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IV IV
Jab ? ? J0a ?
2
Замечание 2. Алгебра Пуанкаре, заданная интегродифференциальными операто-
рами (21), порождает нелоренцовские преобразования для координат и времени.
Действительно,

t = exp iJ0a ?a t exp ?iJ0a ?a = t,
IV IV


т.е. время не меняется при переходе от одной инерциальной системы к другой.
Это означает, что релятивистские уравнения движения допускают, помимо лорен-
цовских преобразований, при которых время изменяется, еще и преобразование
с абсолютным временем, т.е. релятивистские уравнения допускают двойственную
инвариантность. Более подробно см. [3, 5].
Замечание 3. Нетривиальная а.и. уравнения (1) может быть установлена с помо-
?
щью диагонализации и в том случае, когда L(x, p) — абстрактный самосопряжен-
ный оператор, обладающий кратным (непростым) спектром. Чем больше кратность
?
спектра оператора L(x, p), тем шире а.и. уравнения (1).
О новом методе исследования групповых свойств уравнений 5

1. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978.
2. Фущич В.И., Теор. и мат. физ., 1971, 7, № 1, 3; препринт Ин-та теоретич. физики АН УССР,
№ 70-32, Киев, 1970.
3. Fushchych W.I., Lett. Nuovo Cimento, 1974, 11, № 10, 508;
Фущич В.И., ДАН, 1976, 230, № 3, 570.
4. Фущич В.И., Сегеда Ю.Н., Укр. мат. журн., 1976, 26, № 6, 836;
Fushchych W.I., Nikitin A.G., Lett. Math. Phys., 1978, 2, 471.
5. Фущич В.И., В кн.: Теоретико-групповые методы в математической физике, Киев, 1978.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 6–10.

On the new invariance group
of Maxwell equations
W.I. FUSHCHYCH, A.G. NIKITIN

It was Heaviside [1] who first called attention to the invariance of Maxwell equa-
tions under the transformations
E > ±H, H > ±E.
Then Larmor [2] and Rainich [3] generalized this fact and demonstrated that Maxwell
equations well invariant under the one-parametrical group of transformations of a kind
E > E cos ? + H sin ?,
H > H cos ? ? E sin ?.
It is well-known also, that Maxwell equations are invariant under 15-parametrical
conformal group C4 , which includes 10-parametrical Poincar? group and 5-parametri-
e
cal conformal transformations [4].
Within the framework of Lie approach [5] the 16-parametrical group of the above-
mentioned transformations is maximally extensive symmetry group of Maxwell equa-
tions.
It was demonstrated in works [6, 7] that all relativistic motion equations possess
an additional (non obvious) invariance which in principle could not be found by the
classical Lie method. Specifically in [7] there was formulated the theorem about the
additional invariance of Maxwell equations
?E ?H
p ? H = ?i p?E =i
, ,
?t ?t
(1)
?
p · H = 0, p · E = 0, pa = ?i ,
?xa
under the group U2 ? U2 . This group is generated not by the local co-ordinate
transformations, but by the transformations of vectors of electric and magnetic fi-
elds of the kind
?E ?H ? 2 E ?2H
E>E =f E, H, , , , ,... ,
?xa ?xa ?xa ?xb ?xa ?xb
(2)
?E ?H ? 2 E ?2H
H > H = g E, H, , , , ,... ,
?xa ?xa ?xa ?xb ?xa ?xb
where the functions f and g in general depend on any-order derivatives of E and H
and on eight arbitrary parameters. However the explicit form of the functions f and g
had not been found in [7].
The aim of the prevent work is to find the explicit form of the transformations (2),
by which eqs. (1) remain invariant, and which form the representation of the group
Lettere al Nuovo Cimento, 1979, 24, № 7, P. 220–224.
On the new invariance group of Maxwell equations 7

U2 ? U2 . The result will be obtained with the help of non-Lie method for investigation
of the group properties of differential equations, which has been proposed and develo-
ped in [6–10].
We shall prove the following assertion:
Theorem 1. The Maxwell equations (1) are invariant with respect to the eight-
parametrical transformations

Ha > Ha cos ?1 + iDab Eb sin ?1 ,
(3a)
Ea > Ea cos ?1 + iDab Hb sin ?1 ,

Ha > Ha cos ?2 + i?abc pb Dcd Ed sin ?2 ,
?
(3b)
Ea > Ea cos ?2 + i?abc pb Dcd Hd sin ?2 ,
?

Ha > Ha cos ?3 ? ?abc pb Hc sin ?3 ,
?
(3c)
Ea > Ea cos ?3 ? ?abc pb Ec sin ?3 ,
?

Ha > Ha cos ?4 ? i?abc pb Dcd Hd sin ?4 ,
?
(3d)
Ea > Ea cos ?4 + i?abc pb Dcd Ed sin ?4 ,
?

Ha > Ha cos ?5 + Dab Hb sin ?5 ,
(3e)
Ea > Ea cos ?5 ? Dab Eb sin ?5 ,

Ha > Ha cos ?6 + Ea sin ?6 ,
(3f)
Ea > Ea cos ?6 ? Ha sin ?6 ,

Ha > Ha cos ?7 + i?abc pb Ec sin ?7 ,
?
(3g)
Ea > Ea cos ?7 ? i?abc pb Hc sin ?7 ,
?

