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in the Schr?dinger picture: a representation space for an irreducible representation is
o
constituted from the solutions ?(x0 ) of the Schr?dinger–Foldy type equation
o

(8)
i?0 ?(x0 ) = P0 ?(x0 ).

The solutions ?(x0 ) are vector functions ?(x0 ) = ?(x0 , x1 , . . . , xn ) in x-representa-
tion for eq. (6) or ?(x0 ) = ?(x0 , p1 , . . . , pn ) in p-representation for eq. (6) etc., and
their components are also functions of auxiliary variables s3 , t3 , . . . (“spin” variables)
— eigenvalues of generators of Cartan’s subalgebra of the algebra O(n) in the case (5)
or O(1, n ? 1) in the case (7).
Eq. (8) is P (1, n)-invariant: the manifold of all the solutions of eq. (8) is invari-
ant under transformations from the group P (1, n). This is the consequence of the
condition

[(i?0 ? P0 ), Q]? = 0 (9)

being valid for any generator Q of P (1, n) defined by eqs. (5) or (7).
In the Heisenberg picture where vector functions ? of a representation space for
a representation of P (1, n) do not depend on the time x0 (and are the solutions of the
equation P0 ? = E?), formulae for the generators Pµ , Jµ? are obtained by dropping
the terms with x0 , and eq. (8) is replaced by

(10)
i?0 Q = [Q, P0 ]?

for any operators Q as functions of xk , pk , S.
Since in class I the little group of the group P (1, n) is 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 , . . .), and the solutions of the
corresponding equation (8) have here finite number of components. In accord with the
representations of the little group O(1, n ? 1), in class III the group P (1, n) has both
finite- and infinite-dimensional representations. We emphasize that all the unitary
representations are here infinite-dimensional, and the solutions of the corresponding
eq. (8) have here infinite number of components.
Note at the end of this section that the problems of classification and realization of
representations of an arbitrary inhomogeneous group P (m, n) can similarly, without
principal difficulties, be reduced to problems of classification and realization of corre-
sponding representations of homogeneous group of the types O(m , n ).
3. Physical interpretation
Here we deal only with the inhomogeneous de Sitter group P (1, 4) which is a
minimal extention of the Poincar? group P (1, 3). We discuss the main topics of the
e
physical interpretation of a quantal scheme based on the group P (1, 4). This group is
the most attractive one because it will succeed to give a clear physical meaning to a
complete set of commuting variables.
On representations of the inhomogeneous de Sitter group 117

ln the p-representation for eq. (6) a component of the wave function ? — a solution
of eq. (9) with n = 4 — is a function of six dynamical variables of the corresponding
complete set: ?(x0 , p, p4 , s3 , t3 ) where p and p4 are eigenvalues of the operators P
and P4 and their physical meaning has been discussed in the introduction; s3 and
t3 are eigenvalues of the third components of the operators S = (S1 , S2 , S3 ) and
T = (T1 , T2 , T3 ) where
1 1
Sa ? Ta ? (Sbc ? S4a ), (11)
(Sbc + S4a ),
2 2
(a, b, c) = cycl(1, 2, 3). These operators satisfy the relations

[Sa , S 2 ] = [Ta , T 2 ] = [Sa , Tb ] = 0. (12)
[Sa , Sb ] = iSc , [Ta , Tb ] = iTc ,

The operators
W V W V
?
S2 = T2 = (13)
+ ,
2 2
4p 4p
2 p2 2 p2
are invariant both of P (1, 4) and O(4) (in class I) or O(1, 3) (in class III). Note that
in irreducible representations of class I we have

