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relations for the generators of the algebra O(1, n ? 1), as soon as the Skl satisfy the
commutation relations for the generators of the algebra O(n).
Rewrite now the operators (3) in the form

P = (p0 , pa , pn ), Jab = x[a pb] + Sab , J0n = x[a pn] + San ,
(5 )
Sab pb + San pn Sna pa
J0a = x[0 pa] ? J0n = x[0 pn] ?
, .
p2 ? p2 ? p2 p2 ? p2 ? p2
p0 + a n a n
0 0

Using the definitions (9) and (10) corresponding to the schematic substitution “i0 ?
n” when the operators with the symbol 2 “?” are getting from the operators without
the symbol “?”, we obtain from (5):

P ? (p0 , pa , pn ) = (?ipn , pa , ip0 ),
Jab = x[a pb] + Sab , iJa0 = x[a p0] + iSa0 ,
Sab pb ? Sa0 p0
?iJna = ?ix[n pa] ? (8 )
,
?ipn + ?p2 ? p2 + p2
n a 0

iS0a pa
Jn0 = x[n p0] ? .
?ipn + ?p2 ? p2 + p2
n a 0

By virtue of Eqs. (10a), we have P 2 = ?P 2 < 0, so that (8 ) realizes a representations
of the class III as soon as (5) realizes a representation of the class I. Omitting in (8 )
the symbol “?”, we obtain (8).
Since all the representations of the class I are finite-dimensional, such a recipe
allows to obtain only finite-dimensional representations of the class III (i.e., not all
the representations of this class). If, however, getting (8) from (8 ), the operators
(S0a , Sab ) will be substituted by operators (S0a , Sab ) realizing an infinite-dimensional
representation of the algebra O(1, n ? 1), we obtain the corresponding infinite-di-
mensional representation of the algebra P (1, n). Thus it is shown that the formula (8)
defines all the representations of the class III of the algebra P (1, n).
The representations of the class II (P 2 = 0, P = 0) requires a special treatment.
However, in the case when one of invariants of the algebra P (1, n), namely, the
invariant
1
W? Pµ J?? ? Pµ P? Jµ? J?? ,
2
2
vanishes, the representations of the class II are particular cases of representations
of the class I, and formulaes for the generators Pµ , Jµ? are obtained from (5) by
the limit procedure p2 > 0. The detailed discussion of all the representations of the
class II is not given here. As to the class IV (P = 0), in this case the group P (1, n)
reduces to the group O(1, n), therefore the problem of classification and realization
of representations of the algebra P (1, n) reduces to the problem of classification and
realization of representations of the algebra O(1, n) already studied in ref. [7].
Let us discuss now a role of the variable x0 . If we mean possibility to Interpret
vectors ? constituting the representation space for the group P (1, n), as state vectors
of the physical system (see below Section 3), we must interpret x0 as the time, i.e.,
as a parameter which is not an operator and which therefore is not to be included in a
On representations of the inhomogeneous de Sitter group 131

complete set dynamical variables. It means that, for instance, in the x-representation
a vector-function ? is a function of only n dynamical variables: ? = ?(x1 , . . . , xn ).
If the condition C of the Section 1 is not to be violated, the number of independent
dynamical variables in p-representation coincides with those in x-representation, i.e.,
not all the dynamical variables p0 , p1 , . . . , pn are independent. For the representations
in which the invariant P 2 is a fixed constant, the latter are connected by the relation
p2 ? p2 ? p2 = ? 2 > 0, p2 ? p2 = ?? 2 < 0 (11)
0 k 0 k

for the class I and III respectively. One can, for example, chose
p0
p2 + ? 2 p2 ? ? 2 ,
and (12)
p0 = ? p0 = ? ?=
|p0 |
k k


for I and III. Then in p-representation ? = ?(p1 , . . . , pn ). Of course, one can accept
that ? = ?(p0 , p1 , . . . , pn ), but under the condition (11), so that
2p0 ?(p1 , . . . , pn )?(p2 ? a2 ), a = ? 2 , ?? 2 . (13)
? = ?(p0 , p1 , . . . , pn ) =
In the space of vector-functions ? discussed the scalar product can be defined by
P (1, n)-invariant way:

d1+n p ?+ (p0 , p1 , . . . , pn )? (p0 , p1 , . . . , pn ) =
(?, ?) =
(14)
n +
= d p ? (p1 , . . . , pn )? (p1 , . . . , pn ).

