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. 27
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D+ (25)
.
2 2 2 2

We have found that such an equation is of the form

(i?µ ?µ ? ?)? ? (i?0 ?0 + i?k ?k ? ?)? = 0, (26)

where the 8 ? 8 matrices are
0 ?k 10
(27)
?k = , ?0 = .
0 ?1
?k 0

In the case of eq. (26) the explicit form of the generators of P (1, 4) is obtained
from eq. (19) by the replacement ?µ > ?µ . One can see from the explicit form the
the 8 ? 8 matrices S 2 , T 2 , S3 , T3 and ? = ?0 that eq. (26) actually realizes the
representation (25), i.e., that the wave function ? (eight-component spinor) has the
form
?+?
? 1 ,0
?+? 2

?? 1 ?
? 0, ?
? = ? ?2 ? , (28)
?? 1 ?
? 0, 2 ?
? ?,0
1
2

4 Note that it would be more appropriate to call the boson-like K a spinosinglet-isodoublet, and the
fermion-like ? a spinodoublet-isosinglet.
On representations of the inhomogeneous de Sitter group 121

Note, that in the P (1, 4) scheme just the eight-component equation (26) (but not
the four-component equations (18)) symmetrically describes particles and antiparticles
and is therefore P T C invariant (more detailed see refs. [11, 16]).
It is easy to see that the eight-component equation (26) is the unification of the
four-component equations (18a) and (18b). Of course, in the P (1, 3) scheme such
a unification of the Dirac equations is trivial. However, in the P (1, 4) scheme the
unification is not trivial: the matrices ?0 , ?k obey the relations (16), but they are not
matrices of a reducible representation of the Dirac algebra eq. (16) since, in particular,
?0 = ?1 ?2 ?3 ?4 , i.e., the condition (17) is not satisfied. The 8 ? 8 matrices ?0 , ?k
together with the two other matrices
0 ?0 01
(29)
?5 = i , ?6 =
?1 0
?0 0
obey the commutation relations of Clifford algebra in six-dimensional space, the addi-
tional condition
? = ?i?1 ?2 ?3 ?4 ?5 ?6 (17 )
being valid, and realize its irreducible representation. It is of interest to note that the
eight-component equation of the Dirac type
(i?µ ?µ ? ?)? ? (i?0 ?0 + i?1 ?1 + · · · + i?6 ?6 ? ?)? = 0 (30)
realizes a representation of the group P (1, 6).
The wave function of eq. (26) (or even eqs. (18)) describes an unusual multiplet:
it unificates fermions and bosons into a multiplet. For example,
? ?
?
?K?
?=? ? (31)
? ? ?.
K
This circumstance is not unsatisfactory for eq. (26) from the viewpoint of, for
example, the barion number conservation law. The latter only causes some restri-
ctions on possible forms of interactions. In the P (1, 4) scheme the barion number
operator can be defined as usually (as a number of fermions ? +,0 minus a number of
1
2
antifermions ? ?,0 ). It is remarkable that the wave function (28) describes symmetri-
1
2
cally both fermions and isofermions. Therefore in the P (1, 4) scheme we can naturally
+
define the operator of hypercharge as a number of isofermions ?0, 1 minus a number
2
?
of anti-isofermions ?0, 1 . This allows eq. (26) to be considered as a fundamental
2
equation for the dynamical approach to the classification scheme of d’Espagnat and
Prentki [12].
As in the case of the Dirac equation in the P (1, 3) scheme [13], in order to give an
adequate physical interpretation of the wave function ? as a function of x, x4 , one has
to transit from the Dirac representation to the Foldy representation. The transition is
performed by the unitary transformation
Ak pk p
U = exp ?i p2 ,
arctg (32)
, p=
? k
2p
122 W.I. Fushchych, Yu.I. Krivsky

where Ak = i?k for eqs. (18) and Ak = i?k for eq. (26).
In the Foldy–Shirokov representation eqs. (18) and (26) are of the form

