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0 0 a22
коммутирует с оператором Lc . В пространстве матриц размерности 3 ? 3 выберем
базис, состоящий из матриц
?ab = sa sb + sb sa ? ?ab , (29)
I, sa , a, b = 1, 2, 3,
где sa определены формулами (7), I — единичная матрица, причем
(30)
?11 + ?22 + ?33 = I.
Используя этот базис, в множестве матриц (28) выбираем базисные алгебры GL(2)
в виде
1
?1 = ??12 , (?22 ? ?11 ), (31)
?0 = I, ?2 = ?3 = s3 .
2
? D(0) алгебры GL(2) и удовлетво-
1
Матрицы (31) реализуют представление D 2
ряют коммутационным соотношениям
(32)
[?a , ?b ]? = 2i?abc ?c , [?0 , ?a ]? = 0, a, b, c = 1, 2, 3.
Осуществив над матрицами {?0 , ?a } обратное преобразование W ?1 , найдем яв-
ный вид операторов (26), удовлетворяющих условию инвариантности (8).
Следствие. Уравнение Ламе (1) инвариантно относительно преобразований из
группы SU (2) ? GL(2)
1?
1 + ?11 (cos ?a ? 1) + i?a sin ?a u, (33)
u = exp(i?a ?a )u =
2
?
a = 1, 2, 3; по a нет суммирования, где ?a , ?11 определены формулами (25),
(26).
В заключение приведем теорему о негеометрической симметрии уравнения ти-
па (3) (обобщенного уравнения Ламе), когда (2s + 1) ? (2s + 1) матрицы sa ре-
ализуют конечномерное представление D(s) алгебры Ли группы SU (2). Число s,
характеризующее неприводимое представление SU (2), может быть целым или по-
луцелым.
Теорема 4. Алгеброй инвариантности уравнения типа Ламе (3) является ал-
гебра GL(2) ? GL(2) ? · · · ? GL(2), где число слагаемых в прямой сумме равно
целой части числа 1 (2s + 1), s > 1.
2
Доказательство этой теоремы аналогично доказательству теоремы 3.
34 В.И. Фущич, В.В. Наконечный

1. Фущич В.И., О новом методе исследования групповых свойств систем дифференциальных урав-
нений в частных производных, В кн.: Теоретико-групповые методы в математической физике,
Киев, 1978, 5–44.
2. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М.: Наука, 1978, 400 с.
3. Чиркунов Ю.А., Групповое свойство уравнений Ламе, В сб.: Динамика сплошной среды, вып.
14, Новосибирск, 1973, 128–130.
4. Фущич В.И., О новом методе исследования групповых свойств уравнений математической фи-
зики, ДАН СССР, 1979, 246, № 4, 846–850.
5. Fushchych W.I., Nikitin A.G., Conformal invariance of relativistic equations for arbitrary spin
particles, Lett. Math. Phys., 1978, 2, № 2, 471–475.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 35–46.


Reduction of the representations
?
of the generalised Poincare algebra
by the Galilei algebra
W.I. FUSHCHYCH, A.G. NIKITIN

The realisations of all classes of unitary irreducible representations of the generalised
Poincar? group P (1, 4) have been found in a basis in which the Casimir operators of its
e
important subgroup, i.e. the Galilei group, are of diagonal form. The exact form of the
unitary operator which connects the canonical basis of the P (1, 4) group and the Galilei
basis has been established.


