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О дополнительной инвариантности уравнений для векторных полей 163

Теорема 2. Калибровка Кулона эквивалентна калибровке Лоренца. Эквивален-
тность устанавливается с помощью интегродифференциального оператора
1
W (p) = 2?1/2 p0 p?2 p0 I ? Sp ? [p(S ? N ) ? p(N ? S)] ,
2
(21)
1
W ?1 (p) = 2?1/2 p0 p?2 p0 I + Sp + [p(S ? N ) ? p(N ? S)] ,
2
L2 (p)W (p)? = 2?1/2 L1 (p)?,
? ? (22)
где ? = W ?1 (p)?.
?
Теорема 3. Система уравнений (19) инвариантна относительно 9-мерной ал-
гебр Ли группы GL(3), базисные элементы которой имеют вид
Dab = 2?1 p0 p?2 p0 2Sb Sa ? S 2 ?ab + pa (N ? S)b .
? (23)
Доказательство. Доказательство этой теоремы не отличается от доказательства
теоремы 1. Для получения формулы (23) нужно использовать вместо U (p) оператор
v 2
Sp
?
W1 (p) = exp (ln 2) 1 +
|p|
(24)
? ?1
? · 2 p0 p?2 [p(S ? N ) ? p(N ? S)] .
4
Теорема 4. Система (7) инвариантна относительно конформной алгебры, ба-
зисные элементы которой задаются интегродифференциальными оператора-
ми
?
D = xµ pµ + i,
pµ = i µ ,
?x
Jab = xa pb ? xb pa + 2?1 ip0 p?2 [(N ? S)a pb ? (N ? S)b pa ],
J = x p ? x p + ip (2p)?2 {2p (N ? S) ? [p(N ? S)][p(S ? N )p }, (25)
0k 0k k0 0 0 k k

K? = 2x? D ? x? x? p? + 2?1 ip0 p?2 {[p(N ? S)][p(S ? N )](x0 p? ? Dg? )?
0


?2(N ? S)k (g? D ? xk p? )},
k
a, b, k = 1, 2, 3, ?, ? = 0, 1, 2, 3.
Доказательство. Воспользовавшись оператором (24), получим каноническую си-
стему незацепляющихся интегродифференциальных уравнений для вектор-фун-
µ
?1
? ?
кции (?)µ = W1 (p)? , причем компоненты (?)j , j = 1, 2, 3, удовлетворяют
уравнению Даламбера без дополнительных условий. Представление конформной
алгебры в канонической форме можно реализовать с помощью операторов
? ?
Jµ? = (xµ p? ? x? pµ )?3 ,
pµ = i
? ?3 ,
?xµ (26)
? ? ?
K? = (2x? D ? x? x? p? )?3 ,
D = (xµ pµ + i)?3 ,
где ?3 — матрица вида
? ?
000 0
?0 1 0 0?
?3 = ? ?.
?0 0 1 0?
000 1
164 В.И. Фущич, В.А. Владимиров

Совершив обратное преобразование, получаем формулу (25).
? ?
Замечание 3. Операторы Jab , J0k и K? задаются на множестве решений системы
(19), поэтому слагаемые вида R0 (p)L0 (p)+R1 (p)L1 (p) в форму ле (25) опускаются.

1. Fushchych W.I., On additional invariance of relativistic equations of motion, Preprint 70-32E, Inst.
for Theor. Phys., Kiev, 1970; Теор. и матем. физ., 1971, 7, № 1,3; Lett. Nuovo Cimento, 1974, 11,
№ 10, 508.
2. Фущич В.И., В кн.: Теоретико-групповые методы в математической физике, Киев, 1978.
3. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., 1978.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 165–169.

On the new symmetries of Maxwell equations
W.I. FUSHCHYCH, A.G. NIKITIN
The known symmetries of the Maxwell equations are summarized. Then new symmetri-
es of these equations found by the authors are reviewed. These symmetries are generated
by infinitesimal integrodifferential operators of the eight-dimensional Lie algebra. Their
physical meaning is not clear.

Before considering the new symmetries of the Maxwell equations
?E ?H
= ?rot H, = rot E,
?t ?t (1)
div E = div H = 0
it is natural to remind shortly the well-known data on symmetry properties of eqs. (1).
In 1890 Heaviside wrote the original Maxwell equations in the modern form (1)
(independently it has been done by Hertz) and paid attention to their invariance under
the transformation
E > H, H > ?E. (2)
Larmor [1] and Rainich [2] demonstrated that this symmetry may be extended to
one-parametrical group of the following transformations
E > E cos ? + H sin ?,
(3)
H > H cos ? ? E sin ?.
In 1904 Lorentz found the linear transformations of space and time variables
and the corresponding transformations for E and H, under which the free Maxwell
equations remain invariant. In 1905 Poincar? and Einstein have demonstrated that
e
the Maxwell equations are invariant under the Lorentz transformations also in the
presence of charges and currents. Poincar? first established and studied in detail one
e
of the most important properties of these transformations — their group structure
and demonstrated that “Lorentz transformations are nothing but the rotations in the
v
space of four dimensions, a point of which has coordinates (x, y, z, ?1t)”. So it was
Poincar? who united the space and time into the four-dimensional space-time three
e
years before Minkowski [3].
In 1909 Bateman [4] and Cunningham [5] proved eqs. (1) to be invariant under
non-linear conformal transformations. Bateman demonstrated the conformal group
C(1, 3) to be the maximal symmetry group of Maxwell equations with charges and
currents.
One hundred years ago, S. Lie created the mathematical methods of group analysis
of differential equations [6]. It is a surprising fact that these methods have only
recently been used for the investigation of group properties of eqs. (1). It turned
Czechoslovak Journal of Physics B, 1982, 32, P. 476–480.
Invited talk presented at the International Symposium “Selected Topics in Quantum Field Theory and
Mathematical Physics”, Bechyne, Czechoslovakia, June 14–19, 1981.
166 W.I. Fushchych, A.G. Nikitin