p2 + ? 2 ?, (33)
i?0 ? = B k

where B = ?0 , ??0 , ?0 for eqs. (18a), (18b) and (26) correspondingly.
After the transformation (32), the formulae for the generators Pµ , Jµ? coincide
with eq. (5) for n = 4, if the replacement ? > B is made there.
5. The Kemmer–Duffin type equations
Let us consider now an analogue of equations describing bosons with spin 0 and
1 in the P (1, 3) scheme, namely, the equations in Minkowski five-space which are of
the form
(?µ ?µ + ?)? = 0, (34)
µ = 1, 2, 3, 4, 5,
where five Hermitian matrices ?µ obey the algebra of the Kemmer-Duffin-Petiau type
(KDP):
(35)
?µ ?? ?? + ?? ?? ?µ = ?µ? ?? + ??? ?µ .
This algebra has three irreducible representations. The lowest representation is
realized by 6 ? 6 matrices. The non-zero element of these matrices are schematically
written down in table 1 where, for example, “1,6” denotes (?1 )1,6 = 1. Remind for
comparison that the lowest representation of KDP algebra in the P (1, 3) scheme (i.e.,
when µ ? 4) is realized by 5 ? 5 matrices.
Table 1
The unit elements of 6 ? 6 matrices

?1 ?2 ?3 ?4 ?5

1,6 2,6 3,6 4,6 5,6
6,1 6,2 6,3 6,4 6,5

It can be shown by means of the method used in sect. 4 that eq. (34) with the
6 ? 6 matrices (35) realizes the representation
11
D+ (0, 0) ? D? (0, 0) ? D (36)
, ,
22
where the first two representations are realized by principal components of the vector
function ?, on which the energy operator has non-vanishing eigenvalues, and the last
representation is realised by redundant components of the vector function ?, on which
the eigenvalues of the energy operator are equal to zero. Thest last have no physical
sense but they are presented in all linear, with respect to ?µ , equations, except for the
Dirac-type equations. In such cases the Foldy–Wouthuysen transformation does not
only split the states with positive and negative energies, but also makes it possible to
omit the redundant components by an invariant way.
Thus, eq. (34) with the 6?6 matrices (35), which can be obtained by a linearization
procedure of the Klein–Gordon equation in the Minkowski five-space for a scalar,
describes particles and antiparticles with s = t = 0.
On representations of the inhomogeneous de Sitter group 123

Consider now a very interesting case: eq. (34) with the 15 ? 15 matrices ?µ
realizing an irreducible representation of algebra (35). These matrices can be taken,
for example, in the form schematically given by table 2 where only the non-zero
elements of the matrices ?µ equal to ±1, are written down.
Now we shall find out what representation of the groups O(4) and P (1, 4) is
realized by solutions ? of eq. (34) with matrices ?µ of table 2.
Using the method proposed in ref. [14] for reducing the Kemmer-Duffin equati-
ons in the P (1, 3) scheme to the Schr?dinger form, one can verify that eq. (34) is
o
equivalent to the equation

H = S5k pk + ?5 ?, (37)
i?0 ? = H?,

where

S5k ? i(?5 ?k ? ?k ?5 ), k = 1, 2, 3, 4.

Using eq. (35) it is easy to verify that the Hermitian matrices

Sµ? ? i(?µ ?? ? ?? ?µ ), (38)
µ, ? = 1, 2, 3, 4, 5,

obey the commutation relations for the generators of algebra, i.e., they realize a
fifteen-dimensional representation of this algebra. The explicit form of the matrices
?µ given by table 2 allows to find the quantity corresponding to the invariant P 2 of
P (1, 4)
?4 ?
1
? ?
06
?5 = ? ?,
P 2 ? H 2 ? p2 = ? 2 ?5 ,
2 2
(39)
? ?
14
k

01

where the upper indexes denote the dimensionality of the matrices. Table 2 allows us
also to find the matrices Skl .

