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причем ограниченное при ? > ? решение получается из (33) при c3 = c4 = 0.
В этом случае полуширина волны [81 определяется формулой X = ?? (T ? t)1/2 ,
из которой следует, что X > 0 при t > T ? , т. е. поступающая в среду энергия
сосредотачивается в зоне с сокращающимся эффективным размером. При этом
тепловой поток записывается в виде
?
q(?, t) = ?µ(T ? t)???3/2 (? + 1/2)? (?) + ? (?) ? ? 2 ? (?) .
2
Для финитного решения вида (32) уравнения (21) должны, как и ранее, выпол-
няться условия (29), поэтому для определения постоянных c1 , и c2 , из (33) при
фиксированном ? = ?? получаем однородную алгебраическую систему, опреде-
литель которой ? должен быть равен нулю. Однако оказывается, что если в ?
воспользоваться асимптотическими представлениями функций Эрмита при боль-
ших ?? /2? приходим к равенству 12(? + 1)? 2 + ?? = 0, которое невозможно ввиду
2

? > 0. Следовательно, фронт тепловой волны не может находиться в конечной
точке ?? , и, таким образом, в режиме с обострением уравнение (21) описывает
распространение возмущений с бесконечной скоростью (см. также [8]).

1. Морс Ф.М., Фешбах Г., Методы теоретической физики, в 2 т., М., Изд-во иностр. лит., 1958,
Т. 1, 930 с.
2. Лыков А.В., Теория теплопроводности, М., Высш. шк., 1967, 599 с.
3. Толубинский Е.В., Теория процессов переноса, Киев, Наук. думка, 1969, 259 с.
4. Подстригач Я.С., Коляно Ю.М., Обобщенная термомеханика, Киев, Наук. думка, 1976, 310 с.
80 В.И. Фущич, А.С. Галицын, А.С. Полубинский

5. Фущич В.И., О симметрии и частных решениях некоторых многомерных уравнений математиче-
ской физики, в сб. Теоретико-алгебраические методы в задачах математической физики, Киев,
Ин-т математики АН УССР, 1983, 4–22.
6. Fushchych W.I., Cherniha R.М., The Galilean relativistic principle and nonlinear partial differential
equations, J. Phys. A: Math. and Gen., 1985, 18, 3491–3503.
7. Положий Г.Н., Уравнения математической физики, М., Высш. шк., 1964, 560 с.
8. Самарский А.А., Галактионов В.А., Курдюмов С.П., Михайлов А.П., Режимы с обострением в
задачах для квазилинейных параболических уравнений, М., Наука, 1987, 477 с.
9. Маслов В.П., Данилов В.Г., Волосов К.А., Математическое моделирование процессов тепломас-
сопереноса (эволюция диссипативных структур), М., Наука, 1987, 352 с.
10. Никифоров А.Ф., Уваров В.Б., Специальные функции математической физики, М., Наука, 1984,
319 с.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 81–85.

On superalgebras of symmetry operators
of relativistic wave equations
W.I. FUSHCHYCH, A.G. NIKITIN

It is well known that the classical Lie approach does not make it possible to
describe completely the symmetry of systems of partial differential equations. Actually
it gives the possibility of finding only such symmetry operators which are the first
order differential operators.
Using the non-Lie approach, in which the invariance group generators may be
differential operators of any order and even integro-differential operators, the new
invariance groups of a number of relativistic wave equations have been found [1, 2].
It turns out that even such well studied equations as the Dirac and the Maxwell ones
have more extensive symmetry then the relativistic and the conformal invariance [3].
A numerous examples of non-Lie symmetries had been collected in our book [4].
In this communication we give the description of any order symmetry operators
for some class of relativistic wave equations (including the Dirac and the Kemmer–
Duffin–Petiau equations) and determine superalgebraic structure of sets of symmetry
operators of the Dirac and of the Maxwell equations.
Let us write an arbitrary linear system of partial differential equations in the
following symbolic form

(1)
L? = 0,

where L is a linear differential operator defined on H, ? ? H.
Let Q be a linear operator defined on H. We say that Q is the symmetry operator
of the equation (1), if

