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Jkl = xk ?xl + xl ?xk + E k ?E l + E l ?E k + H k ?H l + H l ?H k , (29)

G(1) = x0 ?xa + ?E a + ?H a , (30)
a

G(2) = xa ?x0 ? E a E k ?E k ? H a H k ?H k , (31)
a

D0 = x0 ?x0 ? E l ?E l ? H l ?H l , (32)

D1 = x1 ?x1 + E 1 ?E 1 + H 1 ?H 1 , (33)

D2 = x2 ?x2 + E 2 ?E 2 + H 2 ?H 2 , (34)

D3 = x3 ?x3 + E 3 ?E 3 + H 3 ?H 3 . (35)
276 W.I. Fushchych, I.M. Tsyfra, V.M. Boyko

Proof. To prove theorem 1 we use Lie’s algorithm. The condition of invariance of the
system L(E, H), i.e. (25), with respect to operator X has the form

(36)
XL = 0,
L=0
1

where

X = X + [D? (? l ) ? Ej D? (? j )]?E? + [D? (? l ) ? Hj D? (? j )]?H? ,
l l
l l
1
?E l ?H l
l l
E? = , H? = , l = 1, 2, 3; ? = 0, 1, 2, 3
?x? ?x?

is the prolonged operator. From the invariance condition (36) we obtain the system of
equations which determine the coefficient functions ? µ , ? l , ? l of the operator (26):
µ µ
l l l l µ
?k = 0, ?0 = 0, ?k = 0, ?0 = 0, ??? = 0, ?E a = 0, ?H a = 0,
? k = ?E k ?0 + ?0 + E a ?a ? E a E k ?a ,
0 k k 0
(37)
? k = ?H k ?0 + ?0 + H a ?a ? H a H k ?a ,
0 k k 0

where

?? l ?? l ?? µ ?2?µ
µ
l l µ
?k = , ?0 = , ?E a = , ??? = .
?E a
?xk ?x0 ?x? ?x?

Having found the general solution of system (37), we get the explicit form of all
the linear independent symmetry operators of system (25), which have the structure
(27)–(35). Operators of Lorentz rotations J0k is given by the linear combination of
(1) (2)
the Galilean operators Gk and Gk :

(1) (2)
(38)
J0k = Gk + Gk .

All the following statements, given here without proofs, can be proved in analogy
with the above-mentioned scheme.

4. The finite transformations and invariants
We present some finite transformations which are generated by the operators J0k :

J01 : x0 > x0 = x0 ch ?1 + x1 sh ?1 ,
x1 > x1 = x1 ch ?1 + x0 sh ?1 , (39)
x2 > x2 = x2 , x3 > x3 = x3 ,

E 1 ch ?1 + sh ?1 H 1 ch ?1 + sh ?1
E >E H >H
1 1 1 1
=1 , =1 ,
E sh ?1 + ch ?1 H sh ?1 + ch ?1
E2 H2
E2 > E2 H2 > H2 (40)
=1 , =1 ,
E sh ?1 + ch ?1 H sh ?1 + ch ?1
E3 H3
E3 > E3 H3 > H3
=1 , =1 .
E sh ?1 + ch ?1 H sh ?1 + ch ?1
Nonlinear representations for Poincar? and Galilei algebras
e 277

The operators J02 , J03 generate analogous transformations. ?1 is the real group
(2)
parameter of the geometric Lorentz transformation. Operators Gk generate the
following transformations:
(2)
G1 : x0 > x0 = x0 + ?1 x1 , xk > xk = xk ,
Ek Hk
Ek > Ek = , Hk > Hk = .
1 + ?1 E 1 1 + ?1 H 1
(2) (2) (1)
Analogous transformations are generated by the operators G2 , G3 . Operators Gk
generate the following transformations:
(1)
G1 : x0 > x0 = x0 , x1 > x1 = x1 + x0 ?1 ,
x2 > x2 = x2 , x3 > x3 = x3 ,
E 1 > E 1 = E 1 + ?1 , H 1 > H 1 = H 1 + ?1 ,
E2 > E2 = E2, E3 > E3 = E3,
H 2 > H 2 = H 2, H 3 > H 3 = H 3.
(1) (1)
The operators G2 , G3 generate analogous transformations.
It is easy to verify that
2
1 ? EH
E 2 = 1, H2 = 1 (41)
I1 = ,
1 ? E2 1 ? H2

is invariant with respect to the nonlinear transformations of the Poincar? group which
e
are generated by representations (28), (38).
The invariant of the Galilei group which is generated by representations (28), (31)
has the form:
E2H 2
(42)
I2 = 2,
EH

whereas the Galilei group which is generated by representations (28), (30) has the
invariant
I3 = (E ? H)2 . (43)

