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If 0 < k < 1 (for (8)) or k > 1 (for (10)) then Tµ? has a non-integrable singularity
4 6
in the point xµ = ??µ , in other points of the Minkowsky space R(1, 3) expression (17)
being integrable. In the case k > 1 (for (9)) Tµ? has a singularity on the cone
4

ay = ±[(by)2 + (cy)2 ]1/2
while at other points it is integrable.
Ans?tze (3), (4) proved to be very useful while constructing solutions of the
a
system (2). We shall write down some of the families of exact solutions obtained,
omitting intermediate calculations.
(i) k1 > 1, k1 > 1
4

?(x) = ? ?1/2k2 {?(4k2 ? 1)1/2 (?iC1 + ?4 C2 ) + (C1 ? i?4 C2 ) ?
? [(?a)(ay) ? (?b)(by) ? (?c)(cy)]? ?1 }?, (18)
u(x) = E? ?1/k1 , ? = [(ay)2 ? (by)2 ? (cy)2 ]1/2
Ci = const,
and the following conditions hold:
?2
(1 ? k1 )k1 + {µ1 |E|k1 + µ2 (??)k2 [(C1 ? C2 )4k2 ]k2 }2 = 0,
2 2
?
±(4k2 ? 1)1/2 ? 2k2 {?2 |E|k1 + ?2 (??)k2 [4k2 (C1 ? C2 )]k2 }2 = 0.
2 2
?
(ii) k1 = 2/(m ? 1), k2 = 1/m, m = 2, 3
?(x)(1 + ?2 ? 2 )?(m+1)/2 ?[iC1 + ?4 C2 ? ?(C1 + i?4 C2 )] ?
(?a)(ay) ? (?b)(by) ? (?c)(cy), m = 2,
?
?y, m = 3, (19)
[(ay)2 ? (by)2 ? (cy)2 ]1/2 , m = 2,
u(x) = E(1 + ?2 ? 2 )(1?m)/2 , ?=
(yy)1/2 , m = 3,
where ?, Ci and E are constants satisfying conditions
?2 (m2 ? 1) = [µ1 |E|2/(m?1) + µ2 (??)1/m (C1 ? C2 )1/m ]2 ,
2 2
?
?(m + 1) = [?1 |E|2/(m?1) + ?2 (??)1/m (C1 ? C2 )1/m ].
2 2
?
In (18), (19) we have used notations of (14).
360 W.I. Fushchych, R.Z. Zhdanov

1. Heisenberg W., Z. Naturf. A, 1954, 9, 292.
2. Kortel F., Nuovo Cimento, 1956, 4, 210.
3. Akdeniz K.G., Smailagic A., Nuovo Cimento A, 1979, 51, 345.
4. Merwe P.T., Phys. Lett. B, 1981, 106, 485.
5. Kurdgelaidze D.F., Zh. Eksp. Teor. Fiz., 1959, 36, 842.
6. Fushchych W.I., Shtelen W.M., J. Phys. A: Math. Gen., 1983, 16, 271.
7. Fushchych W.I., Zhdanov R.Z., in Group Theoretical Studies of the Mathematical Physics Problems,
Kiev, Math. Inst., 1985, 20.
8. Fushchych W.I., Zhdanov R.Z., J. Phys. A: Math. Gen., 1987, 20, 4173.
9. G?rsey F., Nuovo Cimento, 1956, 3, 988.
u
W.I. Fushchych, Scientific Works 2001, Vol. 3, 361–366.

?
Non-local ansatze for the Dirac equation
W.I. FUSHCHYCH, R.Z. ZHDANOV
Using non-local (non-Lie) symmetry of the linear Dirac equation we have constructed a
number of new ans?tze reducing it to systems of ordinary differential equations.
a

It is well known (see e.g. [1]) that the Poincar? group P (1, 3) is a maximal local
e
(in Lie’s sense) invariance group of the linear Dirac equation

m = const, (1)
(i?µ ?µ + m)?(x) = 0,

where ? = ?(x0 , x) is a four-component spinor, ?µ ? ?/?xµ , µ = 0, 3 and ?µ are
imaginary 4 ? 4 matrices satisfying the Clifford algebra
?
? 1, µ = ? = 0,
?1, µ = ? = 1, 3,
?µ ?? + ?? ?µ = 2gµ? I ? 2I
?
0, µ = ?.

