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of operators Bµ can be found by the method proposed in ref. [15].
For the case of the group P (1, 4) eqs. (44) referred to either Bµ or Sµ? describe
particles with either values of spin s or isospin t. These equations, however, contain
a lot of redundant components. The analysis of eqs. (44) with matrices Bµ answering
the question which representation of the group P (1, 4) is realized by the equation,
and the P (1, 4)-invariant split of the equation in principal and redundant parts, can be
made with the help of the method illustrated here for the case of the Kemmer-Duffin
equations (34).
As was shown in ref. [16], eq. (44) or any equation on the group P (1, 4) is invariant
under the discrete operators P , T , C if ? transforms by the following representation
of the group P (1, 4)

D+ (s, t) ? D? (s, t) ? D+ (t, s) ? D? (t, s). (46)
1. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1968, 7, 79.
2. Wigner E.P., Ann. Math., 1939, 40, 149.
3. Shirokov Yu.M., JETP (Sov. Phys.), 1957, 33, 1196.
4. Joos H., Fortschr. Phys., 1962, 10, 65.
5. Foldy L., Phys. Rev., 1956, 102, 568.
6. Gelfand I.M., Zeitlin M., DAN USSR, 1950, 71, 1017.
7. Gelfand I.M., Grayev M.I., Trudy Moskovskogo Matematicheskogo obshchestva, 1959, 8; Izv. AN
USSR, Ser. Math., 1965, 29, 5.
8. Dirac P.A.M., Proc. Roy. Soc. A, 1936, 155, 447.
9. Gelfand I.M., Minlos R.A., Shapiro Z.Ya., Representations of the rotation and Lorentz group and
their applications, Moscow, Fizmatgiz, 1958 (in Russian).
10. Umezawa H., Quantum field theory, North-Holland, Amsterdam, 1956.
11. Fushchych W.I., Preprint ITF 69-17, Kiev, 1969.
12. d’Espagnat B., Prentki J., Phys. Rev., 1955, 99, 328.
13. Foldy L., Wouthuysen S., Phys. Rev., 1950, 78, 29.
14. Case K.M., Phys. Rev., 1955, 100, 1513.
15. Fushchych W.I., Ukrain. Phys. J., 1966, 8, 907.
16. Fushchych W.I., J. Theor. Math. Phys., to be published.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 126–138.

On representations of the inhomogeneous
de Sitter group and on equations
?
of the Schrodinger–Foldy type
W.I. FUSHCHYCH, I.Yu. KRIVSKY
This paper is a continuation and elaboration of our works [1] where some approach to
the variable-mass problem were proposed. Here we have found a concret realization
of irreducible representations of the inhomogeneous group P (1, n) — the group of
translations and rotations in (1 + n)-dimensional Minkowski space in two classes (when
2 2 2 2
P0 ? Pk > 0 and P0 ? Pk < 0). All the P (1, n)-invariant equations of the Schr?dinger–
o
Foldy type are written down. Some questions of a physical interpretation of the quantum,
mechanical scheme based on the inhomogeneous de Sitter group P (1, n) are discussed.
Report presented at the Conference on Composite Models of Elementary Particles
(Institute for Theoretical Physics, Kiev, Ukrainian SSR, June 1968).

Данная работа является продолжением и развитием работ [1], где был предложен
определенный подход к проблеме переменной массы. Здесь построена конкретная
реализация неприводимых представлений неоднородной группы P (1, n) вращении
и трансляций в (1 + n)-мерном пространстве Минковского в двух классах (когда
2 2 2 2
P0 ? Pk > 0 и P0 ? Pk < 0). Выписаны P (1, n)-инвариантные уравнения ти-
па Шредингера–Фолди. Рассмотрены некоторые вопросы физической интерпрета-
ции квантовомеханической схемы, основанной на неоднородной группе де Ситтера
P (1, 4).
Работа была доложена на Рабочем совещании по составным моделям элементар-
ных частиц, состоявшемся в ИТФ АН УССР в июне 1968 г.

1. Introduction
Recall here the initial points of our approach to the variable mass problem proposed
in ref. [1]:
A. The square of variable mass operator is defined as an independent dynamical
variables

M 2 ? ? 2 + P4 ,
2
(1)

where ? is a fixed parameter and P4 is an operator lice the components of three-
momentum P , which commutes with all the generators of the algebra P (1, 3) of the
Poincar? group.
e
B. The relation between the energy P0 , three-momentum P and variable-mass M
of a physical system is remained to be conventional (here everywhere = c = 1):

P0 = P 2 + M 2 ? P k + ? 2 ,
2 2
(2)
k = 1, 2, 3, 4.

