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h2 =
2
Потребуем, чтобы оператор (Д.10) удовлетворял уравнению (Д.1д) и получим
14 122 14
p ? p k1 (S · p)2 + k1 (S · p)4 +
4 2 4
(Д.11)
2 2
(s) (s) (s) (s)
(S · p) + · p) = 0.
4 4
2c2 d2 p2 (S 2
+ e2 d2 p+

При s > 3 операторы p4 , (S · p)4 и p2 (S · p)2 линейно независимы. Но тогда из
2
(Д.11) следует, что
2
k1 i
c2 = ?i d2 = ± . (Д.12)
,
2 2
Подставив (Д.12) в (Д.10), приходим к формуле
1
(s)
(?1 ± i?2 ) p2 ? k1 (S · p)2 .
2
(Д.13)
h2 =
2
Таким образом, мы показали, что с точностью до матричных преобразований
(Д.4), (Д.6), все возможные решения системы соотношений (Д.1) для операторов
(1.8) задаются формулами (Д.2), (Д.7), (Д.13) (если k1 = 0 и s > 3 ). Аналогично
2
можно показать, что формулы (Д.2), (Д.7), (Д.13) задают все решения системы
(Д.1) и для k1 = 0, s > 3 . Используя тождества
2

(S · p)3 = p2 (S · p), s = 1,
(Д.14)
5 9 3
(S · p)4 = (S · p)2 p2 ? p4 , s=
2 16 2
3
получаем решения (Д.1) для s = 1 и s = в виде (1.15)–(1.19).
2
392 В.И. Фущич, А.Г. Никитин

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On the new invariance groups of the Dirac
and Kemmer–Duffin–Petiau equations
W.I. FUSHCHYCH, A.G. NIKITIN

In works [1–6] the canonical-transformation method has been proposed for the investi-
gations of the group properties of the differential equations of the quantum mechanics.
This method essence in that the system of differential equation is first transformed
to the diagonal or Jordan form and then the invariance algebra of the transformed
equation is established. The explicit form of this algebra basis elements for the starting
equations is found by the inverse transformation.
The main distinguishing feature of this method from the intensively developed duri-
ng last years classical Lie method [7, 8] is that the basis elements of the invariance
algebra of the corresponding equations do not belong to the class of the differential
operators, but are as a rule integrodifferential operators. The new invariance algebras
of the Dirac [1, 2]1 , Maxwell [2], Klein–Gordon [3], Kemmer–Duffin–Petiau (KDP)
and Rarita–Schwinger [4] equations have been found just in the class of integrodi-
fferential operators.
The aim of this note is to establish the Dirac and the KDP equation invariance
algebras in the class of differential operators. The theorems given below (which es-
tablish new group properties of the Dirac and of the KDP equations) are proved with
the help of the canonical-transformation method.
To establish an invariance of the equation
?
L(p0 , p1 , p2 , p3 )?(x0 , x) ? L? = 0, (1)
pµ = i
?xµ
under the set of transformations ? > ?A = QA ? is to found a set of operators
Q ? {QA } such that

? QA ? Q, (2)
[L, QA ]? ? = 0,

where ? is a function which satisfies eq. (1). Condition (2) may be written in the
operator form

[L, QA ]? = F · L, (3)

where F is some set of operators, which are defined in the space of equation (1)
solutions.
Theorem 1. The Dirac equation

L 1 ? ? (?µ pµ + m)? = 0 (4)
2


Lettere al Nuovo Cimento, 1977, 19, № 9, P. 347–352.
1 The results of the work [2] have been generalized by Jayaraman (J. Phys. A, 1976, 9, 1181) to the case
of the equation without redundant components for any spin. See also [1].
394 W.I. Fushchych, A.G. Nikitin

is invariant under the 16-dimensional Lie algebra A16 , whose basis elements are the
differential operators
p i
Jµ? = xµ p? ? x? pµ + ?µ ?? , (5)
P µ = pµ = i ,
?xµ 2
i
(1 + i?4 )(?µ p? ? ?? pµ ), (6)
Qµ? = i?µ ?? + ?4 = ?0 ?1 ?2 ?3 .
m
Proof. If one does not ask himself about the way to found the operators (6) (the
operators (5), whith form the P1,3 algebra, are well-known), the theorem validity may
be established by direct verification. Indeed, one obtains by direct calculation that
Qµ? satisfies the invariance condition (3)
i
1 1
(?µ p? ? ?? pµ )
2 2
(7)
[Qµ? , L 1 ]? = Fµ? L 1 , Fµ? =
m
2 2