1. Introduction
Some years ago it was proposed to use the generalised Poincar? group P (1, 4)
e
the group of displacements and rotations in five-dimensional Minkovsky space, for
the description of particles with variable masses and spins (Fushchych and Krivsky
[9, 10], Fushchych [8]). This and other generalised groups P (1, n), P (2, 3) etc were
considered and used successively by Castell [4], Aghassi et al [1], Barrabes and
Henry [3], Elizalde and Gomish [5] and many others.
The main property of the P (1, 4) group is that it contains the Poincar? groupe
1
P (1, 3) as well as the Galilei group G(3) as its subgroups . So the P (1, 4) group
unified the groups of motion of relativistic and non-relativistic quantum mechanics.
For the elucidation of the physical grounds of the generalised quantum mechanics
based on the P (1, 4) group (Fushchych and Krivsky [9, 10, 11] the important problem
is the reduction of the irreducible representations IR of the P (1, 4) group, or the Lie
algebra of the P (1, 4) group, by the IR of its subgroups, or its subalgebras2 . The
problem of the reduction of IR of the P (1, 4) algebra corresponding to the time-like
five-momenta by its subalgebra P (1, 3) has been solved (Fushchich et al [12], Nikitin
et al [15]), i.e. the type of representations of the P (1, 3) algebra contained in the IR
of the P (1, 4) algebra has been investigated and the unitary operator was found which
connects the canonical basis of the P (1, 4) group representation with the P (1, 3)
basis, in which the Casimir operators of the Poincar? group have the diagonal form
e
(the spectrum of these operators is nondegenerate).
In this paper we find the realisation of the IR of the P (1, 4) algebra in the “Galilei
basis” namely, in the basis in which the invariant operators of the Galilei subalgebra
are diagonal ones. We also obtain the explicit form of the unitary operator, which
carries out the reduction P (1, 3) > G(2) which plays an important role in the null-
plane approach (see e.g. Leutwyler and Stern [13]).

J. Phys. A: Math. Gen., 1980, 13, P. 2319–2330.
1 The paper of Fedorchuck [6] is devoted to the classification and the description of all subgroups of the
P (1, 4) group.
2 We will indicate the groups and the corresponding Lie algebras by the same indices.
36 W.I. Fushchych, A.G. Nikitin

2. Statement of the problem
The Lie algebra of the P (1, 4) group is specified by the fifteen generators Pµ , Jµ?
(µ, ? = 0, 1, 2, 3, 4) which satisfy the commutation relations
[Pµ , J?? ] = i(gµ? P? ? gµ? P? ),
[Pµ , P? ] = 0,
(2.1)
[Jµ? , J?? ] = i(gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? ).
The algebra (2.1) has three main invariant (Casimir) operators (Fushchych and Krivs-
ky [9, 10])
1 1
P 2 = Pµ P µ = P0 ? P 2 ? P 4 , V2 = ? Jµ? ?µ? ,
2 2
(2.2)
V1 = ?µ? ?µ? ,
2 4
where
1
?µ???? J ?? P ? .
?µ? =
2
As in the case of the Poincar? group, one can specify four different classes of the
e
representations of the algebra (2.1), corresponding to P 2 > 0, P 2 = 0, P 2 < 0 and
Pµ ? 0 (in the last case one arrives at the representations of the homogeneous group
SO(1, 4), which are not considered here).
Algebra (2.1) contains the Lie algebras of the Poincar? and of the Galilei groups
e
as subalgebras. In order to select the subalgebra P (1, 3) it is enough to consider the
relations (2.1) for µ, ? = 4. The subalgebra G(3) may be obtained by the transition to
the new basis
1
? ?
P0 = (P0 ? P4 ), M = P0 + P 4 , Pa = Pa , K = J04 ,
2 (2.3)
1 1
G? = (J0a ? J4a ).
G+ = J0a + J4a ,
Ja = ?abc Jbc , a a
2 2
The operators (2.3) satisfy the commutation relations
?? ? ? ??
[P0 , Pa ] = [P0 , M ] = [Pa , M ] = [Pa , Pb ] = 0,
?
[P0 , Ja ] = [M, Ja ] = [G+ , G+ ] = [M, G+ ] = 0,
a a
b
(2.4)
? ? +?
[Pa , Jb ] = i?abc Pc , [Ga , Pb ] = i?ab M,
? ?
[P0 , G+ ] = iPb ,
[Ja , Jb ] = i?abc Jc , b


[P0 , G? ] = [G? , G? ] = 0, [G? , M ] = ?iPa , [G? , Jb ] = i?abc G? ,
? ?
a a a a c
b
?
?? ? ? ?
[Ga , Pb ] = ?i?ab P0 , [P0 , K] = ?iP0 , (2.5)
+
[Ga , Gb ] = i(?abc Jc + ?ab K),
[G± , K] = ±G± .
?
[Pa , K] = [Ja , K] = 0, [M, K] = iM, a a

The commutation relations (2.4) specify the Lie algebra of the extended Galilei
group (Bargman [2]). The invariant operators of this algebra are given by the formulae
?
?
C1 = 2M P0 ? P 2 , C2 = (M J ? P ? G+ )2 , (2.6)
C3 = M.