C. The spaced p ? (p0 , p1 , . . . , p4 ) and x ? (x0 , x1 , . . . , x4 ) are assumed to be plane
and reciprocally conjugated. It follows then from А, В and С that the generalized
Препринт ИТФ–69-1, Киев, 1969, № 1, 22 с.
On representations of the inhomogeneous de Sitter group 127

relativistic group symmetry is the inhomogeneous de Sitter group1 P (1, 4) — the
group of translations and rotations in five-dimensional Minkowski space. This group
is a minimal extention of the conventional group of relativistic symmetry — the
Poincar? group P (1, 3).
e
In this paper we shall present a further studying of the approach proposed in
ref. [1]. In particular, the main assertions which were briefly formulated in ref. [1], are
generalized here and their detail proofs are given. In Section 2 a concrete realization
of irreducible representations for the generators Pµ , Jµ? of the algebra P (1, n) with
arbitrary n carried out, which made it possible to give a proof of the P (1, n)-invariance
of the Schr?dinger–Foldy type equations written flown in ref. [1] for n = 4. Some
o
questions of a physical interpretation of quantum mechanical scheme based on the
group P (1, 4) are considered in Section 3.
2. Realizations of the algebra representations
For the sake of generality all the considerations are made here not for the de Sitter
group P (1, 4) but for the group P (1, n) of translations and rotations in dimensional
Minkowski space, which leaves unchanged, the form

x2 ? x2 ? x2 ? · · · ? x2 ? x2 ? x2 ? x2 ,
n µ
0 1 0 k
(3)
µ = 0, 1, 2, . . . , n; k = 1, 2, . . . , n,

where xµ are differences of point coordinates of this space.
Commutation relations for the generators Pµ , Jµ? of the algebra P (1, n) are
choosen in the form

?i [Pµ , J?? ] = gµ? P? ? gµ? P? , (4a)
[Pµ , P? ] = 0,

?i [Jµ? , J?? ] = gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? , (4b)

where g00 = 1, ?gkl = ?kl , Pµ is Kroneker symbol, Pµ are operators of infinitesimal
displacements and Jµ? are operators infinitesimal rotations in planes (µ?).
Authors of refs. [2–5] have studed all the irreducible representations of the Poi-
ncar? group P (1, 3) and have found the concrete realization for the generators of its
e
algebra. Their methods we generalize here for the case of group P (1, n). But all the
treatments are carried out in more general and compact form then it was done even
for the case of P (1, 3).
For representations of the class I (P 2 ? P0 ? Pk > 0) when the group O(n) of
2 2

rotations in a n-dimensional Euclidean space is the little group of the group P (1, n),
the generators Pµ , Jµ? are of the form

P = p ? (p0 , p1 , . . . , pn ) ? (p0 , pk ),
(5)
Skl pl
J0k = x[0 pk] ?
Jkl = x[k pl] + Skl , ,
p2 + p2 + p2
k

where

P 2 ? p2 ? p2 ? p2 > 0, x[µ p?] ? xµ p? ? x? pµ ,
0 k

1 The algebras and groups connected with them are designated here with the same symbols.
128 W.I. Fushchych, I.Yu. Krivsky

operators xµ , pµ are defined by relations

[xµ , pµ ] = ?igµ? , (6)
[xµ , x? ] = [pµ , p? ] = 0,

and Skl are matrices realizing irreducible representations D(s, t, . . .) of the algebra
O(n) which have been completely studied in ref. [6] (here the numbers s, t, . . . are
numbers which identify a correspondence irroducible representations of the algebra
O(n)). Using (6) and relations for the generators Skl (which are not written down
here), one can immediately verify that (5) actually satisfy the relations (4). Since
in this class the little group of the group P (1, n) coincides with the compact group
O(n), all the irreducible representations of the group P (1, n) are here unitary and
finite-dimensional (concerning a set of “spin” indexes s3 , t3 , . . .).
A concrete form of the operators Pµ , Jµ? which are defined by Eqs. (5), depends
on a choice of concrete form of matrices Skl and operators xµ , pµ which are defined
by relations (6). The concrete form of the operators xµ , pµ and Skl depends on what of
operators, constituting a complete set of commuting dynamical variables are operators
of multiplicationd (“diagonal operators”). The sets (P0 , P1 , . . . , Pn , S3 , T3 , . . .) or
(x0 , x1 , . . . , xn , S3 , T3 , . . .) are examples of such a complete sets where S3 , T3 , . . . are
all the independent commuting generators of the algebra O(n). In the general case
a complete set of dynamical variables is constructed from the corresponding number
of commuting combinations of operators xµ , pµ and Skl . Different concrete forms of
operators Pµ , Jµ? which are defined by the choice of other complete set as diagonal,
are connected by unitary transformations. The form (5) for the generators is the most
general in the sense that it is not bound to the choice of concrete complete set as
diagonal.
A few words about a apace of vectors ?, in which the operators (5) are defined. It
is an Hilbert space of vector-functions depending on the eigenvalues of operators of
a diagonal complete sel. For instance, in the x-representation where the operators xµ
are diagonal (i.e., are operators of multiplication) and, as it follows from relations (6),
pµ = igµ? ?? , ?? ? ?/?x? the operators (5) are defined in the Hilbert apace of
the vector-function ? = ?(x) = ?(x0 , x1 , . . . , xn ) of (1 + n) independent variables
xµ . The components of a vector ? are functions not only of x0 , x1 , . . . , xn but also
of auxiliary variables s3 , t3 , . . ., i.e., are functions ?(x0 , x1 , . . . , xn , s3 , t3 , . . .), where
s3 , t3 , . . . are eigenvalues of “spin” operators S3 , T3 , . . . and, as it is known, take
discrete values. In p-representation where pµ are operators of multiplication and,
according to (6), xµ = igµ? ?/?p? vector-functions are ? = ?(p) ? ?(p0 , p1 , . . . , pn )
and their components are ?(p0 , p1 , . . . , pn , s3 , t3 , . . .). The scalar product of vectors ?
is defined as