Ha > Ha exp[i?8 ], Ea > Ea exp[i?8 ], (3h)

where
p2 p2 + p2 p2 ? p2 p2 ?ad + p1 p2 p3 (pb ?cd + pc ?bd ? pa pd ) L?1 ,
Dad = ?
ab ac bc
1 2 2 2
L = v p4 p2 ? p2 + p4 p2 ? p2 + p4 p2 ? p2 ,
1 2 3 2 3 1 3 1 2
2
pa = p0 /p, (a, b, c) is the cycle (1, 2, 3), ?k are arbitrary real parameters,
?
1/2
p = p2 + p2 + p2 .
1 2 3

Proof. The invariance of eqs. (1) under the trivial phase transformations (3h) is
obvious. It is not difficult to make sure by the direct verification, that eqs. (1) are
invariant also under the rest of the transformations (3a)–(3g), and that the infinitesi-
mal operators of transformations (3) satisfy Lie algebra of the group U2 ? U2 . But
such a way is too cumbersome. More constructive approach, which gives the method
for finding the explicit form of transformations (3) consists in reduction of the eqs. (1)
to such a form, for which the theorem statements become obvious.
8 W.I. Fushchych, A.G. Nikitin

Let us write eqs. (1) in the form [11]
?
? ?2 S · p,
L1 ? = 0, L1 = i
?t (4)
L2 = ?2 S · p,
L2 ? = 0,

where
? ?
H ? ?I ?
0 Sa 0
?= , ?2 = , Sa = ,
E ?0 ?
I? ?
0 Sa
? ? ? ? ? ? (5)
0 ?i
00 0 0 0 i 0
?1 = ? 0 0 ?i ? , ?2 = ? 0 0 ?, ?3 = ? i 0 0 ?,
0
S S S
?i
0i 0 0 0 00 0
?
I and ? are the three-row square unit and zero matrices.
0
Following the main algorithm of non-Lie approach of investigation of differential-
equation group properties [6–10], let us transform the eqs. (4) to the canonical
quasi-diagonal form. Using for this purpose the operator
? S a pa
? p?
W = exp (?2 ? 1)DS · p exp ?i arctg (6)
? ,
4 p? p1 + p2 + p3
where
1/2
pa = pb ? pc , p = p 2 + p2 + p2
? ? ?1 ?2 ?3 ,

p2 p2 + p 2 p2 ? p 2 p2 1 ? Sa +
2
D= ab ac bc
(7)
a=b=c


L?1 ,
2
+p1 p2 p3 Sa Sb pc ] ? pp1 p2 p3 1 ? (S · p)
?

and taking into account the relations
2
DS · p = ?S · pD, D (S · p) = D, S · pD2 = S · p (8)
?

one obtains from (5) the following equations:
?
L1 = W L1 W ?1 = i ? ?p,
L1 ? = 0, ? = W ?,
?t
(9)
1
?1
? = v (S1 + S2 + S3 ),
L2 ? = 0, L2 = W L2 W = ?p,
3
Equations (9) by definition are invariant with respect to the transformations ? >
Qk ? if an operator Qk satisfies the conditions

(10)
[L1 , Qk ]? = [L2 , Qk ]? = 0.

Conditions (10) are obviously satisfied by the matrices

Qa+3 = ?a ?2 , (11)
Qa = ?a ?, Q7 = ?, Q8 = I, a = 1, 2, 3.
On the new invariance group of Maxwell equations 9

The operators (11) satisfy the commutation relations
[Qa , Qb ] = [Qa+3 , Q3+b ] = i?abc Qc ,
(12)
[Qb , Q3+b ] = i?abc Q3+c , [Q7 , Qk ] = [Q8 , Qk ] = 0,
i.e. form the Lie algebra of the group U2 ? U2 . Since (Qk ) ? = ? , the operators Qa ,
Q3+a the representation D 1 , 0 ? D 0, 1 of the group SU2 ? SU2 . It is not difficult
2 2
make certain to show, that formulae (11) give the complete set of numerical matrices,
satisfying the conditions (10).
It follows from the above, that eqs. (9) are invariant under an arbitrary transfor-
mation from the group U2 ? U2
? > exp [iQk ?k ] ? = (cos ?k + iQk sin ?k ) ? , (13)
where ?k is an arbitrary parameter, and Qk is any generator, given by formula (11)
(no sum over k!). Returning with the help of the operator (6) to the initial ?-
representation one obtains from (13) the transformations, under which eqs. (4) remain
invariant
? > (cos ?k + iQk sin ?k )?, (14)
where
Qk = W ?1 Qk W, Q2 = i?2 S · pD,
?
Q1 = ?1 D,
(15)
Q3 = S · p, Q3+a = ?2 S · p, Q7 = ?2 S · p,
? ? ? Q8 = I.
Substituting (15), (8), (6) into (14), one comes to formulae (3). The theorem is
proved.
So we have found new eight-parametrical symmetry group of Maxwell equations,
given by the transformations (3). These transformations are unitary with respect to
the scalar product

d3 x ?† ?2 . (16)
(?1 , ?2 ) = 1

Substituting expression (6) for the function ? into (16) and claiming for the E and H
to be Hermitian, one comes to the conclusion, that the transformations (3) preserve
the value
E= d3 x (E 2 + H 2 ),

which determines the energy of an electromagnetic field.
The main property of the transformations (3) is that they are carried out by

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