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легко обобщается и на случай группы P(1, 4). Это связано с наличием пяти ма-
триц ?µ , µ = 0, 1, 2, 3, 4, удовлетворяющих алгебре (5.1), причем в формализме
Рарита–Швингера для P(1, 4) равноправно используются все пять матриц ?µ . За-
метим, кстати, что в случае алгебры КДП (6.2) ситуация иная; не существует
пятой 10 ? 10-матрицы, удовлетворяющей алгебре (6.2).
В заключении отметим, что общий вид линейных по ?µ P(1, 4)-инвариантных
уравнений выглядит как

(Bµ ?µ + ?)? = 0, (6.16)

где эрмитовы матрицы Bµ удовлетворяют алгебре

[Bµ , J?? ] = gµ? B? ? gµ? B? ,
(6.17)
Bµ B? B? ? Bµ B? B? ? B? B? Bµ + B? B? Bµ = ?µ? B? ? ??? Bµ .
В зависимости от конкретной реализации этой алгебры уравнение (6,16) описыва-
ет частицы с теми или иными значениями спина s и изоспина t. Эти уравнения,
однако, содержат много лишних компонент. Выяснение того, какое именно пред-
ставление алгебpы P(1, 4) реализует уравнение (6.16) с теми или иными матри-
цами алгебры (6.17), а также выделением инвариантным образом существенных
компонент проводится с помощью методики (6.1) Кеммера-Дэффина.
106 В.И. Фущич, И.Ю. Кривский

1. Соколик Г.А., Групповые методы в теории элементарных частиц, Атомиздат, М., 1965.
2. O’Reifertaigh L., Phys. Rev. Lett., 1965, 14, 575;
Jost R., Helv. Phys. Acta., 1966, 39, 369.
3. Фущич В.И., УФЖ, 1968, 13, 878.
4. Румер Ю.Б., Исследования по 5-оптике, Физматгиз, М., 1956.
5. Fushchych W.I., Krivski I.Yu., Nucl. Phys., 1968, 17, 79.
6. Швебер С., Введение в релятивистскую квантовую теорию поля, ИЛ, М., 1963.
7. Wigner E.P., Ann. Math., 1939, 40, 149.
8. Широков Ю.М., ЖЭТФ, 1957, 33, 1196.
9. Foldy L., Phys. Rev., 1956, 102, 568.
10. Базь А.И., Зельдович Я.Б., Переломов А.М., Рассеяния, реакции и распады в нерелятивистской
квантовой механике, Физматгиз, М., 1967.
11. Mathews P.T., Salam A., Phys. Rev., 1958, 112, 283.
12. Гельфанд И.М., Минлос Р.А., Шапиро З.Я., Представление группы Лоренца, Физматгиз, М.,
1958.
13. Fronsdal C., Phys. Rev., 1967, 156, 1665;
Nambu Y., Phys. Rev., 1967, 160, 1171;
Takabayasi T., Prog. Theor. Phys., 1967, 37, 767;
Stoyanev D., Todorov I., ICTP, preprint IC/67/58.
14. Newton T.D., Ann., Math., 1950, 51, 730.
15. Foldy L., Wouthuysen S., Phys. Rev., 1950, 78, 29.
16. Case K.M., Phys. Rev., 1955, 100, 1513.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 107–113.

Equations of motion in odd-dimensional
spaces and T -, C-invariance
W.I. FUSHCHYCH
The properties of the equation of Dirac type in three-dimensional and five-dimensional
Minkowski space-time with respect to time reflection (in sense of Pauli and Wigner)
as well as to the operation of charge conjugation are investigated. P -, T -, C-invariance
of Dirac equation for the cases of four components (in three-dimentional space) and
eight components (in five-dimensional space) is established. Within the framework of
the Poincar? group a relativistic equation is suggested wich describes the movement of
e
a particle with non-fixed (indefinite) mass in external electromagnetic field.
Reported at the Seminar in Institute for Theoretical Physics, Kiev, Ukrainian SSR.

Исследованы свойства уравнения типа Дирака в трехмерном и пятимерном прост-
ранствах Минковского относительно отражения времени (в смысле Паули и Ви-
гнера) и операции зарядового сопряжения. Показано, что четырехкомпонентное (в
трехмерном пространстве) и восьмикомпонентное (в пятимерном пространстве) урав-
нения Дирака P -, T -, C-инвариантны. В рамках группы Пуанкаре предложено ре-
лятивистское уравнение, описывающее движение частицы с нефиксированной (нео-
пределенной) массой во внешнем электромагнитном поле.
Работа была доложена на семинаре в Институте теоретической физики АН УС-
СР.

