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AP (2) (1, 3) = Pµ , Jµ? = Jµ? + Qµ? , (3.10)
(1)
?
where Pµ and Jµ? are given in (3.3),
?
? Qµ? 08
?
?0 if (µ?) = (0, 1), (0, 2), (1, 2) ,
,
Qµ?
8
? (3.11)
Qµ? =
? 08 Qµ?
?
? if (µ?) = (0, 3), (1, 3), (2, 3)
,
Qµ? 08
and matrices 8 ? 8 Qµ? are given in (2.22). Invariance of (3.6) with respect to
AP (2) (1, 3) (3.10) means that (3.6) describes not only spinor particles(fermionic fi-
elds) but also a coupled system of vector and scalar particles (bosonic fields).
Now consider the following two sets of symmetry operators of equation (3.6):
?(i) ? ?
SA(i) = Pµ , Jµ? , ?4 , I; Qµ? , (3.12)
i = 1, 2,
where
08 ?4
? (3.13)
?4 = ,
?4 08
?4 is givel in (3.4). These sets of operators form Lie algebras as well as superalgebras.
Superalgebras (3.12) are isomorphic to those from (2.25). The isomorphism is achieved
by means of the transformations
? ? ?
Pµ > Pµ , Jµ? > Jµ? , ?4 > ?4 , I > I, Qµ? > Qµ? .
(i)
(3.14)
In conclusion of this section let us consider a nonlinear generalization of (3.6)
possessing dual Poincar? invariance.
e
Theorem 4. The equation
? ?? ??
[i?µ ?µ ? F (??, ?M ?)]? = 0, (3.15)
?
where ? is defined in (3.5),
08 I8
? ? ?
? = row (?? ?T ?+ ?T ), (3.16)
M=
? + I8 08
On the connection between solutions of Dirac and Maxwell equations 329

and F is an arbitrary smooth function, is invariant under the two Poincar? algebras
e
(3.3) and (3.10).
?
Proof. One can make sure that the operator i?µ ?µ commutes with all generators
?? ? ?
of the considered Poincar? algebras. Further, the quantities ??, ?M ? are absolute
e
invariants of these Poincar? algebras. Thus, the theorem is proved.
e
It will be noted that
?? ? ? ?? ? ? (3.17)
?? = 2(?? ?? + ?+ ?+ ), ?M ? = 2(?? ?+ + ?+ ?? ),
?
where ?? , ?+ are four-component functions, ?± = (?± )+ ?0 .
4. P (i) (1, 3)-inequivalent ans?tze, reduction and solutions
a
of nonlinear Dirac equations
The nonlinear equation (3.15), as we have shown, is dual Poincar? invariant and
e
therefore it unites fermionic and bosonic fields. Such unification opens new ways to
solve the general problem of unification forces and fields.
It is important to find exact solutions of (3.15). Of course, we shall be looking
for classical solutions, but these solutions may be very useful as basic ones in the
corresponding quantum theory. It is to be emphasized that the standard procedure
of quantization, when the complete set of solutions of a given equation is quantized
according to bosonic or fermionic rules, may be misleading because our equation
may have bosonic and fermionic subsets of solutions simultaneously (the simplest
example is the massless Dirac equation considered in section 2). Therefore, it is more
preferable to quantize separate families of solutions, having established beforehand
what representation of the Poincar? algebra is realized on them.
e
To fing exact solutions of equations of the (3.15) we construct P (i) (1, 3)-inequiva-
lent ans?tze of codimension 1. These ans?tze reduce a given equation to ODEs. The
a a
general form of such an anzatz is
? (4.1)
?(x) = A(x)?(?),
where A(x) is 16 ? 16 matrix, ? is 16-component function (column) depending on the
new variable ?. Matrix A(x) and the new independent variable ? are determined from
the equations [3]
Qk A(x) ? (?k (x)?? + ?k (x))A(x) = 0,
?
(4.2)
?
?k (x)?? ?(x) = 0, k = 1, 2, 3,
where Q1 , Q2 , Q3 is a three-dimensional subalgebra of AP (1, 3). The full description
of subalgebras of AP (1, 3) is given in [11] and [7]. Fushchych and Shtelen [4]) (see
also [6]) have used one-dimensional subalgebras of AP (1, 3) to construct ans?tze of
a
codimension 3 for the Dirac spinor field. Ans?tze of codimension 1 for the Dirac spinor
a
field are fully described in [5]. We present the complete set of P (i) (1, 3)-inequivalent
ans?tze of codimension 1 for a 16-component field (3.5) in table 1. Basis elements of
a
AP (i) (1, 3) are given in (3.3) and (3.10).
In table 1 ? and ? are arbitrary non-zero constants,
(i) (i) (i) ?(i) ?(i)
Gk = ?0k + j3k = (x0 + x3 )Pk + xk (P0 ? P3 ) + S0k + S3k , (4.3)
j
?(1) ?(2) ?(1) ?
Sµ? are given in (3.4) and Sµ? = Sµ? + Qµ? see (3.10) and (3.11).
330 W.I. Fushchych, W.M. Shtelen, S.V. Spichak

Table 1. P (i) (1, 3)-inequivalent ans?tze (4.1) of codimension 1 for field (3.5).
a

N Algebra A(x) ?

