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которая сохраняется благодаря свойству симметрии тензора (74), ?µ? = ??µ .
Легко убедиться, что если принять обобщение (125) теоремы Нетер, то
?? > T? ? ?? x? T? ,
µ µ
(126)
µ
где T? — стандартный тензор энергии–импульса (77), так что при переходе к ин-
тегральным сохраняющимся величинам из (126) получаем физически удовлетвори-
тельное соответствие ?? > P? . Другими словами, по формулам (125), (126) получа-
ем, что сохранение энергии–импульса P? электромагвитного поля есть следствие
траясляциойной инвариантности теории. Благодаря выполнению условия (126), со-
ответствие “оператор симметрии — закон сохранения”, даваемое формулой (125),
можно считать физически приемлемым и для всех остальных операторов q пре- ?
образований инвариантности уравнений Максвелла.
Однако обобщение теоремы Нетер, заданное формулой (125), можно трактовать
как существенный и, возможно, мало оправданный отход от стандартных принци-
пов лагранжева подхода. Это замечание подтверждается тем, что результат (126)
естественным образом получается и рамках стандартных принципов L-подхода
для электромагнитного поля F = (E, H) при использовании скалярных функций
Лагранжa для этого поля. Такой подход был предложен нами в [7] (еще до рабо-
ты Садбери [18]) и основывался на скалярной функции Лагранжа, построенной в
виде свертки L = ?µ Lµ , ?µ = xµ ? eµ . Детальное обсуждение такого L-подхода,
включая вычисление и анализ сохраняющихся величин — следствий 32-мерной
?
алгебры инвариантности A32 , приведено в [8, 9]. Использование P -скалярной фун-
кции Лагранжа оказалось предпочтительным не только потому, что стандартная
теорема Нетер приводит к обычному соответствию “оператор симметрии — закон
О векторных лагранжианах для электромагнитного и спинорного полей 221

сохранения”, но и потому, что в таком L-подходе естественным образом вводится
?
P -скалярное взаимодействие электромагнитного (E, H) и спинорного ? полей.

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Физика элементар. частиц и атом. ядра., 1983, 14, вып. 1, 5–57.
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АН УРСР, Киев, 1986, 49 с.
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да к электродинамике в терминах напряженностей, Препринт 86.41, Ин-т ядерных исследований
АН УРСР, Киев, 1985, 39 с.
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85.12, Ин-т ядерных исследований АН УРСР, Киев, 1985, 61 с.
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риантности, в кн: Теоретико-групповые исследования уравнений математической физики, Киев,
Ин-т математики АН УССР, 1985, 134–139.
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?
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W.I. Fushchych, Scientific Works 2001, Vol. 3, 223–235.

On the new invariance algebras
and superalgebras of relativistic wave
equations
W.I. FUSHCHYCH, A.G. NIKITIN
We show that any relativistic wave equation for a particle with mass m > 0 and
arbitrary spin s is invariant under the Lie algebra of the group GL(2s + 1, C). The
explicit form of basis elements of this algebra is given for any s. The complete sets
of the symmetry operators of the Dirac and Maxwell equations are obtained, which
belong to the classes of the first- and second-order differential operators with matrix
coefficients. Corresponding new conservation laws and constants of motion are found.

1. Introduction
The classical Lie approach is the main mathematical apparatus used for the analysis
of the symmetry of partial differential equations [1, 30]. This approach was that from
which it was established that the Poincar? group is the maximal symmetry group of
e
the Dirac equation [2, 22] and that the maximal symmetry of Maxwell’s equations
is determined by the conformal group replenished by the Heaviside–Larmor–Rainich
transformation. However, in spite of its power and universality, the Lie approach
does not make it possible to find all the symmetry operators of the given equation.
Actually it gives the possibility or finding only such symmetry operators which are
the first-order differential operators.
Using the non-Lie approach [5, 6, 8, 9], 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.
It has been demonstrated that the Dirac equation was invariant under the group
SU (2) ? SU (2) [5, 6, 12] and that the Kemmer–Duffin–Petiau equation for the vector
field was invariant under the group SU (3)?SU (3) [29, 12]. The non-Lie approach gave
the possibility of finding the additional symmetry of the Dirac and Kemmer–Duffin–
Petiau equations describing the particles in an external electromagnetic field [13, 27].
The hidden symmetry of Maxwell’s equations has also been found and is described
by the eightparameter transformation group including the subgroup of Heaviside–
Larmor–Rainich transformations [13, 14, 15, 17].
In this paper we continue to study the symmetry of the Dirac, Weyl and Maxwell
equations and of relativistic wave equations for any spin particles. The main results
obtained here may be formulated as follows.
(i) We found that any Poincar?-invariant wave equation for a particle of arbitrary
e
spin s and mass m = 0 is additionally invariant under the 2(2s+1)(2s+1)-dimensional
Lie algebra which is isomorphic to the Lie algebra of the group GL(2s + 1, C). The
explicit form of basis elements of this invariance algebra is found for any value of s.
Thus the additional symmetry of an arbitrary relativistic wave equation is descri-
bed whereas previously one studied, as a rule, the symmetry properties of specific
equations.
J. Phys. A: Math. Gen., 1987, 20, P. 537–549.
224 W.I. Fushchych, A.G. Nikitin