In [2, 3] ans?tze reducing the Dirac equation to systems of ordinary differential
a
equations (ODE) were constructed, the subgroup structure of the group P (1, 3) in-
vestigated in detail by Patera et al [4, 5] being used.
As shown in [1, 6, 7] equation (1) possesses non-local (non-Lie) symmetry. So far
this additional non-local symmetry has not been used to construct ansatze reducing
the Dirac equation to systems of ODE. In the present paper we construct a number
of such ans?tze following an approach suggested in [3, 8].
a
If one puts

?µ = diag(?i?µ , ?i?µ ), ?T = (Re ?, Im ?)T

then equation (1) becomes

(?µ ?µ ? m)?(x) = 0. (2)

It is common knowledge that the complete set of first-order symmetry operators
of the Dirac equation (2) is not a Lie algebra. We have succeeded in picking out the
subset which forms the Lie algebra of the Poincar? group:
e

Pµ = [1 + ?(?4 + ?5 )]? µ + ?m(?4 + ?5 )?µ , (3)

1
Jµ? = ?xµ ? ? + x? ? µ ? (?µ ?? ? ?? ?µ ), (4)
4

where ? = const, ? µ = g µ? ?? for µ, ? = 0, 3 and

0 0
?4 + ?5 = 2 .
?0 ?1 ?2 ?3 0

J. Phys. A: Math. Gen., 1988, 21, L1117–L1121.
362 W.I. Fushchych, R.Z. Zhdanov

It is important to note that operators (3) generate a non-local group ot transfor-
mations
? = [1 ? ?m(?4 + ?5 )?µ ?µ ]? + ?(?4 + ?5 )?µ ?xµ , (5)
xµ = xµ + ?µ ,
where ?µ are group parameters.
According to [4, 8] there exists a correspondence between three-dimensional sub-
algebras of the algebra (3) and (4) and ans?tze reducing the Dirac equation (2) to
a
ODE. Omitting very cumbersome intermediate calculations we write the final result
for the non-local ans?tze for the spinor field.
a
1. P0 + P3 , P1 , P2
1
?(x) = exp ?m?45 ?1 x1 + ?2 x2 + ??03 ?(?).
2
2. P1 , P2 , P3
?(x) = exp [?m?45 (?1 x1 + ?2 x2 + ?3 x3 )] ?(x0 ).
3. P0 , P1 , P2
?(x) = exp [?m?45 (?1 x1 + ?2 x2 ? ?0 x0 )] ?(x3 ).
4. J03 , P1 , P2
1
?(x) = exp [?m?45 (?1 x1 + ?2 x2 )] exp ? ?0 ?3 ln ? ? x2 ? x2 .
0 3
2
5. J03 , P0 + P3 , P1
1 1
exp ? ?0 ?3 ln ? ?(x2 ).
?(x) = exp ?m?45 ?1 x1 + ??03
2 2
6. J03 + ?P2 , P0 , P3
?(x) = exp [?m?45 (?3 x3 ? ?0 x0 )] ?
? exp ??45 ?2 + 2? ?0 ?3 (??45 ? 1) x2 ?(x1 ).
1


7. J03 + ?P2 , P0 + P3 , P1
1
?(x) = exp ?m?45 ?1 x1 + ??03 exp x2 m? 2 ?2 (1 + 2??45 ) ?
2
1
? ??2 ?03 ? 3??m??45 ? ?m? 2 ?2 ?45 ?0 ?3 ) ?(?).
2
8. J12 , P0 , P3
1 x1
?1 ?2 tan?1
?(x) = exp [?m?45 (?3 x3 ? ?0 x0 )] exp ? x2 + x2 .
1 2
2 x2
9. J12 + ?P0 , P1 , P2
?(x) = exp [?m?45 (?1 x1 + ?2 x2 )] ?
1
? exp ?1 ?2 (1 ? ??45 ) ? ?m?45 ?0 x0 ?(x3 ).
2?
Non-local ans?tze for the Dirac equation
a 363

