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According to (2.1) each vector from Rs can be representated as

??s (t, x, m)g s (m2 )dm2 ,
?s (t, x) =
(2.2)
? (t, x, m) ? R (m), x ? (x1 , x2 , x3 ),
2 s
Equations of motion in odd-dimensional spaces and T -, C-invariance 111

Pµ ?s (t, x, m) = m2 ?s (t, x, m),
2
(2.3)
µ = 0, 1, 2, 3,

?m2 ?s (t, x, m)g s (m2 )dm2 .
Pµ ?s (t, x) =
2
(2.4)

The generators of the Poincar? group on vectors (2.2) are defined in such a way
e

?Pµ ?s (t, x, m)g s (m2 )dm2 ,
Pµ ?s (t, x) = (2.5)


?Jµ? ?s (t, x, m)g s (m2 )dm2 .
Jµ? ?s (t, x) = (2.6)

The Dirac equation for the function ?s=1/2 (t, x) is:

i?0 p0 + i?k pk ? p2 ?s=1/2 (t, x) = 0. (2.7)
µ

One can easily see now that (2.7) can be reduced to the usual Dirac equation if one
formaly replaces the function g s=1/2 (m2 ) in (2.2) by ?(m2 ? m2 ). The generators of
0
P(1, 3) group defined by (2.5) and (2.6) on the manifold of solutions of eq. (2.7) are
given by (1.4), where

P0 ? H ? ?k pk + ? p2 . (2.8)
µ

We can write down the equation of motion for a particle with indefinite mass, wich
interacts with the external electromagnetic field in form

i?0 ?0 + i?k ?k ? ?µ ?s=1/2 (t, x) = 0,
2 (2.9)

where ?µ ? pµ ? eAµ . It is clear that equation (2.9) essentialy differs from the
usual Dirac equation wich discribes the motion of a particle with fixed mass in the
electromagnetic field. A detaled analysis if equation (2.9) will be performed in a
forthcoming work.
Lurcat [8] pointed out, that interpretation of function ?s (t, x) as a wave function
of particle is not correct.
More appropriate is to characterize the unstable system by the density matrix
(operator). In the Schr?dinger picture the equation of motion for the density matrix
o
looks like
??
i = [H, ?],
?t
where H is defined by (2.8).
Equation (2.7) as well as the usual Dirac equation, is P -, T -, C-invariant.

§ 3. Equation for the flat particle and T -, C-invariance
To clear up how can extend the obtained above (sec. 1) results upon any arbitrary
group P(1, 2n+1) let us consider in this section equations of motion wich are invariant
under P(1, 2) group (the group of rotations and translations in three-dimensional
Minkowski space).
112 W.I. Fushchych

The simplest equations invariant under P(1, 2) are:
?+ (t, x1 , x2 )
H + ?+ (t, x1 , x2 ) = i (3.1)
,
?t
?? (t, x1 , x2 )
H ? ?? (t, x1 , x2 ) = i (3.2)
,
?t
H ± = ?k pk ± ??, k = 1, 2,
(3.3)
?
pk = ?i
?1 = ?1 , ?2 = ?2 , ? = ?3 , ,
?xk
Here ?(t, x1 , x2 ) is a two-component spinor, and ?1 , ?2 , ?3 are Pauli manrices.
Taking into account that in this case
? p = ap · 1 + a p ?, ? w = aw · 1 + a w ?, ? c = ac · 1 + a c ? (3.4)
and arguing in a way similar to that of sec. 1, we reveal that equation (3.1) or (3.2)
is T p -, T w - and P T C-noninvariant but P - and C-invariant.
Equation
??(t, x1 , x2 )
H?(t, x1 , x2 ) = i ,
?t
(3.5)
?+ (t, x)
?(t) ? ?(t, x) = , H = ?k pk + ??, k = 1, 2,
?? (t, x)

?k 0 ?3 0
(3.6)
?k = , ?=
0 ?k 0 ?3

is T p -, T w - and C-invariant as well as equation (1.16) is, i.e. it is P T C-invariant, and
for matrices and ? p , ? w we have
0 ?3 0 ?2 ?1 0
?p = ?w = ?c = (3.7)
, , .
?3 0 ?2 0 0 ?1
Thus equations of (1.1), (3.1) type and a whole class of equations of Bargman–
Wigner type wich are derived from the equations of (1.1) type are invariant under the
limited groups:
P(1, 2) are T p -, T w -, T w C-noninvariant and C-invariant;
P(1, 4) are T p -, C-, T w C-noninvariant and T w -, T p C-invariant;
P(1, 6) are T p -, T w -, T p C-, T w C-noninvariant and C-invariant;
P(1, 8) are T p -, T w -, T w C-noninvariant and T w -, T p C-invariant.
.
.
.
To prove the assertiona given above in the case of arbitrary P(1, 2n + 1) group one
has to carry out the very similar procedure to that we employed for P(1, 4) group and
to use the fact that Dirac matrices ? (2n+1) of group P(1, 2n) are related with those
? (2n?1) of group P(1, 2n ? 2) by
(2n+1) (2n+1)
? ?2 , 1 ? ?3 , 1 ? ?1 ,
(2n+1) (2n?1)
?µ , ?2n , ?2n+1 = ?µ
µ = 0, 1, . . . , 2n ? 1.
Equations of motion in odd-dimensional spaces and T -, C-invariance 113

Puting in (3.1) and (3.2) ? = 0 one sees that equation (3.1) concides with (3.2)
and such equation is C-, T -invariant, and ? p = ?3 , ? w = ?2 .

