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? ?
? and I are four-row square zero and unit matrices. The matrices S4a and
0
1?
?
Sab = S4c + 2?c ?abc
?
2
realize the representation D 1 , 1 of the algebra O(4). Writing equations (5.5) by
22
components, one obtains the usual form for the Maxwell equation (5.1) and the
conditions for ?1 and ?2 :

?1 = C1 , ?2 = C2 ,
On the new invariance algebras of relativistic equations 19

where C1 and C2 are constants which may be equated to zero without loss of genera-
lity? .
Using the unitary operator

S a pa
? p?
tan?1
U = exp ?i (5.8)
,
p? p1 + p2 + p3

where
? ?
Sbc 0
1/2
pa = p b ? p c , p = p 2 + p2 + p2
? ? ?1 ?2 ?3 , Sa = ,
?
?
0 Sbc

one reduces the equations (5.5) to the symmetrical form

? 1
L1 = U L1 U † = i ? v (?1 + ?2 + ?3 )p;
L1 ? = 0,
?t 3
(5.9)
1
L2 = U L2 U † = v (S41 + S42 + S43 ),
L2 ? = 0, ? = U ?.
3
The operator (5.8) also transforms the helicity operator Sp = Sa pa p?1 to the sym-
metrical matrix form:
v
U Sp U † = (S1 + S2 + S3 )/ 3.

The invariance condition (3.2) for the equations (5.9) takes the form
?1 ?2
1 2
(5.10)
[L1 , QA ]? = fA L1 + fA L2 , [L2 , QA ]? = fA L1 + fA L2 .

The conditions (5.10) are obviously satisfied by any operator which commutes with
the matrices
v v
and (5.11)
A = (?1 + ?2 + ?3 )/ 3 B = (S41 + S42 + S43 )/ 3.

We choose the complete set of such operators in the form
v
Q12 = (S1 + S2 + S3 )/ 3, Q23 = iQ12 Q31 ,
v
(Sb ? Sc )p2 p2 ? p2 L?1 / 3,
Q31 = a b c
(5.12)
a
?0
I?
Q4a = AQbc , Q5 = A, Q6 = ?0 = .
0?
?I

Of course this is not the only possible basis set of the operators commuting with
(5.11). However, we prefer the operators (5.12) because they are invariant under the
permutation

Sa > Sb , pa > pb , a, b = 1, 2, 3.
? The analogous “Dirac-like” formulation of the Maxwell equations (but using a four-component wave
function and subsidiary condition different from (5.5b) has been proposed previously by Lomont [17] and
Moses [21].
20 W.I. Fushchych, A.G. Nikitin

?1
1 2
The operators (5.12) satisfy the invariance condition (5.10) (with fA = fA = fA =
?2
fA = 0) and the commutation relations

[Qkl , Qmn ]? = 2i(?km Qln + ?ln Qkm ? ?kn Qlm ? ?lm Qkn ),
(5.13)
[Q5 , Qkl ]? = [Q6 , Qkl ]? = [Q5 , Q6 ]? = 0.

These operators also satisfy the conditions

(Qkl )2 ? = (Q5 )2 ? = (Q6 )2 ? = ?,

i.e. they realise the representation of the Lie algebra of the group U (2) ? U (2) and
Qkl form the representation D 0, 1 ? D 1 , 0 of the group SU (2) ? SU (2).
2 2
It follows from the above that equations (5.9) are invariant under the arbitrary
transformation from the group U (2) ? U (2):
1 1
cos ? + i??1 ?abc Qab ?c ?,
? > ? = exp i?abc Qab ?c ? =
2 2
sin ?
? > ? = exp(iQ4a ?a )? = cos ? + iS4a ?a ?, (5.14)
?
?>? = exp(iQ5 ?)? = (cos ? + iQ5 sin ?)?,
?>? = exp(iQ6 ?)? = exp(i?)?.

