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?(x0 , p, p4 , s3 , t3 ).

As usually, this component is interpreted as the probability amplitude of finding (by
measuring at the given moment of the time t = x0 ) the indicated values p, p4 , s3 ,
t3 of the complete set P , P4 , S3 , T3 . The physical meaning of the operators P and
P4 is given in Section 1. We discuss below the definition and physical meaning of the
operators S3 , T3 in the class I.
Remind that in the case of P (1, 3) the operators Skl k, l = 1, 2, 3 in (16 ) which
constitute the spin vector S = (S23 , S31 , S12 ), are generators of the group O(3) (the
little group of the group P (1, 3) in the class I) and they are interpreted as an angular
momenta of proper rotations. More exactly they should be interpreted as an angular
momenta which are connected with intrinsic (internal) motion because when P = 0,
the angular momenta Jkl do not vanish but reduce to the spin angular monenta Skl .
In the case of P (1, 4) there are six angular momentum operators, which describe
the internal motion of particle (i.e., the motion when p = p4 = 0): Jkl/p=p4 = Skl ,
k, l = 1, . . . , 4. The operators Skl are generators of the group O(4) (the little group
of the group P (1, 4)) in the class I). They can be combined into two 3-dimensional
vectors S and T defined by components

1 1
Sa ? Ta ? (Sbc ? S4a ), (21)
(Sbc + S4a ),
2 2
where (a, b, c) = cycl(1, 2, 3). These components satisfy the relations

[Sa , Sb ] = iSc , [Ta , Tb ] = iTc ,
(22)
Sa , S 2 = Ta , T 2 = [Sa , Tb ] = 0.
On representations of the inhomogeneous de Sitter group 135

Remined that S 2 and T 2 are the invariants of the algebra O(4) being for irreduci-
ble representations D(s, t) of this algebra

S 2 = s(s + 1)? T 2 = t(t + 1)? (23)
1, 1,
where s, t = 0, 1 , 1, . . . and ? is the (2s + 1)(2s + 1)-dimensional unit matrix. It was
1
2
just the relations (22) and (23) to allow us [1] to interpret 3-vectors S and T as the
spin and isospin operators.
It is clear from (16 ) that in the representations of the class I the generators of
the algebra P (1, 4) are constructed not only from the spin operators but also from the
isospin operators (and, of course, of xk and pk ). In this sense in quantum mechanic
scheme based on the group P (1, 4) the spin and isospin are presented on the same foot,
unlike from the case of conventional theory. Furthermore, unlike from to the latter, in
our case both the spin and isospin are entered dynamically. Indeed, in the conventional
approach the group P (1, 3)?SU (2)T is taken as the total symmetry group, so that the
generators of SU (2)T commute with the generators of P (1, 3) (even in the presence
of interactions). The group P (1, 4) which we taken as a total symmetry group, is not
isomorphic to the group P (1, 3)?SU (2)T furthermore, as in can be seen from (21) and
(16 ), SU (2)T ? O(4) ? P (1, 4) as like as SU (2)S ? O(4) ? P (1, 4), and the isospin
operators (as like as the spin operators) do not commute with P (1, 3) ? P (1, 4).
The manifold of solutions of the equation (19) realizes in the case discussed the
irreducible representation D± (s, t) of the algebra P (1, 4), where the sings “±” refer
to the values ? = ±1 of the invariant — the sign of energy, the numbers as s and t
determine the eigenvalues of the invariants
W V W V
?
S2 = T2= (24)
+ , ,
2 2
4p 4p
2 p2 2 p2
which are invariants both of P (1, 4) and O(4).
In quantum scheme based on P (1, 4), possible states of an “elementary particle”
(when ? = +1) or “antiparticle” (when ? = ?1) with given values of s, t and p2 = ? 2
are states which constitute the representation space for an irreducible representation
D± (s, t) of the group P (1, 4). This is just the definition of the elementary particle
in the P (1, 4)-quantum scheme. The simplest states of this particle are identified by
eigenvalues of complete set of comuting variables. It is clear that the representation
D± (s, t), irreducible with respect to P (1, 4), is reducible with respect to P (1, 3) ?
P (1, 4) therefore the “elementary particle” defined here, is not elementary in the
conventional sense (i.e., with respect to the group P (1, 3)). Indeed, a solution ? of
Eq. (19) with given s and t contains componets identified not only by values of the
3-component s3 of spin but also by values of the 3-components t3 , of isospin, so that
the vector ? describes in fact the whole multiplet – the set of states with different
values of t3 , ?t ? t3 ? t (and, of course, of s3 , s ? s3 ? s). For example, the vector
?± with ? ± 1, s = 0 and t = 1 describes a meson isodublet like
2