Introduction
F. KIein and latter de Broglie pointed out the usefulness of spaces with more than
four dimensions for the construction of the physical theories. This idea was intensively
developed in 1930–1940 years bу many authors who tried to unify the gravitation and
electromagnetic theories. Nowadays it is widely developed in connection with the
extension of the Poincar? group (P(1, 3)) as well as with idea of combining P(1, 3)
e
with group of internal symmetries (a review of this works can be found in [1]).
In the works [2] the mass operator was proposed to be defined as one like mo-
mentum or angular momentum operator, i.e. we proposed to define the mass operator
to be not a Casimir operator but the generator of a group wich has the Poincar? e
group as its subgroup. For such a group in papers [3, 4] the inhomogeneous de
Sitter, group is chosen — a group of rotations and translations in 5-dimensional flat
Minkowski space-time with the square-mass operator beings related to the generator
P4 (of group P(1, 4)) in such a way
M 2 = ? 2 + P4 .
2


In the present work the P -, T -, C-invariance properties of the simplest equations
invariant under the group P(1, 4) are investigated.




Препринт ИТФ–69-17, Киев, 1969, № 17, 13 с.
108 W.I. Fushchych

§ 1. Dirac equation within P(1, 4) scheme
and P -, T -, C-transformations
The simplest equations invariant under P(1, 4) group are Dirac equations wich in
the Hamilton form can be writen down as following:
??+ (t, x)
+ +
(1.1)
H ? (t, x) = i ,
?t
??? (t, x)
H ? ?? (t, x) = i (1.2)
,
?t
?
H ± ? ?k pk ± ??, pk = ?i , k = 1, 2, 3, 4,
?xk (1.3)
x ? (x1 , x2 , x3 .x4 ),
?k = ?0 ?k , ? = ?0 ,

where ?µ are five four-dimensional Dirac matrices (µ = 0, 1, 2, 3, 4).
The invariance of equation (1.1) (or (1.2)) under space-inversion xk > ?xk is
obvious sinse in (1 + 4)-dimensional Minkowski space-time this inversion is reduced
to a rotation.
Let us clear up now the question of the invariance of the equation (1.1) (or (1.2))
under the time reflection (t > ?t) and charge conjugation. To this aim we write
down the generators of the group P(1, 4) defined of the solutions of the equations (1.1)
and (1.2) explicitely
P0 = H + , Pk = p k ,
i
Jkl = xk pl ? xl pk + ?l ?k ,
2
(1.4)
1
J0k = x0 pk ? (xk P0 + P0 xk ),
2
[xk , pl ] = i?kl , [xk , xl ] = [pk , pl ] = 0.
According to Pauli the time-reflection operator T p satisfies the conditions
2
T p ?(t, x) = ? p ?(?t, x), (T p ) = 1, (1.5)

[T p , P0 ]+ = 0, [T p , Pk ]+ = 0, [T p , Jkl ]? = 0, [T p , J0k ]+ = 0, (1.6)

where ? p is a (4 ? 4)-matrix.
According to Wigner the time-reflection operator T w must satisfy the following
conditions
T w ?(t, x) = ? w ?? (?t, x),
2
(T w ) = 1, (1.7)

[T w , P0 ] = 0, [T w , Pk ]+ = 0, [T w , Jkl ]+ = 0, [T w , J0k ] = 0, (1.8)

where ? w is a (4 ? 4) matrix.
Finaly the charge-conjugation operator must satisfy the conditions1
C?(t, x) = ? c ?? (t, x), C 2 = 1, (1.9)
general the squares of operators T p , T w and C are equal to unity to within a multiplicative factor
1 In

of unit modulus.
Equations of motion in odd-dimensional spaces and T -, C-invariance 109

(1.10)
[C, P0 ]+ = [C, Pk ]+ = 0, [C, Jµ? ]+ = 0,

where ? c is a (4 ? 4) matrix.
Matrices ? p , ? w and ? c can be representated in folowing form
? p = ap ?µ + ap ?µ ?? , (1.11)
µ < ?,
µ µ?

? w = aw ?µ + aw ?µ ?? , (1.12)
µ < ?,
µ µ?