Returning with the help of the operator (5.8) to the starting ? function one obtains
from (5.14) the following transformation laws:
1
?>? = cos ? + ?abc Qab sin ? ?,
2?
i
?>? = cos ? + Q4a ?a sin ? ?, (5.15)
?
?>? = (cos ? + iQ5 sin ?)?,
?>? = exp(i?)?.

where
Qkl = W ?1 Qkl W, Q? = W ?1 Q? W, ? = 5, 6,
Q12 = Sa pa ,
? Q23 = ?1 F, Q31 = i?1 Sa pa F,
?
1
Q4a = ?2 Sb pb ?abc Qbc ,
? Q5 = ?2 Sb pb ,
? Q6 = 1,
2
(5.16)
?1
? 1? + p 1 p2 p3 pa S b S c ?
p2 p2 p2 p2 p2 p2 2
F =L + Sa
ac ab bc
a=b=c

2
?pp1 p2 p3 1 ? (Sa pa )
? .

Substituting (5.6) and (5.16) into (5.15), we obtain the formulae (5.4). The theorem
is proved.
On the new invariance algebras of relativistic equations 21

So we have found a new eight-parameter symmetry group of the Maxwell equati-
ons which is given by the transformations (5.4). The main property of such transfor-
mations is that they are carried out by the nonlocal (integro-differential) operators.
It is necessary to emphasise that the transformations (5.4) have nothing to do with
the Lorentz ones, inasmuch as they realise the unitary finite-dimensional representa-
tion of the compact group U (2) ? U (2). If ?1 = ?2 = 0, the formulae (5.4b) give the
Heaviside–Larmor–Rainich transformation (5.3).
The transformations (5.4) are unitary under the usual scalar product (3.5). Sub-
stituing (5.6) into (3.5), we discover that the transformations (5.4) do not change the
quantity

E= d3 x E 2 + H 2 ,

which is associated with the full energy of an electromagnetic field.
If the parameters ?a , ?a , ? and ? in (5.4) are the complex ones, the transformations
(5.4) realise the representation of the group GL(2)?GL(2). Such transformations also
leave the equations (5.1) invariant, but are, of course non-unitary.
Using theorem 1, we can show that equations (5.5) provide the Hermitian repre-
sentation of the Lie algebra of the conformal group. The basis elements of this algebra
have the form
P0 = ? · p, Pa = pa ,
Jab = xa pb ? xb pa + Sab = Xa pb ? Xb pa + pc ?,
?
1 1 1 (5.17)
J0a = tpa ? [Xa , P0 ]+ , D = [xa , pa ]+ ? tP0 ? ? [Xµ , P µ ]+ ,
2 2 2
1 1
Kµ = ?[Jµ? , X ? ]+ + [Pµ , X? X ? ]+ ? Pµ ?2 + /p2 ,
2 4
where
1
?abc Sab pc p?1 , Xa = xa + Sab pb p?2 .
X0 = x0 = t, ?= ?
2
But the generators (5.17) together with (5.16) do not form the closed algebra. The
symmetry of equations (5.5) under the 23-dimensional Lie algebra, which includes the
subalgebras C4 and U (2) ? U (2), is established in the following theorem.
Theorem 4. Equations (5.5) are invariant under the 23-dimensional Lie algebra,
basis elements of which are the operators (5.16) and the generators
?
Jµ? = xµ p? ? x? pµ ,
pµ = p µ ,
?
(5.18)
? ? ?
Kµ = ?x? x µ pµ + 2xµ D,
D = xµ pµ + i,

where
x0 = x0 ,
v v
xa = xa + (Sb ? Sc )( 3p ? p1 ? p2 ? p3 ) + Sd pd ( 3?a + 1)+
? p
v
+(pb ? pc )(S1 + S2 + S3 ){p[3p + 3(p1 + p2 + p3 )]}?1 .
22 W.I. Fushchych, A.G. Nikitin

