strengths
,k
F


. To interpret this, let us turn to identity (7). It
shows that the considered Yang-Mills Equation (1) with
the classical sources of Formula (5) are not independent
and there is a differential relation for them.
Therefore, in order to uniquely determine the field
strengths ,k
F


, we should find an additional equation.
For this purpose, let us represent the Yang-Mills Equa-
tion (1) in the form
,,
(4/ )
kk
F
cJ



 , (33)
where
,, ,
(/4)
kk lm
klm
J
jcg FA
 

 . (34)
Taking into account (33) and the evident identity
,0 k
F




, we find that the components ,k
J

satisfy
the three differential equations of charge conservation
,0 k
J




. (35)
In these equations the values ,k
J

can be interpreted
as components of full current densities. As is seen from
(34), they are the sum of the source components ,k j

and
the second addendum which can correspond to charged
field quanta.
Using the components ,k
J

of full current densities
and the source current densities ,k j

, the following
additional relativistic-invariant equation was proposed
in [4-6] to uniquely determine the field strengths
,k
F


:
33
,,
11
kk kk
kk
J
Jjj






. (36)
The expressions on the left and right of this equation
are proportional to the interaction energy of the full cur-
rents and source currents, respectively, in a small part of
a field source. That is why Equation (36) implies the
conservation of this energy when charged field quanta
are created inside the source [5,6].
Using (33), we can represent Eq. (36) in the form
33
,, 2,
11
(4/)
kk kk
kk
F
Fcjj

 

 

 
 . (37)
Substituting expressions (8) and (11) into Equation
(37), we find
22 2
3
22022
1
[(2) ()
(2)](4/)[()()]
kkkkk
zz
kkk
k
uupuh
hhpcj j

 






. (38)
Using (21) and (23), from Equation (38) we obtain
3222
1
[()2](4 /)
kk
k
cj



  . (39)
Taking into account Formulas (31), Equation (39) can
be represented as
22 22
(2)( )(4/)cj
 
  . (40)

A. S. RABINOWITCH
4
From Equations (32) and (40) we obtain
(4/ ) sincj


  . (41)
Equations (32) and (41) give
2cot



 . (42)
Dividing this equation by

and then integrating it,
we find
2cot dd









. (43)
Equation (43) gives
2
lnlnsinconst    . (44)
In order to have the function ()

 nonsingular, let us
choose the sign ‘+’ in Equation (41) and hence in Equa-
tion (44). Then from Equation (44) we find
2
00
sin/, constDD  . (45)
Since we have chosen the sign ‘+’ in Equation. (41),
from it and Formula (45) we obtain
0
(4/ )/cjD


 . (46)
From (45) and (46) we have the following nonsingular
solution:
02
00
sin 4
, , ()Djdjj
cD
 






. (47)
Formulas (31) and (47) give
10
2
23
2
21
sin, ,
4
21
1cos,
cD
DI D
cD
DI
cD






 




 
(48)
0
2
I
jd



, (49)
where ()II

is the source current in the cylindrical
region of radius

.
From Equations (11), (23), and (48) we find the non-
zero strength components
,13 ,23
,
kk
FF
and
,01,13 ,02,23
00
,
kkkk
FFFF   .
For the components ,13 ,23 and
kk
FFwe have
1,13 1,23
eff eff
22
eff
22
, ,
sin, (),
II
xy
FF
cc
I
ID II
D







2,13 3,13
2
2,23 3,23
2
21cos,
21cos ,
DIx
FF cD
DIy
FF cD




 






 
 


(50)
where D is some constant.
Below we use the terms ‘actual’ and ‘effective’ for the
currents
I
and eff sin( /)
I
DID