?+ 1 ) K+ K0
(0,
?? =
??
+ 2
= , .
K?
K0
?+ 1 )
(0,?2


The parameter ? (the threshold value of the free state energy or the “bare” rest mass)
is the same for all the members of the given multiplet. Of course, the introduction
136 W.I. Fushchych, I.Yu. Krivsky

of a sutable interaction into the equation (19) leads to the mass splitting within a
multiplet.
The approach proposed may be found successful for a consequent description of
unstable systems (resonances, particles or systems with non-fixed mass) already in
the framework of the quantum mechanics3 . As it is known, the conventional quantum
mechanical approach deals a with finding complex eigenvalues of energy operators
which must be Hermitian in a Hilbert space of wave functions, i.e., in fact, one must
go put of the framework of Hilbert space; the latter is connected with breaking of
such a fundamental principles as unitarity, hermiticity etc. [12].
There are no similar difficulties in the quantum mechanical approach proposed.
Indeed, here the mass operator is an independent dynamical variable (1), it is Her-
mitian, defined in the Hilbert space; therefore one can find its eigenvalues m2 and
distributions ?(m2 ) in the same Hilbert space, as like as they find eigenvalues and
distributions for operators of energy, momentum and other dynamical variables. For
example, if we have a stationary wave function ? = {?(x, x4 , s3 , t3 )} of, generally
speaking, an unstable multiplet (we meant: a solution of an equation of the type (19)
with an interaction not depending on the time x0 ) then
2
v
?i m2 ?? 2 x4
2 3
(25)
?(m , s3 , t3 ) = dx dx4 e ?(x, x4 , s3 , t3 ) .

If the distribution (25) with the given s3 , t3 has one maximum, the experimentally
observed mass of the particle with given s3 , t3 is defined either by the position of the
maximum or form

d3 x dx4 ?? (x, x4 , s3 , t3 )(p2 + ? 2 )?(x, x4 , s3 , t3 )
m2 = (26)
? 4


and its mean lifetime ? is defined from
m2 ? 2 = 1. (27)
??
If the distribution (25) has more than one maximum, the position of them defines
an experimentally observed masses of unstable particles and the semi-widths of the
distribution (25) in the regions of maximums define the lifetimes of corresponding
unstable particles. If, finally, ?(m2 , s3 , t3 ) has a ?-like singularity in a point m2 = m2 ,
0
the m0 is identified with the mass of a stable particle.
It is important to emphasize that in accord with our interpretation, the parti-
cles experimentally observed are described not by the free equation (19), but by
an equation of the type (19) with a sutable interaction which may breakdown the
P (1, 4)-invariance, but, of course conserves the P (1, 3)-invariance4 . As for solutions
of the free equation (19) they are some hypothetical (“bare”) states which may not
correspond, to any real particles. From view point of this interpretation there are
two types of interactions: interactions which cause a “dressing” of particles and are
inhierent even in asymptotical states, and usual interactions which cause a scatteri-
ng processes of real (“dressed”) particles. Therefore, in particular, the 5-dimensional
3 The consequent consideration of such problems demands, obviously, the quantum field approach, but a
quantum mechanical approach can be regarded as a half-fenomenology.
4 In this sense the consideration of P (1, 4)-symmetry here presented is only a base for its sutable

