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?p0 ?pa
? Sab pb + S0a ? ?
J4a = ?iP4 + , J04 = iP4 .
|P4 | + ?p0
?pa ?p0
42 W.I. Fushchych, A.G. Nikitin

Substituting (4.3) into (2.3) and going from {pa , p0 } to the new variables {pa , m},
1/2
where m = p0 + p2 + ? 2 ? p2 , one obtains the Galilei group generators in the form
a
0
(3.17a), and the remaining generators G? , K in the form (3.17c), where, however,
a
m0 = ?? 2 /2m, ?? 2 < m < 0, 0 < m < ?, and Sab are the generators of the group
SO(3) ? SO(1, 3).

5. Covariant representation of the P (1, 4) group
Consider an arbitrary covariant representation of the Lie algebra of the P (1, 4)
group. Such a representation is realised by the operators
? ?
? pµ (5.1)
P µ = pµ , Jµ? = i p? + Sµ? ,
?pµ ?p?
where Sµ? are the generators of a representation of the SO(1, 4) group. Let us confine
ourselves to the case where Pµ P µ ? > 0.
Substituting (5.1) into (2.3), we obtain
1 ?
? ?
P0 = (p0 ? p4 ), Ja = ?i p ?
P a = pa , + Sa ,
2 ?p a

G+ = x0 pa ? xa M + ?+ , (5.2)
M = p0 + p4 , ?
a a
1
G? = x4 pa ? xa P0 + ?? ,
? ??
K = x4 M ? x0 P0 + S04 ,
? ?
a
2a
where
? ? ? ?
?± = S0a ± S4a , ?
x0 = 2i
? , x4 = i
? + .
?p0 ?p4 ?p0 ?p4
For the transition of the realisation (5.2) into the Galilei basis we use the operator
U3 = exp[i?+ p/M ]. (5.3)
With the help of the transformation
? ?1 ?1
? ?
Pµ > Pµ = U3 Pµ U3 , Ja > Ja = U3 Ja U3 ,
?1 ?1
G± > (G± ) = U3 G± U3 , K>K = U3 KU3 ,
a a a

one comes to the realisation in which the invariant operators (2.6) of the G(3) subal-
gebra are of diagonal form:
1
? ?
P0 = (p0 ? p4 ), P a = pa , M = M = p0 + p 4 ,
2
Ja = ?i(p ? ?/?p)a + Sa , G+ = x0 pa ? xa M,
?
a
µ
?0 ? Sab pb + S40 pa + 1 ?? ? ?+ pµ p ,
G?= x4 pa ? xa P
?
a
2a M2
M
??
K = x4 M ? x0 P0 + S04 ,
?

where Sa = 1 ?abc Sbc . The operators Ca (2.6) take the form
2

C2 = M 2 S 2 ,
C1 = pµ pµ , C3 = M
Reduction of the representations of the generalised Poincar? algebra
e 43

i.e. the eigenvalues of the operator C1 coincide with the values of P 2 , the eigenvalues
of the operator C2 are characterised by the spectrum of the Casimir operator of the
group SO(3) ? SO(1, 4), and the eigenvalues of the operator C3 lie in the interval
(C1 )1/2 ? C3 < ?.
The results of this section may be used for the diagonalisation of the wave equa-
tions, which are invariant under the P (1, 4) group. As an example we will consider
the five-dimensional generalisation of the Dirac equation

(?µ pµ + ?)? = 0, (5.4)
µ = 0, 1, 2, 3, 4.

On the set of the solutions of the equation (5.4) the generators of the P (1, 4) group
have the form (5.1) where Sµ? = 1 i[?µ , ?? ]. Using the operator (5.3) on equation
4
(5.4), one obtains an equation, which is equivalent to (5.4) but is manifestly invariant
under the Galilei group

?
P0 ?+ = ?/2m + p2 /2m ?+ , (5.5)
?? = 0,

where
1
(1 ± ?0 ?4 )?, ? ? m < ?.
?± = ? = U3 ?,
2
If one imposes the Galilean-invariant subsidiary condition (p0 + p4 )? = m0 ? and
puts ? = 0, then equation (5.4) is reduced to the Levi-Leblond equation for the non-
relativistic particle of spin s = 1 (Levi-Leblond [14]). In this case (5.3) coincides with
2
the operator which diagonalises the Levi-Leblond equation (Nikitin and Salogub [16]).