and form together with Pµ , Jµ? the Lie algebra
[P? , Jµ? ]? = i(g?µ P? ? g?? Pµ ),
[Pµ , P? ]? = 0, [P? , Qµ? ]? = 0,
[Jµ? , J?? ]? = i(gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? ), (8)
1
[Qµ? , Q?? ]? = i(gµ? Q?? + g?? Qµ? ? gµ? Q?? ? g?? Qµ? ).
[Qµ? , J?? ]? =
2
But such calculations are very cumbersome. A more elegant and constructive way,
which shown the method to obtain the operators (6) is to transform eq. (4) to the
diagonal form. After such a transformation the theorem statements become obvious
ones.
Such a transformation may be carried out in two steps. First eq. (4) is multiplied
by the inversible differential operator
1 1
W =1? ?µ pµ ? (1 + i?4 )pµ pµ ,
2
m 2m
(9)
1 1
W = 1 + ?µ pµ ? (1 ? i?4 )pµ pµ .
2m2
m
As a result one obtains the equation
(10)
W L 1 ? = 0,
2


which is equivalent to the starting eq. (4). Then with the help of the isomeric operator
1 1
(1 + i?4 )?µ pµ ? 1 + (1 + i?4 )?µ pµ (11)
V = exp
2m 2m
one reduces eq. (10) to the diagonal form
??
L ? ? V (W L 1 )V ?1 ? = ?+ m + (pµ pµ ? m2 ) ? = 0, (12)
m
2



where ? = V ?, ?± = 1 (1 ± i?4 ).
2
Equation (12) is equivalent to the starting eq. (4) and contains the only matrix ?4 ,
which may be taken in the diagonal form without loss of generality. So it is evident
New invariance groups of the Dirac and Kemmer–Duffin–Petiau equations 395

that the matrices Qµ? = i?µ ?? commute with the operator L1 . These matrices
1
2
satisfy the commutation relations of the Lie algebra of the SU2 ? SU2 group and
satisfy the relations (8) together with the generators Pµ = V Pµ V ?1 = Pµ and
Jµ? = V Jµ? V ?1 = Jµ? .
To complete the proof it is sufficient to find the explicit form of the matrices
Qµ? in the starting ?-representation. Calculating Qµ? = V ?1 Qµ? V , one obtains the
operators (6). The theorem is proved.
Corollary 1. If one makes in (4), (9)–(l2) the substitution
ie
?µ pµ > ?µ = (pµ ? eAµ )?µ , pµ pµ > ?µ ? µ ? ?µ ?? Fµ? ,
2
where Aµ is the vector potential, and Fµ? is the tensor of the electromagnetic field,
the transformations (9)–(12) establish the one-to-one correspondence between the
solutions of the Dirac and of the Zaitsev–Gell–Mann equations [9].
Corollary 2. The above founded operators Qµ? may be used to find the constants of
motion for the particle interacting with external field. For instance the operator the
Q = ?abc Qbc (?)(Ha ? iEa ) is the constant of motion for a particle moving in the
homogeneous constant magnetic field H and the electric field E(Qbc (?)) is obtained
from (6) by the substitution pµ > ?µ .
Corollary 3. In theorem 1 the invariance condition of eq. (4) is formulated by the
language of Lie algebras, i.e. on the infinitesimal level. The natural question arises:
what sort of finite transformations are generated by Qµ? ? Using the explicit form of
the generators (6), one obtains these transformations in the form
?(x) > ? (x) = exp[iQab ?ab ]?(x) = (cos ?ab ? ?a ?b sin ?ab )?(x)+
1 ??(x) ??(x)
? ?b
+ (1 + i?4 ) sin ?ab ?a ,
m ?xb ?xa
?(x) > ? (x) = exp[iQ0a ?ab ]?(x) = (cosh ?0a ? i sinh ?0a ?0 ?a )?(x)+
i ??(x) ??(x)
? ?a
+ (1 + i?4 ) sinh ?0a ?0 ,
m ?xa ?x0 (13)
1
xµ > xµ = exp[iQab ?ab ]xµ exp[?iQab ?ab ] = xµ + (1 + i?4 ) sin ?ab ?
m
? (?a gµb ? ?b gµa )(cos ?ab ? ?a ?b sin ?ab ),
i
xµ > xµ = exp[iQ0a ?0a ]xµ exp[?iQ0a ?0a ] = xµ + (1 + i?4 ) sinh ?0a ?
m
? (?0 gµa ? ?a gµ0 )(cosh ?0a ? i?0 ?a sinh ?0a ),

where ?µ? = ???µ are the six transformation parameters (there is no sum by a, b).
Transformations (13) together with the Lorentz transformations form the 16-parame-
ter invariance group of the Dirac equation.
In the quantum field theory not only the Dirac equation (4) but the system of two
?
four-component equations for the two independent functions ? and ? is considered
usually. Such a system is equivalent to one eight-component Dirac equation

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