1
1 P 0 , P1 , P2 x3
1
2 P1 , P2 , P 3 x0
3 P0 ? P3 , P1 , P2 1 x0 + x3
?(i) ?(i) x2 ? x2
exp[?S03 ln(x0 + x3 )]
4 J03 , P1 , P2 0 3
?(i) ?(i)
J03 , P1 , P0 ? P3 exp[?S03 ln(x0 + x3 )]
5 x2
?(i) ?(i)
exp ? x2 S03
J03 + ?P2 , P0 , P3
6 x1
?

?(i) ?(i)
exp ? x2 S03
7 J03 + ?P2 , P0 ? P3 , P1 ? ln(x0 + x3 ) ? x2
?

?(i) ?(i)
exp S12 tan?1 x2 + x2
x1
8 J12 , P0 , P3 1 2
x2

?(i) x0 ?(i)
9 J03 ? ?P0 , P1 , P2 exp S x3
? 12

?(i) (i)
?
exp ? x3 S12
10 J12 + ?P3 , P1 , P2 x0
?

?(i) (i)
?
1
11 J12 ? P0 + P3 , P1 , P2 exp ? 2 (x3 ? x0 )S12 x0 + x3
(i) (i) (i)
?
x1
12 G1 , P0 ? P3 , P2 exp ? x (S01 + S31 ) x0 + x3
0 +x3
(i) ?(i) (i)
x2 ??x1
13 G1 , P0 ? P3 , P1 + ?P2 exp (S01 + S31 ) x0 + x3
?(x0 +x3 )
(i) ?(i) (i)
14 G1 + P2 , P1 , P0 ? P3 exp ?x2 (S01 + S31 ) x0 + x3
(i) ?(i) ?(i) 2x1 + (x0 + x3 )2
15 G1 ? P0 , P2 , P0 ? P3 exp (x0 + x3 )(S01 + S31 )
(i) ?(i) ?(i) 2(x2 ? ?x1 ) ? ?(x0 + x3 )2
16 G1 ? P0 , P0 ? P3 , exp (x0 + x3 )(S01 + S31 )
P1 + ?P2
?(i) ?(i) 1 ?(i) ?(i)
+ ?S12 ) tan?1 x2 + x2
x1
17 J03 + ?J12 , P0 , P3 exp (S03 1 2
? x2

?(i) ?(i) ?(i) ?(i) x2 ? x2
18 J03 + ?J12 , P1 , P2 exp ?(S03 + ?S12 ) ln(x0 + x3 ) 0 3

(i) (i) ?(i) ?(i)
1
19 G1 , G2 , P0 ? P3 exp ? [x (S + S31 ) + x0 + x3
x0 +x3 1 01

?(i) ?(i)
+ x2 (S02 + S32 )]
(i) (i) ?(i) ?(i)
?x2 ?x1 (x0 +x3 +?)
20 G1 + P2 , G2 + ?P1 + exp (S01 + S31 ) + x0 + x3
(x0 +x3 )(x0 +x3 +?)??

?(i) ?(i)
x1 ?x2 (x0 +x3 )
+?P2 , P0 ? P3 + (S01 + S32 )
(x0 +x3 )(x0 +x3 +?)??

(i) (i) ?(i) ?(i)
x1
21 G1 , G2 + P1 + ?P2 , exp ? (S01 + S31 ) ? x0 + x3
x0 +x3

?(i) ?(i)
x2
P0 ? P3 ? (S02 + S32 ) +
x0 +x3 +?

?(i) ?(i)
x2
+ (S01 + S31 )
(x0 +x3 )(x0 +x3 +?)

(i) (i) ?(i) (i)
?
x1
22 G1 , G2 + P2 , P0 ? P3 , exp ? (S01 + S31 ) ? x0 + x3
x0 +x3

?(i) (i)
?
x2
? (S02 + S32 )
x0 +x3 +1
(i) ?(i) ?(i) ?(i) x2 ? x2 ? x2
x1
exp ? x (S01 + S31 ) ?
23 G1 , J03 , P2 0 1 3
0 +x3

?(i)
? exp[?S3 ln(x0 + x3 )]
On the connection between solutions of Dirac and Maxwell equations 331

Table 1. (continued)

N Algebra A(x) ?
(i) ? (i) ? ln(x0 +x3 )?x1 ?(i) ?(i)
24 J03 + ?P1 + ?P2 , G1 , P0 ? P3 exp (S01 + S31 ) ? x2 ? ? ln(x0 + x3 )
x0 +x3

?(i)
? exp[?S03 ln(x0 + x3 )]

?(i) (i) (i) ?(i) ?(i)
1
25 J12 ? P0 + P3 , G1 , G2 exp ? [x (S + S31 ) + x0 + x3
x0 +x3 1 01

?(i) ?(i)
+ x2 (S02 + S32 ) ?
?(i)
? exp (S12
x·x
2(x0 +x3 )

?(i) ?(i) (i) (i) ?(i) ?(i)
1
26 J03 + ?J12 , G1 , G2 exp ? [x (S + S31 ) + x·x
x0 +x3 1 01

?(i) ?(i)
+ x2 (S02 + S32 )] ?

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