The operators Pµ , Jµ? defined in this space of vector-functions ?, have the form (5)
and (8) where the sudstitution
1
x[0 pk] > ? (xk p0 + p0 xk ), (15)
2
is made, p0 is defined by (12), xk and pk are defined by relations
(6 )
[xk , pl ] = i?kl , [xk , xl ] = [pk , pl ] = 0,
while Skl and (S0a , Sab ) are the same as in the formulae (5) and (8).
Thus, the “quantum mechanical” representation (of the Foldy–Shirokov [3, 5] type)
of the generators Pµ , Jµ? of the algebra P (1, n) is of the form:
For the class I

p0 ? ? p2 + ? 2 ,
P = (p0 , pk ), k
(16)
1 Skl pl
= ? (xk p0 + p0 xk ) ?
Jkl = x[k pl] + Skl , J0k ;
p0 + ?
2
For the class III

p0 ? ? p2 ? ? 2 ,
P = (p0 , pk ), k

Sab pb ? Sa0 p0
Jan = x[a pn] ?
Jab = x[a pb] + Sab , , (17)
pn + ?
1 S0a pa 1
J0n = ? (xn p0 + p0 xn ) ? J0a = ? (xa p0 + p0 xa ) + S0a .
,
2 pn + ? 2
132 W.I. Fushchych, I.Yu. Krivsky

Since operators Q = xk , pµ , S, Pµ , Jµ? of (16) and (17) are defined on the apace
of vectors ? not depending on the time x0 , the representations (16) and (17) are, in
fact, the representations of the algebra P (1, n) in the Heisenberg picture where for
the operators Q as functions of the time x0 , the equation of motion

(18)
i?0 Q = [Q, P0 ]

is postulated.
In the Schr?dinger picture vectors ? depends explicitly on the time x0 as on a
o
parameter (but not as on a dynamical variable!) and for this dependence the equation
of the Schr?dinger–Foldy type is postulated
o

(19)
i?0 ?(x0 ) = P0 ?(x0 ),

where P0 is defined by (12) and in x-representation ?(x0 ) = ?(x0 , x1 , . . . , xn ), in
p-representation ?(x0 ) = ??(x0 , p0 , p1 , . . . , pn ) under the condition (11) or ?(x0 ) =
?(x0 , p1 , . . . , pn ) etc. These functions are vector-functions, the manifold of which con-
stitutes the representation space for irreducible representations of the group P (1, n)
in the Schr?dinger picture. It is clear therefore that their components are functions
o
not only on x0 , x1 , . . . , xn (or x0 , p1 , . . . , pn etc.) but also on “spin” variables s3 , t3 , . . .
discussed above in connectian with representations of homogeneous group O(n) and
O(1, n ? 1). In accordance with the domain of definition of “spin” variables s3 , t3 , . . .
in different classes, the equation (19) is finite-component or infinite-component. In
the class I, where “spin” variables s3 , t3 , . . . take only discrete and finite values,
all the equations (19) are finite-component and their solutions ? realises the uni-
tary representations (i.e., vectors ? are normalizable). In the class III we have both
finite-component and infinite-component equations, but unitary representations can be
realized only on the solutions of the infinite-component equations.
One can suspect that owing to standing out of the time x0 in the equation (19),
the last is not invariant under the group P (1, n) discussed. For the equation (19)
the conventional demand of invariance under the given group is equivalent to the
demand that the manifold of its solutions is invariant under this group (i.e., that
any of its solution under transformations from P (1, n) remains a solution of it but,
generally speaking, another one). The mathematical formulation of this requirement
is to satisfy the condition

[(i?0 ? P0 ), Q]? = 0, (20)

where ? is any of solutions of Eq. (19) and Q is any generator of P (1, n) or any
linear combination of them, i.e., any element of the algebra P (1, n). Therefore the
generators Q = Pµ , Jµ? must have such a form that both the relations (4) and the
condition (20) must be satisfied. One can immediately verify that such operators Pµ ,
Jµ? are given by formulas (5) and (8) where, however, the substitution
1
x[0 pk] > x0 pk ? (xk p0 + p0 xk ) (15 )
2
is made and operators xk , pµ are defined by (6 ) and (12).
Thus, the “quantum mechanical” representation of the generators Pµ , Jµ? of the
algebra P (1, n) in the Schr?dinger picture have the form:
o
On representations of the inhomogeneous de Sitter group 133