? c = ac ?µ + ac ?µ ?? , (1.13)
µ < ?,
µ µ?

where aµ , aµ? are the arbitrary numbers (µ = 0, 1, 2, 3, 4).
Using (1.11) and (1.13) one can immediately verify that the relations (1.6) and
(1.10) are satisfied only for the zero-matrices ? p and ? c . Relation (1.7) is satisfied if
? w = ?1 · ?3 .
Thus, the equation (1.1) or (1.2) is T p -, C-noninvariant but P , T w -invariant. This
means that the four-component Dirac equations in five-dimensional scheme are not
P T C-invariant as it was pointed out in [4, 5].
This result is a consequence of the fact that in contrary to the usual Dirac equati-
on (1.1) (or (1.2)) do not describe a particle and antiparticle. In fact the generators of
the group P(1, 4) given in the form (1.4) defined on the manifold of all solutions of
equations (1.1) and (1.2) realize the representations
D+ (1/2, 0) ? D? (0, 1/2), (1.14)

D+ (0, 1/2) ? D? (1/2, 0) (1.15)

respectively. As it is commonly known, the usual Dirac equation describes a particle
and antiparticle and on the manifold of all its solutions the representation D+ (1/2) ?
D? (1/2) is realized of group P(1, 3).
Starting from the equation (1.1) (or (1.2)) and using Bargman–Wigners method [6]
one can describe some class of equations invariant under the P(1, 4) group and the
time reflection in sense of Wigner, however they are noninvariant under T p and C
operations.
Hence we see that (1.1) (or (1.2)) as well as the class of the Bargman-Wigner type
equations (derived from (1.1) o (1.2)) are T w -invariant, but T p -, C-noninvariant.
It may seen in this connection that any theory wich is built up in five-dimensional
Minkowski space-time is always P T C-noninvariant [5]. Though actualy it is not so.
In fact, let us consider equation
?+ (t, x)
??(t, x)
?(t) ? ?(t, x) = (1.16)
H?(t, x) = i , ,
?? (t, x)
?t

where
H ? ?k pk + ??, k = 1, 2, 3, 4,
(1.17)
?k 0 ? 0
?k = , ?= .
0 ??
0 ?k
On the manifold of solutions of this equations operators T w , T p and C are defined as:
T w ?(t) = ? w ?? (?t), C?(t) = ? c ?? (t),
T p ?(t) = ? p ?(?t), (1.18)
110 W.I. Fushchych

0 ? ?1 ?3 0 0 ?2 ?4
?p = ?w = ?c = (1.19)
, , .
? 0 0 ?1 ?3 ?2 ?4 0

One can immediatly verify that the relations (1.16), (1.8) and (1.10) actually satisfy
for the equation (1.16). In means that the equation (1.16) is T w -, T p -, C- and P T C-
invariant. That is also clear from the fact that equation (1.16) realizes representation
D+ (1/2, 0) ? D? (1/2, 0) ? D+ (0, 1/2) ? D? (0, 1/2). (1.20)
Starting from (1.16) and generalizing the Bargman–Wigner method on P(1, 4)
group one can describe all the equations of Bargman–Wigner type wich are P T C-
invariant [7].
Thus in case of five dimensions one has to choose for the basic equation on eight-
component equation (1.16) but not a four-component equation (1.1) or (1.2).
If cane puts in (1.1) ? = 0, then such four-component equation is T p -, C-invariant
and in this case:
? p = ?0 , ? c = ?2 ?4 , ? w = ?1 ?3 . (1.21)
Equation (1.1)
??± (t, x)
±
(1.1 )
?k pk ? (t, x) = i
?t
describes a particle whose spin is 1/2 but the mass is non-fixed since
?? < p4 < ?, 0 ? m2 ? ?. (1.22)
M 2 ?(t, p ) = p2 ?(t, p ) = m2 ?(t, p ),
4

Here ?(t, p ) is the Fourier-image of function ?(t, x).
From what was performed above it reveales that in P(1, 4) scheme it is possible
to describe a particle with non-fixed mass (i.e. the particles of resonance type) the
spin of wich fixed.

§ 2. Equation for a particle with non-fixed mass on P(1, 3) group
In this section we show how one can write down the relativistic equation of mation
for a particle with the non-fixed mass within the framework of Poincar? group.e
Usually elementary particle either stable or unstable whose spin is s, is associated
with a Hilbert space Rs (m) in wich on irreducible representation of the Poincar? e
group P(1, 3) is realized. Such a correspondence is unjustified one since we cannot
attribute the definite mass to the unstable particle. Following [2, 8] let us attribute
to an unstable particle (resonance) a Hilbert space Rs with is the direct integral of
spaces Rs (m), i.e.

?Rs (m)g s (m2 )dm2 ,
Rs = (2.1)

where function g s (m2 ) is not equal to zero only within the interval [m2 , m2 ] wich
1 2
characterizes the spread (indefinite) of mass of a particle.

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