The proof may be carried out in full analogy with the proof of theorem 2 (but
using the operator (5.8) instead of (3.3)). The operators (5.18) satisfy the algebra
(2.2) and (2.3) and commute with (5.16).
It is not difficult to generalise the statements of theorem 4 to the case of “Dirac-
like” equations for massless particles of any spin (Fushchych and Nikitin [11], Nikitin
and Fushchych [22]).
We note that the generators (5.16) and (5.17) are nonlocal (integro-differential)
ones. This means that the invariance algebra of the Maxwell equations which we
have obtained in principle cannot be obtained in the classical Lie approach, where,
as is well known, the group generators always belong to the class of differential
first-order operators.
1. Bateman H., Proc. London Math. Soc., 1909, 8, 223–264.
2. Bose S.K., Parker R., J. Math. Phys., 1969, 10, 812–813.
3. Cunningham E., Proc. Lond. Math. Soc., 1909, 8, 77–97.
4. Dirac P.A.M., Ann. Math., 1936, 37, 429–435.
5. Fock V.A., Z. Phys., 1935, 98, 145–149.
6. Fushchych W.I., Preprint E-70-32, Institute for Theoretical Phisycs, Kiev, 1970.
7. Fushchych W.I., Teor. Mat. Fiz. 1971, 7, 3–12 (transl. Theor. Math. Phys., 1971, 7, 3–11).
8. Fushchych W.I., Nuovo Cim. Lett., 1973, 6, 133–138.
9. Fushchych W.I., Nuovo Cim. Lett., 1974, 11, 508–512.
10. Fushchych W.I., Nikitin A.G., Nuovo Cim. Lett., 1977, 19, 347–352.
11. Fushchych W.I., Nikitin A.G., Preprint 77-3, Mathematical Institute, Kiev, 1977.
12. Fushchych W.I., Nikitin A.G., Nuovo Cim. Lett., 1978, 21, 541–546.
13. Gross L., J. Math. Phys., 1964, 5, 687–695.
14. Heaviside O., Electromagnetic Theory, London, 1893.
15. Larmor, Collected papers London, 1928.
16. Levi-Leblond, Am. J. Phys., 1971, 39, 502–506.
17. Lomont I.S., Phys. Rev., 1958, 111, 1700–1709.
18. Lomont I.S., Moses H.E., J. Math. Phys., 1962, 3, 405–408.
19. Mack G., Salam A., Ann. Phys., NY, 1969, 53, 174–202.
20. McLennan A., Nuovo Cim., 1956, 3, 1360–1380.
21. Moses H.E., Nuovo Cim. Suppl., 1958, 7, 1–18.
22. Nikitin A.G., Fushchych W.I., Teor. Mat. Fiz., 1978, 34, 319–333.
23. Nitikin A.G., Segeda Yu.N., Fushchych W.I., Teor. Mat. Fiz., 1976, 29, 82–94 (transl. Theor. Math.
Phys., 1976, 29, 943–954).
24. Ovsjannikov L.V., The Group Analyses of Differential Equations, Moscow, Nauka, 1978.
25. Rainich G.Y., Trans. Am. Math. Soc., 1925, 27, 106–125.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 23–27.

О подгруппах обобщенной группы
Пуанкаре
В.М. ФЕДОРЧУК, В.И. ФУЩИЧ

Для решения многих задач современной теоретической и математической фи-
зики важно знать не только группу симметрии рассматриваемой системы, но и
подгрупповую структуру этой группы. Действительно, описание всех подгрупп
однородной группы Лоренца позволило П. Винтерницу и И. Фришу [1] с теорети-
ко-групповой точки зрения подойти к задаче о разделении переменных в уравне-
нии Лапласа. Ими доказана теорема, что каждому разбиению группы Лоренца на
подгруппы, обладающие инвариантами, соответствует одна координатная система,
допускающая разделение переменных.
Среди всех уравнений математики, уравнения математической физики имеют
ту особенность, что все они обладают, как правило, нетривиальной симметрией.
Знание группы симметрии физической системы позволяет определить многие ее
свойства. Взаимодействия обычно нарушают исходную симметрию системы, но
иногда сохраняется симметрия относительно одной из подгрупп исходной груп-
пы. Таким образом, описание всех подгрупп группы симметрии позволяет дать
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