S 2 = s(s + 1)? T 2 = t(t + 1)? (14)
1, 1,

where s, t = 0, 1 , 1, . . . , . . . and ? is the (2s + 1)(2t + 1)-dimensional unit matrix.
1
2
The irreducible representations D± (s, t, ? 2 ) of the group P (1, 4), identified by
fixed numbers s, t, ? 2 and ? = ±1 (i.e., by values of the corresponding invariants of
P (1, 4)), allow us to introduce the concept of “elementary particle” in the quantum
scheme based on the group P (1, 4) possible states of an “elementary particle” (when
? = +1) or “antiparticle” (when ? = ?1) with given values of s, t and ? 2 , are
states which constitute the representation space for the irreducible representation
D± (s, t, ? 2 ) of the group P (1, 4). As it is seen from eq. (2), ? is the boundary value
of the energy P0 ; the physical meaning of s and t is dictated by the relations (12): they
allow to interpret the operators S and T as the spin and isospin operators. Thus, the
components ?(x0 , p, p4 , s3 , t3 ) are interpreted as the probability amplitude of finding
(by measuring at a given instant x0 ) the indicated values of three-momentum p, mass
m = p2 + ? 2 and third components of spin s3 and isospin t3 .
4
It is clear that an irreducible representation D± (s, t, ? 2 ) of the group 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)). The vector function of the representation space for D± (s, t, ? 2 ) describes,
in fact, a multiplet of particles with different t3 , ?t ? t3 ? t (and, of course, with
different s3 , ?s ? s3 ? s); the parameter ? is then a “bare” rest mass of the given
multiplet.
The P (1, 4) quantum scheme in our interpretation 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 quantal approach2 without breaking
down such fundamental principles as unitarity, hermiticity etc. Indeed, here the mass
2 The consequent consideration of such problems demands, obviously, the quantum field approach, but a
quantal approach can be regarded as half-phenomenological.
118 W.I. Fushchych, Yu.I. Krivsky

operator is an independent dynamical variable eq. (1), it is Hermitian, and the problem
of unstable systems is, in fact, reduced to the problem of calculation quantities like
distributions
2
v
?i m2 ?? 2 x4
?(m , s3 , t3 ) ?
2 3
(15)
dx dx4 e ?(x0 , . . . , x4 , s3 , t3 ) ,

where ? are solutions of an equation of the type (8) with a suitable interaction.
The positions and forms of maxima of the distribution ?(m2 ) define experimentally
observed masses and lifetimes of unstable particles, and singularities of ?(m2 ) define
masses of stable particles.
It is important to emphasize that in accord with our interpretation, the particles
experimentally observed are described not by the free equation (8), but by an equation
of the type (8) with a suitable interaction which may break the P (1, 4)-invariance, but,
of course, conserves the P (1, 3)-invariance3 . As for solutions of the free equation (8),
they are some hypothetical (“bare”) states which may not correspond to any real
particles. From the viewpoint of this interpretation there are two types of interactions:
interactions which cause a “dressing” of particles and are inherent even in asymptotical
states, and usual interactions which cause a scattering processes of real (“dressed”)
particles. Therefore, in particular, the five-dimensional conservation law following
from the free P (1, 4)-invariant scheme, may have no real sense.
We emphasize that the interpretation of the P (1, 4)-scheme proposed does not
pretend to be the only one and complete. The more detailed discussions of interpre-
tation problems are possible only in connection with solutions of suitable models of
interactions in this scheme, what is not the subject of this article.
4. The Dirac-type equations
A characteristic feature of eqs. (8) is that they do not contain any redundant
components. However, in this equation the differential operators ?k ? ?/?xk enter
under the square root, therefore they are considered not to be appropriate for introdu-
cing interactions and for theoretical field considerations. Let us consider the simplest
equation of first order in ?µ , manifestly invariant under the group P (1, 4).
Remind that there are five Dirac matrices ?µ satisfying the relations
(16)
?µ ?? + ?? ?µ = 2gµ? , µ, ? = 0, 1, . . . , 4,
where
?0 ? ? = ?1 ?2 ?3 ?4 ?4 = ??0 ?1 ?2 ?3 .
or (17)
The Dirac equation in the Minkowski five-space is of the form
(i?µ ?µ ? ?)? ? (i?0 ?0 + i?k ?k ? ?)? = 0 (18a)
or
(i?µ ?µ + ?)? = 0. (18b)
Eqs. (18) were written down long ago by Dirac [8]. It is clear that they are
invariant under the inhomogeneous de Sitter group. Our aim is to find out which
this sense the consideration of P (1, 4) symmetry here presented is only a base for its suitable
3 In

violation — analogously to considerations and violations of SU (n) symmetries.
On representations of the inhomogeneous de Sitter group 119