(2)
L(Q?) = 0

for any ? satisfying (1).
Below we consider the symmetry operators of relativistic wave equations, the most
famous of which is the Dirac one:
?
L? ? (?µ pµ ? m)? = 0, (3)
pµ = i , µ = 0, 1, 2, 3.
?xµ
Using the equation (3) as an example we shall give the definition of the first
(Q ), the second (Q(2) ), the third (Q(3) ), . . . , order symmetry operator as a linear
(1)

differential operator which satisfies (2) and has the form

Q(1) = aµ Pµ + B, Q(2) = aµ? pµ p? + B µ pµ + B,
(4)
Q(3) = aµ?? pµ p? p? + B µ? pµ p? + B µ pµ + B,
in Selected Topics in QFT and Mathematical Physics, Proceedings of the 5th International conference
(Liblice, Czechoslovakia, June 25-30, 1989), Editors J. Niederle and J. Fisher, World Scientific, 1990,
P. 385–391.
82 W.I. Fushchych, A.G. Nikitin

where B, B µ , B µ? , . . . are matrices depending on x = (x0 , x1 , x2 , x3 ), aµ , aµ? , aµ?? , . . .
are functions on x. For the Dirac equation all matrices, are 4 ? 4 dimensional,
in general the matrices dimension is determined by the number of components of
wavefunction ?.
It is well known that the complete set of first order symmetry operators of the
Dirac equation is exhausted by the Poincar? group generators Pµ , Jµ?
e
i
Jµ? = xµ p? ? x? pµ + [?µ , ?? ], (5)
P µ = pµ ,
4
which satisfy the commutation relations
[Pµ , J?? ] = i(gµ? P? ? gµ? P? ),
[Pµ , P? ] = 0,
(6)
[Jµ? , J?? ] = i(gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? ).
It means that the Poincar? invariance is the most extensive symmetry of the Dirac
e
equation in the Lie sense [5, 6].
Using higher order symmetry operators it is possible to extend the symmetry
group of the Dirac equation to the 16-parametrical Lie group which includes the
Poincar? group as a subgroup [4]. Higher-order symmetry operators are useful in
e
construction of coordinate systems in which the solutions in separated variables exist
[7 ,8]. These operators may be considered also as the generators of Lie–B?cklunda
groups [9].
Below we present some our general results connecting with the symmetry opera-
tors of relativistic wave equations for any spin particles.
Definition. Equation (1) is Poincar?-invariant and describes a particle of mass m and
e
spin s, if it has 10 symmetry operators Pµ , Jµ? which satisfy the algebra (6), and any
solution ? satisfies the conditions
Wµ W µ ? = ?m2 s(s + 1)?,
Pµ P µ ? = m2 ?, (7)
where Wµ = 1 ?µ??? J ?? P ? is the Lubanski–Pauli vector.
2
Besides the Dirac equation the well known examples of relativistic wave equations
satisfying given definition are the Kemmer–Duffin–Petiau equations for particles of
spin 0 and 1 and the Rarita–Schwinger equation for a particle of spin 3 .
2
Theorem 1. Any Poincar?-invariant equation for a particle of mass m and spin
e
s = 0 is invariant under the algebra ASL(2, C) [10].
Proof. Let Pµ , Jµ? be the symmetry operators of the equation (1), satisfying the
commutation relations (6). Then by the definition (2) the following combinations
1
Q± = W ? P ? ± i(Pµ W? ? P? Wµ )] (8)
[?
2 µ???
µ?
m
are also the symmetry operators of this equation.
Using (6), (7) and the relations [Wµ , W? ] = i?µ??? P ? W ? , [Pµ , W? ] = 0 can make
sure that the operators (8) satisfy the conditions
[Q± , Q± ] = i(gµ? Q± + g?? Q± ? gµ? Q± ? g?? Q± ),
µ? ?? ?? µ? ?? µ?
1
?µ??? Q± µ? Q± ?? ? = il0 l1 ?,
Q± Q± µ? ? = 2(l0 ? l1 ? 1)?,
2 2
µ?
4
l0 = s, l1 = ±(s + 1),
On superalgebras of symmetry operators of relativistic wave equations 83