Table 2
Non-zero elements of 15 ? 15 matrices ?µ equal to ±1.
?1 ?2 ?3 ?4 ?5
4, 15 15, 4 3, 15 15, 3 2, 15 5, 12 1, 15 15, 1 ?1, 14 ?14, 1
7, 14 14, 7 6, 14 14, 6 15, 14 14, 55 ?5, 13 ?13, 5 ?2, 13 ?13, 2
9, 13 13, 9 8, 13 13, 8 ?8, 12 ?12, 8 ?6, 12 ?12, 6 ?3, 12 ?12, 3
10, 12 12, 10 ?10, 11 ?11, 10 ?9, 11 ?11, 9 ?7, 11 ?11, 7 ?4, 11 ?11, 4

One can clearly see from the explicit form of diagonal matrices S 2 , T 2 , S3 , T3 ,
? = ?5 that eq. (34) with 15 ? 15 matrices ?µ realize the representation

11 11
? D? ? D(1, 0) ? D(0, 1) ? D(0, 0),
D+ (40)
, ,
22 22

where the first two representations of the group P (1, 4) are realized by eight principal
components of the vector function ? and the last three representations of the group
124 W.I. Fushchych, Yu.I. Krivsky

O(4) are realized by seven redundant components of the vector function ?. Of course,
only the eight components realizing the representation
11 11
? D?
D+ (41)
, ,
22 22
have a physical sense. From the seven redundant components the eight principal can
be separated by transformation of the Foldy–Wouthuysen type
?k pk p
U = exp ?i p? p2 .
arctg (42)
,
? k
2p
This transformation splits in an invariant way eq. (34) with 15 ? 15 matrices into two
independent equations, the first being for the principle components ?(x0 , x, x4 , s3 , t3 ),
s3 , t3 = ± 1 and coincide with eq. (8) in ref. [1], and the second being for redundant
2
components ?(x0 , x, x4 , s3 , t3 ), s3 , t3 = 0, ±1, and ?0 (x0 , x, x4 ) having no physical
sense.
Thus, the Kemmer–Duffin equation (34) in Minkowski five-space with 15 ? 15
matrices describes symmetrically fermions and antifermions with spin and isospin
s = t = 1 (multiplets of the type spinodoublet-isodoublet), i.e., for example, the
2
systems of particles like a nucleon-antinucleon (N, N ). This equation is, of course,
P T C invariant.
As it was mentioned above, in five-space the algebra KDP eq. (35) has three
irreducible representations. The third irreducible representation is realized by 20 ? 20
matrices ?µ . We do not present here the explicit form of the matrices and the analysis
of the equation connected with them. Not only that the principal components of the
wave function of this equation realize the representation
D+ (1, 0) ? D? (1, 0) ? D+ (0, 1) ? D? (0, 1) (43)
of the group P (1, 4), i.e., describe a meson multiplet like (?, ?), and is P T C invariant
as well.
In this paper we have made the analysis of P (1, 4)-invariant equations of the
Dirac and Kemmer–Duffin type. The analysis of another linear on ?µ equations in
five-dimensional space, for example, equations of the Rarita–Schwinger type, Pauli–
Fierz type and other, can be made analogously. It is interesting to note that the
Rarita–Schwinger formalism developed in the P (1, 3) scheme for finding equations
for particles with arbitrary spin, can be generalized on the case of the P (1, 4) scheme
without any difficulties. This is because of there are five matrices ?µ , µ = 0, 1, 2, 3, 4,
obeying the algebra (16), and in the Rarita–Schwinger formalism for the P (1, 4)
scheme all the five matrices are equal in rights. Note that in the case of the KDP
algebra (35) the situation is another: there is no fifth 5 ? 5 or 10 ? 10 matrix ?5
obeying the algebra (35).
It should finally be noted that the general form of the P (1, n)-invariant equation
linear in ?µ is
(Bµ ?µ + ?)? = 0, (44)
µ = 1, 2, . . . , n, n + 1,
where the operators Bµ are defined by the relations
[Bµ , S?? ]? = ?µ? B? ? ?µ? B? , (45)
µ, ?, ? = 1, . . . , n + 1.
On representations of the inhomogeneous de Sitter group 125

For the representation of class I the operators Bµ are finite-dimensional, for those of
class III the operators Bµ can be both finite- and infinite-dimensional. Definite forms

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. 27
( 122 .)



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