For the class2 I

p2 + ? 2 ,
P = (p0 , pk ), p0 = ? k
(16 )
1 Skl pl
= x0 pk ? (xk p0 + p0 xk ) ?
Jkl = x[k pl] + Skl , J0k ;
p0 + ?
2
For the class III

p2 ? ? 2 ,
P = (p0 , pk ), p0 = ? k

Sab pb ? Sa0 p0
Jan = x[a pn] ?
Jab = x[a pb] + Sab , ,
pn + ?
(17 )
1
J0a = x0 pa ? (xa p0 + p0 xa ) + S0a ,
2
1 S0a pa
J0n = x0 pn ? (xn p0 + p0 xn ) ? .
2 pn + ?
It should be emphasized that in the Schr?dinger picture the operators do not
o
depend on the time x0 , except of the operators J0k . These last depend on the time x0
only by due to the presence of the term x0 pk ; it is just the presence of the term x0 pk
to satisfy the invariance condition (20) of the equation (19).
Note in the end of this section that last years the problem of using in physics some
groups like P (m, n), O(m, n) etc. as groups of generasized symmetry, was repeatedly
arised (see, for instance, ref. [8] and refs. in ref. [9]). The consequent physical analysis
of a quantumscheme based on either unificated group, is in fact possible only after
a mathematical analysis of representations of this group and equations connected
with it, like the analysis made here for the group P (1, n). The method used here for
studying the representations of the group P (1, n), is extend on the groups P (m, n)
without special difficulties. Thus the problem of classification of representations and
realization of an inhomogeneous group P (m, n) is in fact reduced to the problem of
classification and realization of homogeneous groups of the type O(m , n ) already
studied in ref. [7].
3. Physical interpretation
Last years many attempts of using different groups like O(m, n), P (m, n) as
relativizing internal symmetry groups like SU (n), were undertaken. The problem of
a relativistic generalization of an internal symmetry group is in fact connected with
finding a total symmetry group G containing non-trivially the Poincar? group P (1, 3)
e
(the group of “external” symmetry) and a group of “internal” symmetry like SU (n).
As it is shown in refs. [10], it is impossible non-trivially to unity the algebra P (1, n)
and the algebra of “internal” symmetries in the framework of a finite-dimensional
algebra Lie G, if the spectrum of the mass operator M 2 ? P0 ? P 2 is discrete. In
2

ref. [11] a non-trivial example of the algebra G ? P (1, 3) was constructed for the
case when the spectrum of the mass operator is already stripe; but the algebra G was
found to be infinite-dimensional in this case too. The consideration of the infinite-
dimensional algebras for the physical purposes is difficult both owing to the absence
formulae (16 ) for generators Pµ , J?? in the case P 2 > 0 coinside with the corresponding
2 Our

formulae (B.25–28) in ref. [5] if the last are rewritten in the tensor form.
134 W.I. Fushchych, I.Yu. Krivsky

of developed mathematical apparatus of such algebras and owing to the necessity
of solving a very difficult problem of physical interpretation of all the commuting
generators, the number of which is infinite. Do not speak about that the question
of writing down equations of motions, invariant under such an algebras, is quite
not clear. All this circumstances prompt that, to find a finite-dimensional algebra
G ? P (1, 3) of a total symmetry group, we have to refuse from the demand of the
discreticity or even stripiticity of the mass spectrum. In this case one can propose
many groups as total symmetry groups (the groups of the type P (m, n)). However,
in a G = P (m, n), O(m, n), as like as in cases of other groups which are groups
of coordinate transformations in spaces of great dimensions, it still arises a diffi-
cult problem of necessity to give a physical interpretation to the great number of
commuting operators.
Below we deal only with the inhomogeneous de Sitter group P (1, 4) which is a
minimal extention of the Poincar? group P (1, 3). Here we discuss a main topics of
e
physical interpretation of a quantum mechanical scheme based on this group. The
group P (1, 4) is the most attractive because of in this case it is a success to give a
clear physical meaning to a complete set of commuting variables.
In p-representation a component of the wave function ? — the a solution of the
equation (19) with n = 4 is a function of six dynamical variables of corresponding
complete set:

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. 29
( 122 .)



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