10. J12 + ?P3 , P1 , P2
?(x) = exp [?m?45 (?1 x1 + ?2 x2 )] ?
1
? exp ?m?45 ?3 + (1 ? ??45 )?1 ?2 x2 ?(x0 ).
2?
11. J12 + P0 + P3 , P1 , P2
?(x) = exp [?m?45 (?1 x1 + ?2 x2 )] ?
1 1
? exp ? ?m?45 ?03 + (1 ? ??45 )?1 ?2 ?(?).
2 2
12. G1 , P0 + P3 , P2
1 x2
exp ?
?(x) = exp ?m?45 ?2 x2 + ??03 ?03 (?1 + ?mx2 ?45 ) ?(?).
2 2?
13. G1 , P0 + P3 , P1 + ?P2
1
?
?(x) = exp ?m?45 x1 (?1 + ??2 ) + ??03
2
?x1 ? x2 1
? exp ?1 ?03 ? ?m??45 (?1 + ??2 ) ?(?).
?? 2
14. G1 + P2 , P0 + P3 , P1
1
?
?(x) = exp ?m?45 x1 ?1 + ??03
2
1
? exp x2 ?m?45 (?2 ? ??1 ) + (??45 ? 1)?03 ?1 ?(?).
2
15. G1 + P0 , P0 + P3 , P2
1
?
?(x) = exp ?m?45 x2 ?2 + ??03
2
? exp [mx1 (?1 + ??03 + 3??1 ?45 ? 4???45 ?03 )] ?(?).
16. G1 + P0 , P0 + P3 , P1
1
?
?(x) = exp ?m?45 x1 ?1 + ??03
2
? exp [m?2 x2 (3??45 + ???45 ?03 ?1 ? 1)] ?(?).
17. G1 + P0 , P1 + ?P2 , P0 + P3
1 x2
?
?(x) = exp ?m?45 ??03 + (?1 + ??2 )
2 ?
m
? exp (1 ? 2??45 )[(?2 ? ??1 ) ? ???03 + ?(??1 ? ?2 )?45 ] ?
?1 +1
1
? ??45 ?0 ?3 + ?1 ?2 + ?03 ?1 ? 1 ? 1 (?x1 ? x2 ) ?(?).
?
364 W.I. Fushchych, R.Z. Zhdanov

18. J03 + ?J12 , P0 , P3

?(x) = exp [?m?45 (?3 x3 ? ?0 x0 )] ?
1 x1
(?0 ?3 + ??1 ?2 ) tan?1
? exp ? x2 + x2 .
1 2
2? x2
19. J03 + ?J12 , P1 , P2
1
?(x) = exp [?m?45 (?1 x1 + ?2 x2 )] exp ? (?0 ?3 + ??1 ?2 ) ln ? ? x2 ? x2 .
0 3
2
20. G1 , G2 , P0 + P3
1
?m??45 ?03 ?
?(x) = exp
2
1m
? exp ? ?03 ? x2 + x2 ?45 + ?1 x1 + ?2 x2 ?(?).
1 2
2?
21. G1 + P2 , G2 + ?P1 + ?P2 , P0 + P3
1
?(x) = exp ?m??45 ?03 [f1 + ??45 (g1 ?1 + g2 ?2 + g3 ?03 ) +
2
+ ?03 (h1 ?1 + h2 ?2 ) + ?u?45 ?03 ?1 ?2 ]?(?),

where
?m m
(?x2 ? x1 ), [?x1 + (?? ? ?)x2 ],
f1 = 1, g1 = g2 =
? ?
1 1 m
[?x2 ? (? + ?)x1 ], h2 = (x1 ? ?x2 ), g3 = ? (? + 1)x1 x2 ,
h1 =
2? 2? 2?
m
u = ? (?x1 + ?x2 + ?x1 x2 ), ? = ?(? + ?) ? ?.
2 2

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