стр. 28 |

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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