, respectively.
It should be noted that when
I
D, the effective
current eff
I
practically coincides with the actual current
I
and we have the Maxwell field expressions for the
strength components 1,13 1,23 and
F
F. The value D
should be a sufficiently large constant. Then Formula (50)
can be regarded as a nonlinear generalization of the cor-
responding Maxwell field expressions for the strengths
components 1,13 1,23 and
F
F when the actual current
I
is sufficiently large.
Formula (50) describe a nonlinear effect of field satu-
ration. Namely, let the absolute value of the actual cur-
rent
I
be increasing from zero. Then when it reaches
the value D  , the strengths components 1,13
F
and
1, 23
F
become equal to zero and after that they change
theirs signs.
This property could be applied to give a new interpre-
tation for the unusual phenomenon of bipolar lightning
that actually changes its polarity (positive becoming
negative or vice versa) [8].
It is also interesting to note that puzzling data for
lightning were recently obtained by the Fermi Gamma-
ray Space Telescope which could be explained by For-
mula (50). Namely, some of lightning storms had the
surprising sign of positrons, and the conclusion was
made that the normal orientation for an electromagnetic
field associated with a lightning storm somehow reversed
[9].
To explain these data, let us note that as follows from
(50), the sign of the effective current eff
I
can differ
from the sign of the actual current
I
when the latter is
sufficiently large.
3. Yang-Mills Fields Inside Thin Circular
Cylinders With Nonstationary Plasma
Consider now a nonstationary thin cylindrical source of
Formula (8) and let us assume that the matter inside it is
in the plasma state.
Besides, let the functions 0 and jj in (8) have the
following form inside the thin source:
00
0
(,), (,), 0jjzjjz



 (51)

A. S. RABINOWITCH
5
where 0

is its radius.
Our objective is to describe the Yang-Mills field inside
the source and for this purpose let us apply Eqs.
(12)-(16). In the considered case we seek the functions
, , and
kkk

 , describing the field potentials
 ,k
A
,
in the following special form:
0
0, (,), (,), 0
kkk kk
zz

    (52)
Then substituting (52) into Formula (12) and using (20)
and the identity 0
lm
klm
 , since klm

are anti-
symmetric, we find
0, 0
kk
uh, (53)
0
,
kkkk k
z
pjj


  (54)
It should be noted that in the examined case, as fol-
lows from (11) and (53), the values ,03 ,30kk
F
F are
the only nonzero field strengths. That is why the chosen
Formula (52) for the field potentials (10) provides the
absence of currents in the directions orthogonal to the
axis z of the considered cylindrical source in the
plasma state, in accordance with the used Formula (8) for
the source.
Substituting now expressions (52) and (53) into Eqs.
(13)-(15), we obtain
0
(4/ )
klm k
zklm
pg pcsj    , (55)
(4/ )
klm k
klm
pg pcsj

   (56)
where, as indicated in (16), 123 1, 0sss

.
Let us multiply Equations (55) and (56) by 0 and jj,
respectively, and add the products. Then using (20):
00 mm
jj  , we obtain
00 kk
z
jp jp

. (57)
To find solutions ),( zpk  to Equation (57), let us
introduce the function
0
00
(,) (0,)
z
qjzd jzdz

 

. (58)
Consider its partial derivatives. Using expression (58)
and Equation (17): 0
z
jj

 , we obtain
(,)qjz


, (59)
0
0
(,) (0,)
zz
qjzdjz

 
.
000
0
(,)(0,)(,)jzdjz jz



  
 (60)
From Formulas (59) and (60) we find
0
0
z
jq jq

. (61)
Taking this into account, we come to the following
solutions to the partial differential Equation (57) of the
first order for (,)
k
pz

:
)(qpp kk, (62)
where ()
k
pqare arbitrary differentiable functions.
Indeed, substituting (62) into Equation (57) and taking
into account equality (61), we find
00
(/)( )0
kkk
zz
jpjpdpdqj qjq


, (63)
and hence, Equation (57) are satisfied.
Thus, as follows from (11), (62), and the first term in
(58), the nonzero field strengths ,03k
F
inside the cylin-
drical source under consideration depend on all charge
passing through unit area of a cross section of the cylin-
drical source from beginning of the current flow.
Let us turn to Equation (54) and seek the functions
and
kk

 , satisfying them, in the form
0[()(,) ()],
[( )(,)( )],
kk k
kk k
jbqzpq
jbqzpq



 (64)
where ()
k
bqare arbitrary differentiable function and
(, )z

 is some differentiable function.
Then substituting expressions (64) into Equation (54)
and taking into account equality (17): 00
z
jj