Our aim is to find the realisations of the generators (2.3) for any class of IR of the
P (1, 4) algebra, in a basis where the Casimir operators (2.6) have a diagonal form.
This enables us to answer the question what IR of the G(3) algebra are contained
Reduction of the representations of the generalised Poincar? algebra
e 37

in the given representation of the P (1, 4) algebra and to establish the connection
between the vectors in the Poincar? and in the Galilei bases.
e
The realisations of all IR of the P (1, 4) algebra have already been found (Fushchych
and Krivsky [9, 10, 11], Fushchych [8]). So the problem of the description of the IR of
the P (1, 4) algebra in the Galilei basis reduces to transforming the known realisation
to a form in which the operators (2.6) are diagonal.

3. The representations with P 2 ? 0
Let us consider the IR of the P (1, 4) algebra, which corresponds to the positive
values of the invariant operator P 2 = ? 2 > 0. The generators Pµ , Jµ? in the canonical
basis |pk , j3 , ?3 ; ?, j, ?, ? have the form (Fushchych and Krivsky [9, 10])
1/2
P0 = ?E ? ? p2 + p2 + ? 2 , Pk = p k ,
4

? ?
? pk
Jkl = i pl + Skl , k, l = 1, 2, 3, 4, (3.1)
?pk ?pl
? Skl pl
J0k = ?i?E ?? ? = ±1,
,
E+?
?pk
where Skl (k, l = 1, 2, 3, 4) are the generators of the IR D(j, ? ) of the SO(4) group.
The basis of the realisation (3.1) is formed by the vectors |pk , j3 , ?3 ; ?, ?, ? , which
are the eigenfunctions of the complete set of the commuting operators
1 1
(?12 ? ?43 ), ? = P0 /|P0 |,
T = Pk , J3 = (?12 + ?43 ), T3 = ?
2 2
1 1
(V1 ? 2??V2 ),
J2 = T2 = P 2,
(V1 + 2??V2 ),
2 2
4? 4?
with the eigenvalues pk , j3 , ?3 , ?, j(j + 1), ? (? + 1) and ? 2 correspondingly, where j
and ? are the integers or half-integers labelling the IR of the SO(4) group,
j3 = ?j, ?j + 1, . . . , j, ?3 = ??, ?? + 1, . . . , ?, ? = ±1, ?? < pk < ?.
The basis vectors may be normalised according to
pk , j3 , ?3 ; ?, j, ?, ? | pk , j3 , ?3 ; ?, j, ?, ? = 2E?(pk ? pk )?j3 j3 ??3 ?3 ,
and the generators (3.1) are Hermitian with respect to the scalar product

d4 p/E ?† (pk , j3 , ?3 )?2 (pk , j3 , ?3 ). (3.2)
(?1 , ?2 ) = 1

The basis of the IR of the P (1, 4) algebra, in which the invariant operators (2.6) of
the G(3) algebra and the operators Pa (a = 1, 2, 3) and S3 = J3 ?(1/m)(P2 G+ ?P1 G+ )
1 2
have the diagonal form, will be called “Galilei basis” (or “G(3) basis”) and denoted by
|pa , m, s, s3 ; ?, j, ?, ? .
We will normalise the basis vectors as
pa , m, s, s3 ; ?, j, ?, ? | pa , m , s , s3 ; ?, j, ?, ? = 2m?(m ? m )?(pa ? pa )?ss ?s3 s3 .
This will lead us to the scalar product
?
dm
d3 p ?† (s, s3 , m, p)?2 (s, s3 , m, p). (3.3)
(?1 , ?2 ) = 1
m
|j?? |?s?j+? ?
38 W.I. Fushchych, A.G. Nikitin

Our task is to establish the explicit form of the generators of the P (1, 4) group
in the Galilei basis and to find the transition operator, which connects the canonical
and Galilei bases. First we substitute (3.1) into (2.3) and (2.6) and obtain the Galilei
generators Pµ , Ja , G+ the invariant operators Ca and the remaining generators G? ,
?
a a
K in the canonical basis in a form
1
?
P0 = (?E ? p4 ), Ja = ?i(p ? (?/?p))a + Sa ,
M = ?E + p4 ,
2
(3.4)
?Sab pb ? S4a (E + ? + ?p4 )
= x4 pa ? M xa ?
G+ ,
E+?
a


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