and so form the basis of the finite dimensional irreducible representation D(s, ±(s+1))
of the algebra ASL(2, C). Thus the theorem is proved.
We see that any relativistic wave equation for a particle of nonzero spin and mass
is automatically invariant under the algebra ASL(2, C) basis elements of which belong
to the enveloping algebra of the Lie algebra of the Poincar? group. The operators (8)
e
form the basis of the 16-dimensional Lie algebra together with Pµ and Jµ? . For the
Dirac equation they take the form [4]
i i
Q± = [?µ , ?? ] + (?µ p? ? ?? pµ )(1 ± i?4 ). (9)
µ?
4 m
The operators (5), (9) generate the 16-parametrical invariance group of the Dirac
equation. The corresponding finite transformations mix ? and ??/?xµ and can be
easily calculated using the relation (Q± )2 = 1/4 [4].
µ?
The following statement gives the basis of any order symmetry operators for
a class of relativistic wave equations of a type
(?µ pµ ? m)? = 0 (10)
where ?µ are numerical matrices, ?0 is diagonalizable.
Theorem 2. Any finite order symmetry operator of Poincar?-invariant equation for
e
a particle of mass m = 0 and spin s (10) belongs to the enveloping algebra of the
algebra AP (1, 3).
The proof can be carried out using the Theorem 1 and bearing in mind that the
necessary conditions for the symmetry operators of the equation (10) is to be the
symmetry operators of the equation (7).
Let us note that relativistic wave equations (10) also possess such additional
invariance algebras which belong to the class of integro-differential operators [4] and
generally speaking are not membered among the enveloping algebra of the algebra
AP (1, 3).
In contrast to the first order symmetry operators the higherorder ones in general
do not form the basis of the Lie algebra. But as a rool the higher order symmetry
operators have the structure of superalgebra. We shall demonstrated it for the Dirac
and for the Maxwell equations.
Let us consider the complete set of the second order symmetry operators of the
equation (3) commuting with Pµ . Using the Theorem 2 it is not difficult to find such
a set in the form
i
I, Pµ , ?µ? = pµ p? , Wµ = ?4 (?µ m ? pµ ), W µ? = ?4 (?µ p? ? ?? pµ ), (11)
4
where I is the unit matrix.
Direct verification can make sure that the operators (11) do not form the basis of
the Lie algebra. But these operators together with Jµ? (5) form the Lie superalgebra
with the basis elements (12)
{Wµ , Wµ? ; Jµ? , P? , ?µ? , I}. (12)
The operators Wµ , Wµ? satisfy the anticommutation relations
1
[Wµ , W? ]+ = Wµ W? + W? Wµ = (?µ? ? gµ? I),
2
[Wµ , W?? ]+ = i(gµ? P? ? gµ? P? ),
[Wµ? , W?? ]+ = 2(gµ? ??? + g?? ?µ? ? gµ? ??? ? g?? ?µ? ),
84 W.I. Fushchych, A.G. Nikitin

the commutation relations Wµ , Wµ? with Pµ , Jµ? , ?µ? , I and between Pµ , Jµ? , ?µ? ,
I are obvious.
So the Dirac equation is invariant under the 27-dimensional Lie superalgebra which
contains the subalgebra AP (1, 3). Basis elements of this superalgebra are second order
symmetry operators.
Consider the Maxwell equations with currents and charges

?E ?H
= ? ? H + j, = ?? ? E, ? · E = j0 , ? · H = 0. (13)
?t ?t
The symmetry superalgebra of the equations (13) is formed by the set of the operators

{Qab ; Pµ , Jµ? , ?ab = ?a ?b D, ?abcd = ?a ?b ?c ?d }

where Pµ , Jµ? are the Poincar? group generators, a, b, c, d = 1, 2, 3, and Qab , D are
e
the additional symmetry operators of the Maxwell equations [11] which act on Ea ,
Ha ja and j0 as follows
Qab : Ec > qcd Ed , Hc > ?qcd Hd ,
ab ab

jc > qcd jd , j0 > (?ab ? ? ?a ?b )j0 ;
ab


D : Ec > ?c ?d Ed , Hc > ?c ?d Hd ,
jc > ?c ?d jd , j0 > ?j0 ,
where
ab ab ba ab ba
qcd = fcd + fcd + fdc + fdc ;
1 1 1
fcd = ?ad ?b ?c + ?cd (?ab ? ? ?a ?b ) ? ?ac ?bd ? ?ab ?c ?d .
ab
4 2 2
The operators Qab satisfy the anticommutation relations

[Qab , Qa b ]+ = fklnm ?klnm + gkl b ?kl ,
aba b aba


where
fklnm = 2(?aa ?kl ? ?ak ?a l )(?bb ?nm ? ?bn ?b m )?
aba b

?(?ab ?kl ? ?ak ?bl )(?a b ?nm ? ?a n ?b m ) + (a - b),
gkl b = 2(?aa ?bk ?b l ? ?a b ?ak ?bl )
aba

+(?ab ?a b ? ?ab ?a b )?kl + (a - b) + (a - b ) + (a - b, a - b ).
The remaining commutation relations for the operators (14) can be easily calculated.
It is interesting to note that the symmetry operators Qab do not belong to the
enveloping algebra of the Lie algebra of the conformal group. These and other prob-
lems connecting with the symmetry of relativistic and nonrelativistic wave equations,
the description of classes of equations with given symmetry, the exact solutions of
linear and nonlinear wave equations are discussed in our book [11] which will be
published this year.
On superalgebras of symmetry operators of relativistic wave equations 85

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