1. Hegerfeldt G.C., Henning J., Fortschr. Phys., 1968, 16, 9.
2. .., . ., 1968, 13, 363; 1967, 12, 741.
3. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1968, 7, 79.
4. .., .., 5- ,
-68-72, , 1968.
5. Rosen S.P., J. Math. Phys., 1968, 9, 1593.
6. Bargman Y., Wigner E., Proc. Nat. Acad. Sci., 1948, 34, 211.
7. .., .., -69-33, , 1969.
8. Lurcat F., Phys. Rev., 1968, 173, 1461.
9. Brauer B., Weyl H., Am. J. Math., 1935, 57, 447.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 114–125.

On representations of the inhomogeneous
de Sitter group and equations in
five-dimensional Minkowski space
W.I. FUSHCHYCH, I.Yu. KRIVSKY
This paper is a continuation and elaboration of our brief notice [1] where some approach
to the variable-mass problem was proposed. Here we have found a definite realizati-
on 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 P (1, n)-invariant equations of the Schr?dinger– o
Foldy type are written down. Some equations of physical interpretation of the quantal
scheme based on the inhomogeneous de Sitter group P (1, 4) are discussed.
The analysis of the Dirac and Kemmer–Duffin type equations in the P (1, 4) scheme
is carried out. A concrete realization of representations of the algebra P (1, 4) connected
with this equations, is obtained. The transformations of the Foldy-Wouthuysen type for
this equations are found. It is shown that in the P (1, 4) scheme of the Kemmer–Duffin
type equation describes a fermion multiplet like the nucleon-antinucleon.

1. Introduction
We recall here the initial points of our approach of the variable-mass problem
proposed in ref. [1]:
(i) The square of the variable-mass operator is defined as an independent dynamical
variable:
M 2 ? ? 2 + P4 ,
2
(1)
where ? is a fixed parameter and P4 is an operator similar to the components of the
three-momentum P , which commutes with all the generators of the algebra P (1, 3)
of the Poincar? group.
e
(ii) The relation between the energy P0 , three-momentum P and variable-mass M
of a physical system remains conventional (here = c = 1):
P0 = P 2 + M 2 ? P 2 + ?k ,
2 2
(2)
k = 1, 2, 3, 4.
(iii) The spaces p ? (p0 , p1 , . . . , p4 ) and x = (x0 , x1 , . . . , x4 ) are assumed to be
plane and reciprocally conjugated. It follows then from (i), (ii) and (iii) that the
generalized relativistic group symmetry is an inhomogeneous de Sitter group1 P (1, 4),
i.e. 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 sect. 2 a definite realization of irreducible representations for the generators
Pµ , Jµ? of the algebra P (1, n) with arbitrary n is carried out, which made it possible
to give a proof of the P (1, n)-invariance of the Schr?dinger-Foldy type equations
o
Nuclear Physics B, 1969, 14, P. 573–585.
1 Algebras
and groups connected with them are designated here with the same symbols.
On representations of the inhomogeneous de Sitter group 115

given in ref. [1] for n = 4. Some questions of a physical interpretation of a quantal
scheme based on the group P (1, 4), are considered in sect. 3. Sects. 4 and 5 answer
the question which representations of the group P (1, 4) are realized by two types of
equations linear in ?µ ? ?/?xµ — the Dirac and Kemmer–Duffin type equations.
2. Realizations of the algebra P (1, n) representations
For the sake of generality all 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 (1 + n)-
dimensional Minkowski space which leaves 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,
unchanged, 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 are operators of infinitesinal displacements and Jµ? are
operators of infinitesimal rotations.
In refs. [2–5] all irreducible representations of the Poincar? group P (1, 3) are
e
studied and the concrete realization for the generators of its algebra is found. The
methods are generalized here for the case of the group P (1, n).
For representations of the class I (P 2 ? P0 ? Pk > 0) when the group O(n) of
2 2

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

P 0 = p0 ? ? p 2 ? ? 2 , Pk = p k ,
k
x[k pl] ? xk pl ? xl pk ,
Jkl = x[k pl] + Skl , (5)
1 Skl pl
J0k = x0 pk ? (xk p0 + p0 xk ) ? ,
p0 + ?
2
where the operators xk and pk are defined by the relations
(6)
[xk , pl ] = i?kl , [xk , xl ] = [pk , pl ] = 0,
and Skl are matrices realizing irreducible representations of the algebra O(n) which
have been studied in ref. [6].
For representations of the class III (P 2 = P0 ? Pk < 0) when the little group
2 2

of the group P (1, n) is already a non-compact group O(1, n ? 1) of rotations in
[1 + (n ? 1)]-dimensional pseudo Euclidean space, the generators Pµ , Jµ? are of the
form

P 0 = p0 ? ± p 2 ? ? 2 , Pk = p k , Jab = x[a pb] + Sab ,
k
Sab pb ? Sa0 p0 1
Jan = x[a pn] ? J0a = x0 pa ? (xa p0 + p0 xa ) + S0a ,
, (7)
pn + ? 2
1 S0a pa
= x0 pn ? (xn p0 + p0 xn ) ?
J0n ,
2 pn + ?
116 W.I. Fushchych, Yu.I. Krivsky

where a, b = 1, . . . , n ? 1, ? is a real constant, the operators xk , pk 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 studied by
Gelfand and Grayev [7].
Formulae (5) and (7) give the irreducible representations of the algebra P (1, n)

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