violation — analogously to considerations and violations of SU (n)-symmetries.
On representations of the inhomogeneous de Sitter group 137

conservation law following from the free P (1, 4)-invariant scheme; may have not a
real sense.
In the interpretation of the P (1, 4)-scheme proposed we automatically have the
SU (2)T -systematic of particles. In contrast to the conventional systematics, our one
is a dynamical in the sense that for compaund model like those of Fermi–Yang,
Goldhaber–Gy?rgyi–Kristy and others we can write down an equation in which spin
o
and isospin variables are mixed non-trivially.
Emphasize, that the interpretation of the P (1, 4)-scheme proposed and, in particu-
lar, of the complete set of commuting variables mentioned above, was mainly based on
the definition (1) of the varlable-mass operator as an independent dynamical variable.
This interpretation does not pretend, of course, to be the only one and complete. In
particular, the problem of giving the “fifth coordinate” x4 the more immediate physical
sense than that one laying under its definition as a dynamical variable canonically
conjugated to the mass variable p4 , and the same problem refers to operators like
J04 , Ja4 , a = 1, 2, 3 we do not discusse here. The more detail discussion of these
problem is possible only in connection with considerations of solutions of equations
like Eq. (19) with sutable interactions what is not a subject of this article.
Here we have considered the P (1, n)-invariant equations of the Schr?dinger–Foldy
o
type in an arbitrary dimensional Minkowski space, in which the differential operators
?k ? ?/?xk of the “space” variables are presented under square root. This equations
describing the positive and negative states separatelly, are suitable for quasirelativistic
quantum mechanical considerations (e.g., for calculations of spin-isospin effects in
P (1, 4)-invariant equations with interactions included). Theoretic-field considerations
are usually based on equations of first order on ?µ . The general form of P (1, n)-
invariant linear on ?µ equation is
(Bµ ?µ ± ?)?± = 0, (28)
µ = 1, 2, . . . , n, n + 1,
where the operators Bµ are defined by the relations
[Bµ , J?? ] = ?µ? B? ? ?µ? B? , (29)
(µ, ?, ? = 1, . . . , n + 1).
For the representation of the class I the operators Bµ are finite-dimensional; for those
of the class III the operators Bµ can be both finite- and infinite-dimensional. Concrete
forms of operators Bµ can be founed by the method proposed in ref. [13].
In this paper we have not considered the problem of invariance of the equation (28)
as to discrete transformations, that is relatively to
xk = ?xk , x0 = ?x0 . (30)
This problem has been investigated by one of us [14]. As it is shown in [14] the equa-
tion (28) for n = 2m, m = 1, 2, 3, . . ., is neither invariant as to transformations (30)
nor
x0 = ?x0 , (31)
xk = xk , k = 1, 2, . . . , 2m.
Thus, in the field theory constructed on the basis of the groups P (1, 2), P (1, 4),
P (1, 6) and so on the theorem CP T may be broken down. It should be emphasized,
however, that the direct of the manifold of solutions {?+ } and {?? } T -, CP T -
invariant.
138 W.I. Fushchych, I.Yu. Krivsky

1. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1968, 7, 79;
Фущич В.И., Кривский И.Ю., Препринт ИТФ-68-72, Киев, 1968.
2. Wigner E.P., Ann. Math., 1939, 40, 149.
3. Широков Ю.М., ЖЭТФ, 1957, 33, 1196.
4. Jоos H., Fortschr. Phys., 1962, 10, 65.
5. Fоldу L., Phys. Rev., 1956, 102, 568.
6. Гельфанд И.М., Цейтлин М., ДАН СССР, 1950, 71, 1017.
7. Гельфанд И.М., Граев М.И., Труды Московского математического общества, 1959, 8; Изв.
АН СССР. Сер. матем., 1965, 29, 5.
8. Нederfledt G.C., Henning J., Fortschr. Phys., 1968, 16, 491.
9. Соколик Г.А., Групповые методы в теории элементарных частиц, Атомиздат, М., 1965.
10. O’Raifeartaigh L., Phys. Rev. Lett., 1965, 14, 575;
Jost R., Helv. Phys. Acta, 1966, 39, 369.
11. Фущич В.И., Укр. физ. журн., 1968, 13, 362.
12. Mathews P.T., Salam A., Phys. Rev., 1958, 112, 283.
13. Фущич В.I., Укр. фiз. журн., 1966, 8, 907.
14. Фущич В.И., Письма ЖЭТФ, 1969.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 139–163.

Уравнения Баргмана–Вигнера на
неоднородной группе де Ситтера
В.И. ФУЩИЧ, Л.П. СОКУР
In the paper the equations of the Bargman–Wigner type invariant under the
inhomogeneous de Sitter group P (1, 4) (the rotation and translation group in the
five-dimensional plane Minkovski space) are constructed. A detaled group-theoretical
analysis of these equations is carried out. The explicit form of the generators of the group
P (1, 4) is derived for the case when the basic invariants of this group P 2 = W = V = 0.
The question of the invariance of the Bargman–Wigner equations under the P -, T -, C-
transformation is considered.
Report presented at the 3d Conference on Axiomatical Field Theory and Theory
of Elementary Particles (Institute for Theoretical Physics, Kiev, Ukrainian SSR, Aрril
1969).
В работе построены уравнения типа Баргмана–Вигнера, инвариантные относитель-
но неоднородной группы де Ситтера (группы вращений и трансляций в пятимер-
ном плоском пространстве Минковского) P (1, 4). Проведен детальный теоретико-
групповой анализ этих уравнений. Найден явный вид генераторов группы P (1, 4) в
том случае, когда основные инварианты этой группы P 2 = W = V = 0. Рассмо-
трен вопрос об инвариантности уравнений Баргмана–Вигнера относительно T -, P -,
C-преобразований.
Работа была доложена на 3-ем Рабочем совещании по аксиоматической теории
поля и теории элементарных частиц, состоявшемся в ИТФ АН УССР в апреле 1969 г.

Введение
Идея использования пространств размерности выше, чем четыре, интенсивно
обсуждалась в физике довольно давно в связи с объединением теории тяготения
и электричества, а также в связи с построением волновой оптики в пятимерном
пространстве (обзор этих работ см. [1]). В настоящее время эта идея вновь ши-
роко обсуждается в результате объединения группы Пуанкаре P (1, 3) с группами
“внутренних симметрий” [2]. Полагают, что на этом пути удастся получить спектр
масс и другие характеристики элементарных частиц. Расширение четырехмерно-
го пространственно-временного континуума может оказаться также плодотворным
для описания нестабильных систем (резонансов) [3]. В связи с этим представляет
интерес детально рассмотреть одно из таких расширений.
В настоящей работе обсуждается минимальное расширение четырехмерного
пространства — 5-мерное пространство Минковского, в котором в качестве груп-
пы симметрии выбирается группа вращений и трансляций. Таким образом, группой
симметрии в 5-мерном плоском пространстве Минковского является неоднородная
группа де Ситтера, которую обозначим через P (1, 4). Ясно, что группа P (1, 4)
содержит в качестве подгруппы группу Пуанкаре P (1, 3), поэтому теория, постро-
енная на основе группы P (1, 4), будет определенным обобщением релятивистской
квантовой механики. Для построения основ квантовой механики, основывающей-
ся на группе P (1, 4), в качестве первого шага необходимо написать уравнения
Препринт ИТФ–69-33, Киев, 1969, № 33, 38 с.
140 В.И. Фущич, Л.П. Сокур

движения “частицы” (системы относительно P (1, 3)), инвариантные относительно

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