, we
come to the equations
0
0
[/ /](
)( )
kk k
k
zz
pdbdqdpdqjq
jqjj p





 . (65)
Using Formula (61), from (65) we find
01
z
jj



 (66)
When 00j

, from (17) and (66) we have
0
() and /()()jj zj



, where 0()

 is an
arbitrary function.
Consider the case 00j

. Then in order to solve Eq.
(66), it is convenient to choose the variable q instead
of the variable z and put
(,), (,)qqqz

 

. (67)
Indeed, using (67) and Formulas (59) and (60), we
find
0
,
qz q
jj


  , (68)
and substituting (68) into Equation (66), we derive
01j

 . (69)
Therefore, we obtain
0
(, )/(,)qdjq   , (70)
where 0 jis represented as a function of q and

.
Let us now substitute Formulas (62) and (64) into Eq-
uations (55) and (56). Then using Formulas (59) and (60)
and the evident identity, we find that Equations (55) and
(56) give the same equations of the following form:

A. S. RABINOWITCH
6
123
/(4/), 1, 2, 3,
(), (), 1, 0.
klmk
klm
kk kk
dpdqgp bcsk
ppqbbqs ss
  

(71)
Multiplying Equation (71) by k
p2 and summing the
products over, taking into account the antisymmetry of,
we obtain
321
1
() (8/)
k
k
dpcp
dq



. (72)
Besides this equation, from the second and third equa-
tions in (71) we also find relations of the functions
23
() and ()bq bq to the functions 1() and ()
k
bqp q.
Let us put
1
23 1/2
(4/)()cos(),
(4/)2()sin(),
pcqq
ppcq q




 (73)
where condition (30) is taken into account, and
() and ()qq

are some functions.
Then substituting expressions (73) into Equation (72),
we find
cos

. (74)
Let us now turn to Equation (38). From it and (53) we
have
322 2022
1
[()() ](4/)[()() ]
kk
z
k
pp cjj



 
. (75)
Using Formulas (59), (60), and (62), from Equtaion
(75) we find
322
1
(/)(4/)
k
k
dp dqc 


. (76)
Substituting Formula (73) into Equation (76), we ob-
tain
22
() ()1 

 , (77)
where () and ()qq



.
Substituting now expression (74) for

 into Equa-
tion (77), we find
sin
 
 . (78)
Equations (74) and (78) give
/cot




 . (79)
Let us integrate Equation (79) and choose the sign ‘+’
in it and hence in (78), in order to have its nonsingular
solution. Then we obtain
0sinB

, (80)
where 0 B is some constant.
Substituting expression (80) into Equation (78) and
taking into account that the sign ‘+’ has been chosen in it,
we find
000
1/, /BqB



, (81)
where 0

is some constant.
From Formulas (80) and (81) we obtain
000
sin( /)BqB


, (82)
where q is defined by Formula (58).
Using Formulas (11), (58), (73), and (82), we find
1,03
000
2,03 3,03
000
0
00
00
(2/)sin(2/),
(2/)[1cos(2/)],
2, (,)(0,).
z
FcBqB
FFcB qB
qjzdjzdz



 

 
 

