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?± = ?1 ± i?2 , 2 2 3
? = ?1 + ?2 + ?3 ,
? = (?x0 ? ? · x)(?x0 + ? · x)(???)/(?+?) , s = ln(?x0 + ? · x)/? + ?,
1/2
?+? ?? ± ?
?c1 (23)
? > ?,
?= , ?± = ,
???
2? 2
1/2
?+?
(F0 F0 + F1 F1 ? G? G0 ? G? G1 ).
? ?
c1 = 4
??? 0 1

This solution is also analytic in the coupling constant and
1/2
?
?(x)?(x) = c1 / (?x0 )2 ? (? · x)2 (24)
,
?
i.e. ?? dies off as was previously the case (see (20)).
When k = 1 some particular exact solutions of equation (2) analogous to those
given in (22) are provided by the ansatz (4) with A(x) and ?(?) having the form:
? ?? µ+ s ?
?3 µ? s ?? µ? s ?3 µ+ s
?e ?e e e
? ?
? ? ? ?
? ?
? ?+ µ? s ?3 µ+ s ?
?3 µ? s ?+ µ+ s
? ?
A(x) = ? ? ? e ?e
e e ?,
? ? ?
? ?
? ?
µ? s µ+ s
? ?
e 0 e 0
(25)
eµ? s eµ+ s
0 0
? ?
?0 (?)
? ?
?1 (?)
? ?
?(?) = ? µ+ /(???) ?,
?? ?2 (?) ?
? µ+ /(???) ?3 (?)
On some exact solutions of the nonlinear Dirac equation 329

where ?0 , ?1 , ?2 , ?3 are defined from the following system of ordinary differential
equations
1
?? ?2 + ?0 ?? + ?? ?3 + ?1 ?? = ? c2 = constant,
0 2 1 3
2
d?0 i? k µ+ (k+1)/(???)
= c? ?2 (?),
2? 2
d?
i? k [µ+ (k?1)/(???)]?1 ? + ?
d?2
(26)
= c? ?0 (?),
???
2? 2
d?
d?1 i? k µ+ (k+1)/(???)
= c? ?3 (?),
2? 2
d?
i? ? + ? k [µ+ (k?1)/(???)]?1
d?3
= c? ?1 (?),
2? ? ? ? 2
d?
µ± = (?? ± ?k)/2k, c2 is an arbitrary real constant, and s, ?, ?± and ?3 ? are defined
in (23).
Below we present the explicit form of transformations admitted by equations
(1) or (2). They can be used to generate new exact solutions of the equations in
accordance with the formula (10).
The conformal transformations
xµ = xµ ? cµ x2 /?(x), ?(x) ? 1 ? 2cx + c2 x2 ,
(27)
?
Rconf = ? ?2 (x)(1 ? ?x?c).
? (x ) = Rconf ?(x) = ?(x)(1 ? ?c?x)?(x),

The transformation dilatation
?1
? (x ) = RD ?(x) = e??/2k ?(x),
xµ = e? xµ , RD = e?/2k . (28)

The transformations of rotations
x?? ?(? · x)
x = x cos ? + (1 ? cos ?),
x0 = x0 , sin ? +
?2
?
1 i 1
? (x ) = Rrot ?(x) = cos ? + (? · ?) sin ? ?(x), (29)
2 ? 2
1 i 1
Rrot = cos ? ? (? · ?) sin ?,
2 ? 2
1/2
where ? = (?1 , ?2 , ?3 ), ? = ?1 + ?2 + ?3 , ? = ?0 ? ?, ? = {?1 , ?2 , ?3 } are Pauli
2 2 2

matrices, ?0 is the identity 2 ? 2 matrix.
The Lorentz transformations
x0 = x0 cosh ?1 ? x1 sinh ?1 ,
x1 = x1 cosh ?1 ? x0 sinh ?1 ,
1 1 (30)
? (x ) = RL1 ?(x) = cosh ?1 + ?0 ?1 sinh ?1 ?(x),
2 2
1 1
?1
RL1 = cosh ?1 ? ?0 ?1 sinh ?1 .
2 2
The rest of the Lorentz transformations are analogous to those given above.
330 W.I. Fushchych, W.M. Shtelen

The transformations of displacements

(31)
xµ = xµ + aµ , ? (x ) = ?(x).

In formulae (26)–(31) cµ , ?, ?a , ?a , aµ are arbitrary real constants.
In conclusion we would like to give a simple example of the transformation of
the well known plane-wave solution of the free massless Dirac equation into the new
solution using formulae (10) and (27).

k 2 ? kµ k µ = 0,
?pw (x) = exp(ikx)?,

? is a space-time independent spinor, ?? = constant:
?
1 ? ?x?c
?pw (x) > ?(x) = xµ ? cµ x2 /?(x)
exp ik µ kµ k µ = 0.
?,
2 (x)
?
It is easy to verify that it is a solution of the free massless Dirac equation but it is
no longer a plane-wave solution because of the nonlinear character of the conformal
transformations. Moreover,
3
?
?(x)?(x) = ??/? 3 (x) ? constant/ 1 ? 2cx + c2 x2
?

and dies off very fast when x2 ? x? x? > ? while
?
?pw (x)?pw (x) = ?? = constant.
?