representation of the group P (1, 4) is realized in the representation space of solutions
of eq. (18). Here we shall not follow the conventional method which is ordinary used
(see, e.g.. refs. [9, 10]) and which in fact answers only the question which representati-
on of the homogeneous Lorentz group O(1, 3) is realized by the Dirac equation in the
Minkowski four-space but does not answer the question of representation of the Poi-
ncar? group P (1, 3).
e
Here we deal with the method suitable for analysis both of the Dirac equation
and of other wave equations (linear and non-linear with respect to ?µ ) and besides in
arbitrary (1 + n)-dimensional Minkowski space. The method is based on definition (9)
of the invariance of the wave equation. It is clear from this definition that to answer
the question whether a wave equation is invariant under the group P (1, n), one has
to find an explicit form of generators Pµ , Jµ? of the algebra connected with the
equation in such a way that its Hamiltonian H and the operator i?o ? i?/?t must
commute with the generators Pk , Jµ? exactly just as the generator P0 does. Further,
if the explicit form of the generators are found, one can find the invariants of the
group P (1, n) in the explicit form; their eigenvalues will answer the question which
representation of the group is realized by solutions of this equation.
Let us illustrate the method for the case of eq. (18a). Rewrite eq. (18a) in the
Hamiltonian form

H ? ?k pk + ??, (18a )
i?0 ? = H?, ?k = ??k .

It can, be immediately verified that in this case the explicit form of generators Pµ ,
Jµ? satisfying the relations of the algebra P (1, 4), is given by

P0 = H ? ?k pk + ??, Pk = pk ? ?i?k ,
(19)
1
= x0 pk ? (xk P0 + P0 xk ),
Jkl = x[k pl] + Skl , J0k
2
where
i
Skl ? (?k ?l ? ?l ?k ). (20)
4
We choose ?µ in the form

0 ?a 01 10
?0 ? ? = (21)
?a = , ?4 = i , .
??a 0 ?1
0 10

Then the spin and isospin operators for the particle described by eqs. (18), are of the
form
1 1 1 1
?a 0 00
Sa ? Ta ? (Sbc ?S4a ) =
(Sbc +S4a ) = , ,(22)
0 0 0 ?a
2 2 2 2
and their squares coinciding with the invariants of the group P (1, 4), are of the form
3 3
1 0 00
S2 = T2 = (23)
, .
0 0 01
4 4

Further, the invariant P 2 = ? 2 and the invariant ? is the sign of energy coincides (in
the “rest frame” pk = 0) with the matrix ?.
120 W.I. Fushchych, Yu.I. Krivsky

It is clearly seen from S 2 , T 2 , S3 , T3 and ? = ? that the manyfold of solutions
of eq. (18a) constitutes the representation space for the four-dimensional reducible
representation D+ ( 1 , 0) ? D? (0, 1 ) of the group P (1, 4). Thus, in accord with our
2 2
interpretation of the numbers s and t the Dirac equation (18a) describes a multiplet

? +,0
1
2
(24)
?= ,
?
?0, 1
2

where ? +,0 is a spinor-isoscalar describing a fermion with the spin s = 1
and isospin
1 2
2
?
t = 0 (a particle like the ? hyperon) and ?0, 1 is a scalar-isospinor describing an
2
antiboson with s = 0 and t = 1 (an antiparticle like the K meson)4 .
2
It can analogously be shown that eq. (18b) realizes the representation D? ( 1 , 0) ?
2
1
+
D (0, 2 ) of the group P (1, 4), i.e., describes a multiplet like (K, ?). In this case the
explicit form of the operators Pµ , Jµ? is obtained from eq. (19) by the replacement
? > ?? or ? > ??.
Thus, in contrast to the Dirac equations in the P (1, 3) scheme, the Dirac equations
(18) in the P (1, 4) scheme do not describe particles and antiparticles symmetrically
and therefore they will not be invariant under transformations of type P T C.
It can be perceived from the analysis of eqs. (18a) and (18b) that in the P (1, 4)
scheme the equation describing particles and antiparticles symmetrically, must realize
the representation

1 1 1 1
? D? , 0 ? D? 0,
, 0 ? D+ 0,
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