(83)
As follows from (83), when the value 00 2(qB



)n


, where n is an integer, the strength component
1,03
F
is zero and when n is an even integer,
,03 0, 1, 2, 3 k
Fk .
Let us apply obtained results to the puzzling pheno-
menon of current pause which takes place in exploding
wires [10]. The phenomenon proceeds in three stages. At
the instant of closure of the circuit, sufficiently large
current flows through the wire and causes its explosion.
Then in some time the current flow ceases and the period
of current pause begins. After a certain period of time the
current pause can end and the current flow can continue.
The origin of the current pause is not well understood
within the framework of the Maxwell electrodynamics
[10,11]. That is why let us apply its nonlinear generaliza-
tion based on the Yang-Mills equations which we have
studied. For this purpose, let us turn to Formula (83) and
apply them to an exploding wire. As follows from For-
mula (83), after some period of time the strength com-
ponent 1, 0 3
F
becomes zero. At this moment the current
in the wire should cease. Therefore, Formula (83) allow
one to interpret the origin of current pause in exploding
wires. The pause could end and the current flow could
continue after some redistribution of charges in explod-
ing wires.
4. Conclusions
We have studied classical Yang-Mills fields with SU(2)
symmetry generated by charged circular cylinders with
currents. Our objective was to find solutions to the non-
linear Yang-Mills equations that could generalize the
corresponding solutions to the linear Maxwell equations
for sufficiently powerful sources.
We considered two cases. In the first of them we stu-
died a Yang-Mills field generated by a stationary current
flowing through a circular cylinder. In this case we found
a particular exact solution to the Yang-Mills equations.
In the obtained solution the strength components 1,13
F

A. S. RABINOWITCH
7
1,2 3
and
F
have the form 1,13 2
eff
(2/ )/FIcx

, 1,2 3
F

2
eff
(2/ )/Icy

, eff sin(/)
I
DID, where () and II  
effeff ()II

 are the actual and effective currents in the
cylindrical region of radius

, respectively, and D is
a sufficiently large constant. When the actual current
I
is not large and /1ID, the effective current eff
I
is very close to the actual current
I
and the found ex-
pressions for 1,13 1,23 and
F
F are practically coinciding
with the corresponding Maxwell field expressions. At the
same time, when the actual current
I
is sufficiently
large, the effective current eff
I
can substantially differ
from the actual current
I
and, moreover, the values
eff
and
I
I can have different signs. Using this result, we
gave a new interpretation for the phenomenon of bipolar
lightning and explained the puzzling inversion of the
normal orientation for electromagnetic fields associated
with some lightning storms which was recently detected
by the Fermi Gamma-ray Space Telescope.
In the second case we considered a Yang-Mills field
inside a thin circular cylinder with nonstationary plasma.
We sought field potentials in Formula (52) and came to
the partial differential Equations (54-56). Solving these
equations, we found expressions for the field strengths
inside the cylindrical source under consideration. It was
shown that the strengths could depend on all charge
passing through unit area of a cross section of the cylin-
drical source from beginning of the current flow. The
obtained Formula (83) shows that the field strengths in-
side the cylindrical source can become zero after some
period of time. This property of the found solution was
above used to explain the puzzling phenomenon of cur-
rent pause in exploding wires.
5. References
[1] L. H. Ryder, “Quantum Field Theory”, Cambridge
University Press, Cambridge, 1987.
[2] L. D. Faddeev and A. A. Slavnov, “Gauge Fields:
Introduction to Quantum Theory,” Benjamin-Cummings
Publishing, 1990.
[3] P. Frampton, “Gauge Field Theories,” Wiley-VCH, 2008.
[4] A. S. Rabinowitch, Russian Journal of Mathematical
Physics, Vol. 12, No. 3, 2005, pp. 379-385.
[5] A. S. Rabinowitch, Russian Journal of Mathematical
Physics, Vol. 15, 2008, pp. 389-394.
[6] A. S. Rabinowitch, Physics Letters B, Vol. 664, 2008, pp.
295-300.
[7] A. S. Rabinowitch, “Nonlinear Physical Fields and
Anomalous Phenomena,” Nova Science Publishers, New
York, 2009.
[8] V. A. Rakov and M. A. Uman, “Lightning: Physics and
Effects,” Cambridge University Press, Cambridge, 2003.
[9] R. Cowen, “Signature of Antimatter Detected in Lighten-
ing,” Science News, Vol. 176, No.12, December 2009, p.
9.
[10] W. G. Chace and H. K. Moore (Eds.), “Exploding
Wires,” Plenum Press, New York, 1962.
[11] L. I. Urutskoev, V. I. Liksonov and V. G. Tsinoev,
“Observation of Transformation of Chemical Elements
during Electric Discharge,” Applied Physics Reports/
Prikladnaya Fizika, Vol. 4, 2000, pp. 83-100.