Acknowledgment
We would like to express our gratitude to the referees for their comments and
useful suggestions.

1. Bjorken J.D., Drell S.D., Relattivistic quantum mechanics, New York, McGraw-Hill, 1964.
2. Boerner H., Representations of groups, Amsterdam, North-Holland, 1970, ch. 8, § 3.
3. Fushchych W.I. (ed.), The symmetry of mathematical physics problems, in Algebraic-Theoretic
Studies in Mathematical Physics, Kiev, Institute of Mathematics, 6–28.
4. Fushchych W.I., Moskaliuk S.S., Lett. Nuovo Cimento, 1981, 31, 571–576.
5. Fushchych W.I., Serow N.I., Dokl. Akad. Nauk USSR, 1982, 263, 582–586.
6. G?rsey F., Nuovo Cimento, 1956, 3, 988.
u
7. Heisenberg W., Z. Noturf. A, 1954, 9, 292–303.
8. Merwe P.T., Phys. Lett. B, 1981, 106, ¹ 6, 485–486.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 331–334.

Conformal symmetry and new exact solutions
of SU2 Yang–Mills theory
W.I. FUSHCHYCH, W.M. SHTELEN
Conformally invariant and some others exact solutions of the SU2 Yang–Mills (YM)
theory are found. The final conformal transformations for the YM potentials and the
formulae of generating new solutions from known ones are presented.

The field equations of the SU2 invariant YM theory are formidable system of
partial differential equations yet possessing a wide symmetry group of local transfor-
mations. This group is known to be the 15-parametrical conformal group C1,3 and the
gauge group SU2 . Recently [1] it was shown that SU2 YM equations do not allow
any other Lie symmetries.
By now the interest in classical YM theory has become so widespread that many
workers are involved in the seach for new solutions. A comprehensive review of the
known exact solutions of SU2 YM theory is presented in [2]. These solutions are
obtained by introducing various ans?tze for the YM potentials. But the very question
a
of finding relevant ans?tze is rather obscure, although there can be no doubt that
a
successes achieved are connected with the symmetry properties of YM equations.
In this note we exploit the symmetry of SU2 YM equations to obtain some new
exact solutions. Besides that we present the final conformal transformations for the
YM potentials and construct formulae allowing us to generate “new” solutions of the
equations, starting from an “old” known one.
The SU2 -invariaut YM equations have the form

? ? Ga = e?abc g ?? Gb W? ,
c
(1)
a, b = 1, 2, 3, µ, ? = 0, 1, 2, 3,
µ? µ?

where

Ga = ?µ W? ? ?? W? + e?abc Wµ W? , g?? = (1, ?1, ?1, ?1)???
a a b c
µ?

or at greater length
2W? ? ?? (? µ Wµ ) + e?abc (? µ Wµ )W? ? g ?µ W? (?? Wµ ) + 2Wµ (? µ W? ) +
a a b c b c b c
(2)
+e2 Wµ g µ? (W? W? ? W? W? ) = 0.
b b a b a

As was previously mentioned these equations are invariant under the conformal group
C1,3 the special conformal transformations having the form

xµ ? cµ x2
?(x) = 1 ? 2cx + c2 x2 ,
xµ = ,
?(x)
(3)
cx ? c? x? , c2 ? c? c? , x2 ? x? x? ,
Wµ (x ) = ?(x)?µ + 2 xµ c? ? cµ c? + 2cxcµ x? ? c2 xµ x? ? x2 cµ c?
a ? 2
W? (x).
Lettere al Nuovo Cimento, 1983, 38, ¹ 2, P. 37–40.
332 W.I. Fushchych, W.M. Shtelen

One can directly verify that (3) leaves eq. (2) invariant. Expressions (3) can be
obtained by solving Lie equations for the infinitesimal generator of conformal trans-
formations
Qconf = 2cxx? ? x2 c? + 2(cxI4 + Sµ? cµ x? ) ? I3 , (4)
where I3 , I4 are unit matrices 3 ? 3 and 4 ? 4, respectively, ? = {?/?x? }, c? are
constants, Sµ? = ?S?µ are (4 ? 4)-matrices realizing the D 1 , 1 representation of
22
the SO1,2 algebra
? ? ? ? ? ?
0100 0010 0001
?1 0 0 0? ? ? ? ?
? , S02 = ? 0 0 0 0 ? , S03 = ? 0 0 0 0 ? ,
S01 = ? ?0 0 0 0? ?1 0 0 0? ?0 0 0 0?
0000 0000 1000
? ? ? ? ? ? (5)
00 0 0 000 0 0 0 00
? 0 0 ?1 0 ? ? ? ? ?
? , S23 = ? 0 0 0 0 ? , S31 = ? 0 0 0 1 ? .
S12 = ? ?0 1 0 0? ? 0 0 0 ?1 ? ?0 0 0 0?
0 ?1 0 0
00 0 0 001 0
The general form of the operator from the C1,3 invariance group of eq. (2) is
Q = ? µ (x)?µ + ?(x) ? I3 , (6)
where ? µ (x) are scalar functions, ?(x) denotes a (4 ? 4)-matrix. Following [3, 4] we
set for the solutions to be found
W? (x) = a? (x)?a (?),
a
(7)
? ?
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