5. Complex Euler equation for the electromagnetic field
Let us consider the system of equations
??k k
l ??
?k = E k + iH k . (44)
+? = 0,
?x0 ?xl
The complex system (44) is equivalent to the real system of equations for E and H
?E k k k
l ?E l ?H
?H
+E = 0,
?x0 ?xl ?xl
(45)
?H k k k
l ?E l ?H
+H +E = 0.
?x0 ?xl ?xl
278 W.I. Fushchych, I.M. Tsyfra, V.M. Boyko

The following statement has been proved with the help of Lie’s algorithm.
Theorem 2. The maximal invariance algebra of the system (45) is the 24-dimensio-
nal Lie algebra whose basis elements are given by the formulas

Pµ = ?xµ ,
(1)
Jkl = xk ?xl ? xl ?xk + E k ?E l ? E l ?E k + H k ?H l ? H l ?H k ,
(2)
Jkl = xk ?xl + xl ?xk + E k ?E l + E l ?E k + H k ?H l + H l ?H k ,
(1)
Ga = x0 ?xa + ?E a ,
(2)
Ga = xa ?x0 ? (E a E k ? H a H k )?E a ? (E a H k + H a E k )?H k , (46)
D0 = x0 ?x0 ? E k ?E k ? H k ?H k ,
Da = xa ?xa + E a ?E a + H a ?H a (no sum over a),
K0 = x2 ?x0 + x0 xk ?xk + (xk ? x0 E k )?E k ? x0 H k ?H k ,
0
Ka = x0 xa ?x0 + xa xk ?xk + [xk E a ? x0 (E a E k ? H a H k )]?E k +
+ [xk H a ? x0 (H a E k + E a H k )]?H k .

The algebra, engendered by the operators (46), include the Galilei algebras
AG(1) (1, 3), AG(2) (1, 3) and Poincar? algebra AP (1, 3), and conformal algebra
e
(2)
AC(1, 3) as subalgebras. Operators Ga generate the linear geometrical transforma-
tions in R(1, 3)

x0 > x0 = x0 + ?a xa xl > xl ,
(no sum over a), (47)

as well as the nonlinear transformations of the fields
E l + iH l
E + iH > E + iH =
l l l l
(no sum over a),
1 + ?a (E a + iH a )
(48)
E l ? iH l
E l ? iH l > E l ? iH l = .
1 + ?a (E a ? iH a )

The invariant of the group G(2) (1, 3) is
2
E2 ? H 2 + 4 EH
(49)
I4 = .
2
E2 H2
+

Operators J0k generate the linear transformations in R(1, 3)

x0 > x0 = x0 ch ?k + x0 sh ?k ,
xk > xk = xk ch ?k + x0 sh ?k (no sum over k), (50)
xl > xl = xl , if l = k,

as well as the nonlinear transformations of the fields
(E k + iH k ) ch ?k + sh ?k
E + iH > E
k k k k
+ iH = ,
(E k + iH k ) sh ?k + ch ?k
(E k ? iH k ) ch ?k + sh ?k
E k ? iH k > E k ? iH k = .
(E k ? iH k ) sh ?k + ch ?k
Nonlinear representations for Poincar? and Galilei algebras
e 279

If l = k, then

E l + iH l
E l + iH l > E l + iH l = ,
(E k + iH k ) sh ?k + ch ?k
(51)
E l ? iH l
E ? iH > E ? iH =
l l l l
(no sum over k).
(E k ? iH k ) sh ?k + ch ?k

The invariant of group P (1, 3) is

1 ? 2 (E 2 ? H 2 ) ? 1 (E 2 ? H 2 )2 ? 2(E H)2
2
E 2 + H 2 = 1. (52)
I5 = ,
2
1? (E 2 H 2)
+

The operator K0 generates the following nonlinear transformations in R(1, 3) and
linear transformations of the fields

xµ > xµ = ,
1 ? ?0 x0
(53)
E k > E k = E k + ?0 (xk ? x0 E k ),
H k > H k = H k (1 ? ?0 x0 ).

The operators Ka generate nonlinear transformations in both R(1, 3) and of the fields
x0 xa
x0 > x0 = xa > xa =
, .
1 ? xa ?a 1 ? xa ?a
If k = a, then
xk
xk > xk = ,
1 ? xa ?a
E a + iH a
E + iH > E
a a a a
+ iH = ,
1 + ?a [x0 (E a + iH a ) ? xa ]
E a ? iH a
E a ? iH a > E a ? iH a = .
1 + ?a [x0 (E a ? iH a ) ? xa ]
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