out that the maximal local invariance group of eqs. (1) is the direct product of the
conformal group and of the Heaviside–Larmor–Rainich one (3), i.e. C(1, 3) ? H [7].
In connection with the facts mentioned above the impression may arise that the
symmetry properties of eqs. (1) have been completely investigated and there is no
hope to discover any new symmetry of these equations. But it is not true. It appears
that all the relativistic equations of motion for spinning particles possess an additional
non-geometrical symmetry under the group U (2) ? U (2) (the only exception is the
Weyl equation) [9–11]. The basis elements QA (A = 1, 2, . . . , 8) of Lie algebra of
this group are not the first-order differential operators, but the integro-differential
ones. That is why this non-geometrical symmetry could not be discovered by the
infinitesimal Lie method.
In general case the non-geometrical symmetry of the relativistic equations of
motion is more extensive than the symmetry U (2) ? U (2) and increases with the
rise of spin [9, 15].
Let us denote by {QA } the basis elements of a finite-dimensional Lie algebra. This
algebra is by the definition the invariance algebra of Maxwell equations if QA are
defined on the set of solutions of eqs. (1), i.e. satisfy the conditions

?
? ?2 S · p, ? = column (E, H),
L1 QA ? = 0, L1 = i
?t (4)
L2 = p1 ? S · pS1 ,
L2 QA ? = 0,

where
? ?
Sa 0 0 I
(5)
Sa = , ?2 = i ,
? ?
I 0
0 Sa

? ?
I and ? are unit and zero 3 ? 3 matrices, Sa are the spin matrices, (Sa )bc = i?abc .
0
Theorem 1 [11, 14, 16]. The Maxwell equations (1) are invariant under the 8-
dimensional Lie algebra A8 , basis elements of which are the integro-differential
operators of the form

S·p S·p
Q1 = ?3 D, Q2 = i?2 , Q3 = ?1 D,
p p
(6)
S·p S·p
= ?i?2
Q3+a Qa , Q7 = I, Q8 = i?2 ,
p p

where
?
?
p2 p2 + p 2 p2 ? p 2 p2 1 ? S a + p 1 p2 p3 S a S b pc ?
2
D=
? ab ac bc
a=b=c

2
S·p (7)
??1 ,
?pp1 p2 p3 1 ?
p
1/2
1 2 2 22
? = v p4 p2 ? p2 ? ?
p4 p2 p2 p4 p2
+ + p2 ,
1 2 3 2 3 1 3 1
2
On the new symmetries of Maxwell equations 167

1/2
p2
?a are the Pauli matrices, commuting with Sa , p = . The operators (6)
a
satisfy the algebra

[Qa , Qb ] = ?[Q3+a , Q3+b ] = ??abc Qc ,
(8)
[Q3+a , Qb ] = ?abc Q3+c , [Q7 , QA ] = [Q8 , QA ] = 0,

which is isomorphic to the Lie algebra of the group U (2) ? U (2).
Proof. The correctness of the Theorem, i.e. that the operators (6) satisfy the com-
mutation relations (8) and the conditions of invariance (4), may be verified by the
direct calculations. But it is necessary to note that such a verification is very compli-
cated. There exists another, more constructive proof of Theorem 1, which explains
the nature of the non-geometrical symmetry of the relativistic equations of motion.
This proof consists in the diagonalization (decomposition) of the Maxwell equations
into independent subsystems [11, 14, 15]. Such a diagonalization of the operator L1 is
carried out by the operator

W ?1 = W † , (9)
W = U4 U3 U2 U1 ,

where
S·p
1 ?
1 + ?2 ? (1 ? ?2 )D
U1 = , U2 = exp(?iSa ?a ),
2 p
v v
U3 = 1 ? i S1 S2 + S2 S1 + 1 ? S3 U4 = exp (S2 ? S1 )i?/4 2 ,
2
2,
?
?a = ?abc (pb ? pc ) arctg [?/(p1 + p2 + p3 )] /2?,
p p
1/2
p = (p1 ? p2 )2 + (p3 ? p1 )2 + (p2 ? p3 )2
? .

As a result one obtains
?
L1 = W L1 W ?1 = i ? ?0 p, (10)
?t
where ?0 is the diagonal matrix

?0 = ?i(S1 S2 + S2 S1 )S3 = diag (1, ?1, 0, 1, ?1, 0). (11)

Now it is not difficult to find eight linearly independent matrices QA , which
commute with the operator L1 . These matrices may be chosen in the form

Q3+a = ?i?0 Qa , (12)
Qa = i?a , Q7 = I, Q8 = i?0 .

The matrices (12) satisfy the algebra (8) and are connected with QA (6) by the
transformation QA = W ?1 QA W . It is obvious that the matrices (12) operating on
the vector-function ? = W ? change their components but do not alter the variables
(t, x). The theorem is proved.
Since QA (6) are integro-differential operators, we give the finite transformations,
generated by QA , for the Fourier-components of Ea and Ha
? ? ? ?
Ea > Ea = Ea cos ?1 + i?abc pb Dcd Ed sin ?1 ,
?
(13a)

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