(ii) In our earlier work we restricted ourselves to studying symmetry operators of
relativistic wave equations which belong to a finite-dimensional Lie algebra [17]. Here
we also consider the symmetry operators belonging to the classes of firstand second-
order differential operators with matrix coefficients which, generally speaking, are
not the basis elements of any finite-dimensional Lie algebra, but are closely connected
with conservation laws. The complete set of symmetry operators of the Dirac equation
in the class of first-order differential operators with matrix coefficients (class M1 ) is
found. We also obtain the symmetry operators of the Weyl and Maxwell equations
which form the basis of the Lie superalgebra.
(iii) The new conservation laws and motion constants, which are connected with
hidden symmetry of the Dirac and Maxwell equations, are found.
The results of this paper supplement and in some sense complete, those obtained
by us and expanded by a number of other authors [3, 31, 24, 32] by studying the
additional symmetry of Poincar?-invariant wave equations.
e

2. The additional symmetry of Poincar?-invariant wave equations
e
for arbitrary spin particles
In this section we demonstrate that any relativistic wave equation for a particle of
non-zero mass and spin s = 0 has more extensive symmetry than Poincar? invariance,
e
and describe this additional symmetry exactly.
Let us write an arbitrary linear (differential or integro-differential) equation in the
following symbolic form

(2.1)
L? = 0,

where L is a linear operator defined on a vector space H, ? ? H.
Let Q be an operator defined on H. We say that Q is the symmetry operator of
the equation (2.1), if

(2.2)
L(Q?) = 0

for any ? satisfying (2.1).
Definition. Equation (2.1) is Poincar?-invariant and describes a particle of mass
e
m and spin s if it has 10 symmetry operators Pµ , Jµ? , µ, ? = 0, 1, 2, 3, which form
the basis of the Lie algebra of the Poincar? group, and any solution ? satisfies the
e
conditions

Wµ W µ = ?m2 s(s + 1)?,
Pµ P µ ? = m2 ?, (2.3)

where Wµ is the Lubansky–Pauli vector
1
?µ??? J ?? P ? . (2.4)
Wµ =
2
Below we consider only such equations (2.1) which satisfy the given definition and
so may be interpreted as equations for a relastivistic particle of spin s and mass m.
The symmetry operators Pµ , Jµ? of such a equation satisfy the commutation relations

[Pµ , J?? ] = i(gµ? P? ? gµ? P? ),
[Pµ , P? ] = 0,
(2.5)
[Jµ? , J?? ] = i(gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? )
On the new invariance algebras and superalgebras 225

which characterise the Lie algebra of the Poincar? group P (1, 3). The eigenvalues
e
of the corresponding Casimir operators Pµ P and Wµ W µ are fixed and given by the
µ

relations (2.3). Let us emphasise that we do not make any supposition with regards to
the explicit form of the operators Pµ and Jµ? — they can be as differential operators
of first order as non-local (integro-differential) ones.
Theorem 1. Any Poincar?-invariant equation for a particle of mass m and spin s is
e
invariant under the algebra1 GL(2s + 1, C).
Proof. Let Pµ , Jµ? be the symmetry operators of the equation (2.1) satisfying the
commutation relations (2.5). Then by the definition (2.3) the following combinations
1
Q± = W ? P ? ± i(Pµ W? ? P? Wµ )] (2.6)
[?
2 µ???
µ?
m
are also the symmetry operators of these equations.
Using (2.5) and the relations
[Wµ , W? ] = i?µ??? P ? W ? (2.7)
[Wµ , P? ] = 0,
can make sure that the operators (2.6) satisfy the conditions
[Q± , Q± ] = i(gµ? Q± + g?? Q± ? gµ? Q± ? g?? Q± )m?4 (Pµ P µ )2 , (2.8)
µ? µ? ??
?? ?? µ?

1 ± ±µ?
= ?m4 W? W ? P? P ? ,
C1 = Qµ? Q
4
(2.9)
1
C2 = ?µ??? Q±µ? Q±?? = ?im?4 Wµ W µ P? P ? .
4
It follows from (2.3) and (2.8) that on the set of the equation (2.1) solutions the
operators (2.6) satisfy the commutation relations
[Q± , Q± ]? = i(gµ? Q± + g?? Q± ? gµ? Q± ? g?? Q± )?, (2.10)
µ? µ? ??
?? ?? µ?

which characterise the Lie algebra of the group SL(2, C). From (2.3) and (2.9) one
obtains the eigenvalues of corresponding Casimir operators
12
l0 + l1 ? 1 ?,
2
(2.11)
C1 ? = C2 ? = il0 l1 ?,
2
where l0 = s, l1 = ±(s + 1).
So we have demonstrated that any Poincar?-invariant equation for a particle of
e
non-zero mass and spin s = 0 is additionally invariant under the algebra SL(2, C),
the basis elements of which belong to the enveloping algebra of the P (1, 3) and are
given exactly by the relations (2.6). According to (2.11) the operators (2.6) realise
the representation D(l0 , l1 ) = D(s, ±(s + 1)) of the algebra GL(2, C). Now we see
that this invariance algebra may be extended to 2(2s + 1)-dimensional Lie algebra
isomorphic to the algebra GL(2s + 1, C). Exactly the basis elements of the algebra
GL(2s + 1, C) have the following form on the set of the equation (2.1) solutions:
?n+k n = ank (Q+ ? Q+ )Pn , ?n n+k = akn Pn (Q+ + Q+ ),
s s

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