(?, ? ) ? d1+n x ?+ (x)? (x) =

?? (x, s3 , t3 , . . .)? (x, s3 , t3 , . . .) =
d1+n x
=
(7)
s3 ,t3 ,...


?? (p, s3 , t3 , . . .)? (p, s3 , t3 , . . .),
d1+n p ?+ (p)? (p) = d1+n p
=
s3 ,t3 ,...


where d1+n x = dx0 dx1 . . . dxn , ? and ? being connected by Fourier-transformations.
On representations of the inhomogeneous de Sitter group 129

For representations of the class III (P 2 = P0 ? Pk < 0) when the little group of
2 2

the group P (1, n) is already uncompact group O(1, n ? 1) of rotations in 1 + (n ? 1)-
dimensional pseudo-Euklidean space, the generators Pµ , Jµ? are of the form
P = p ? (p0 , pk ) = (p0 , pa , pn ),
Sab pb ? Sa0 p0
Jan = x[a pn] ?
Jab = x[a pb] + Sab , ,
?p2 ? p2 + p2 + ?p2 (8)
a 0
S0a pa
J0n = x[0 pn] ?
J0a = x[0 pa] + S0a , ,
?p2 ? p2 + p2 + ?p2
a 0

where a, b = 1, 2, . . . , n ? 1 the operators xµ , pµ are defined by relations (6) as before
and the operators (S0a , Sab ) are generators of the algebra O(1, n ? 1) in corresponding
irreducible representations which have been well studied by Gelfand and Grayev [7].
Components of vector-functions, in the space of which the operators (8) are defi-
ned, are the functions of variables s3 , t3 , . . . (besides of variables xµ or pµ , of course)
which are the eigenvalues of the corresponding independent commute generators of the
algebra O(1, n ? 1). In contrast to the case I, in this case the variables s3 , t3 , . . . may
take both discrete and continual valuea. Remind (see ref. [7]) that the group O(1, n?1)
has both unitary and nonunitary representations, all the unitary representations bei-
ng infinite-dimensional (in the last case the “spin” variables s3 , t3 , . . . take continual
values). In accordance with this, among the representations of the group P (1, n) in
the class III there will be both unitary (only infinite-dimensional) and nonunitary
(finite- and infinite-dimensional) irreducible representations.
Now we shall give here a recipe of constructing the representations of the class III
from those of the class I.
Note first that if operators P , J realize representation of the algebra P (1, n), then
operators P , J being defined by means of

(9a)
(P0 , Pa , Pn ) = (?iPn , Pa , iP0 ),
? ?? ?
?iJna Jn0
J0a J0n
? ?=? ?, (9b)
Jab Jan Jab iJa0

realize a representation of the algebra P (1, n) too. To proof this assertion, it is enough
to verify that from the commutation relations (4) for P , J and from definitions (9), it
follows that the operators P , Jsatisfy the commutation relations (4) too.
Define, further, the operators x, p and s by means of the following relations
(10a)
(x0 , xa , xn ) = (?ixn , xa , ix0 ),
from
(10b)
(p0 , pa , pn ) = (?ipn , pa , ip0 ),

(10c)
(Sab , San ) = (Sab , iSa0 ).

From (6) and (10a), (10b) it follows that operators x, p satisfy the relations (6)
too, whereas the operators (S0a , Sab ) defined by Eqs. (10c) satisfy the commutation
130 W.I. Fushchych, I.Yu. Krivsky

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