6. IR of the Poincar? group in the G(2) basis
e
The transition of the IR of the P (1, 3) group to the basis of a two-dimensional Gali-
lei group G(2) may be made by complete analogy with the reduction P (1, 4) > G(3).
Here we consider only the representations of the P (1, 3) group, which correspond to
time-like four-momenta. The generators of such a representation in a Shirokov–Foldy
realisation (Shirokov [17, 18], Foldy [7]) have the form (3.1) where µ, ? = 0, 1, 2, 3;
k, l = 1, 2, 3. With the help of the transformation

Pµ > Pµ = U Pµ U ?1 , Jµ? > Jµ? = U Jµ? U ?1 ,
? ?

where

U = exp (iS3? p? /|p|) tan?1 [|p|/(|P0 | + ?p3 + ?)] ,
1/2
|p| = p2 + p2 , ? = 1, 2,
1 2

and the following replacement of the variables {p1 , p2 , p3 } > {p1 , p2 , m}, where m =
1/2
?p3 + p2 + p2 + ? 2 , one obtains the generators of the Poincar? group in the G(2)
e
1 2
basis:
1?
? ? ?
P0 = (P0 + P3 ) = ? 2 /2m + |p|2 /2m, P ? = p? ,
2 (6.1)
J3 = i[p2 (?/?p1 ) ? p1 (?/?p2 )] + S12 , M = ?m,
44 W.I. Fushchych, A.G. Nikitin

?
? ?
G+ = J0? + J3? = ?i?m |?| ? m < ?,
,
?
?p?
1? (6.2)
G? = (J0? ? J3? ) = i[p? (?/?m) ? P0 (?/?p? )] ? ?(S?? p? + S3? ?)/m,
? ?
?
2
?
K = J03 = ?im(?/?m).

The operators (6.1) coincide with the “kinematical group generators”, which are
used in the null-plane formalism (see e.g. Leutwyler and Stern [13]).
Using the results of §§ 3–5, it is not difficult to make the transition into the G(2)
basis of the representations of the P (1, 3) algebra which corresponds to light-like and
space-like four-momenta.

7. Connection between the Galilei and the Poincar? bases
e
We now consider the connection between the realisations of the generators of
the P (1, 4) group (corresponding to time-like five-momenta) in both the Galilei and
Poincar? bases.
e
The generators of the P (1, 4) group in the Poincar? basis (i.e. in the basis where
e
the Casimir operators of the P (1, 3) group are of diagonal type) have the form
(Fushchych et al [12], Nikitin et al [15])
1/2 1/2
P4 = ? 4 m 2 + ? 2
P 0 = E = p 2 + m2
? , Pa = p a , ? ,
Jab = i[pb (?/?pa ) ? pa (?/?pb )], ?4 = ±1,
J0a = ?ip0 (?/?pa ) ? Sab pb /(E + m),
? a, b = 1, 2, 3,
1/2
J04 = ?iE ?4 1 ? ? 2 /m2 , ?/? m ? (?/m)(S4a pa /m), (7.1)
? ? ? ?
1/2 1/2
J4a = ipa ?4 1 ? ? 2 /m2 , ?/? m ? i?m 1 ? ? 2 /m2
? ? ? ? ?/?pa +
?pa S4b pb ?S4a
1/2
+ ?4 1 ? ? 2 /m2
+ ? [Sab pb /(E + m)] +
? ,
2 (E + m)
m m
?
where
{A, B} = AB + BA, |?| ? m < ?.
?
The generators (7.1) are Hermitian with respect to the scalar product
?
j+?
d3 p †
(?1 , ?2 ) = dm
? ? (p, m, s, s3 )?2 (p, m, s, s3 ).
? ?
2E 1
s=|j?? | ?

As soon as the operators (7.1) and (3.17) realise the same IR D+ (?, j, ? ) of the
P (1, 4) group, the equivalence transformation, which connects these two realisations,
exists. In order to come from (7.1) to (3.17), we make the isometric transformation
Pµ > W Pµ W ?1 , Jµ? > W Jµ? W ?1 (7.2)
and the following replacement of variables
pa > pa , m > m(m, p), (7.3)
? ?
Reduction of the representations of the generalised Poincar? algebra
e 45

where
1/4
W = 1 ? ?/m2 exp[i(S4a pa /p)(?1 ? ?2 )],
?
1/2
?1 = 2 tan?1 p/ E + ?4 m2 ? ? 2 +?
? ,
(7.4)
1/2
?2 = 2 tan?1 ?4 p m2 ? ? 2 /(E + m)(m + ?) ,
?
1/2
2
m2 ? ? 2 ? p 2 + 4m2 ? 2
m = (1/2m)
? .

One can ensure by direct verification that the transformations (7.2)–(7.4) reduce
the generators (7.1) into the Galilei basis (i.e. that the transformed generators coincide
with (3.17) after substitution into (2.3)). We do not give the detailed calculations here
because the transformations (7.2)–(7.4) may be represented as two consequent ones:
namely, the transition from the Poincar? to the canonical basis (Nikitin et al [15])
e
Pµ > V Pµ V ?1 , Jµ? > V Jµ? V ?1 ,
1/2
m > m(p4 ) = ?4 p2 + ? 2 (7.5)
? ? ,
4
1/4
V = 1 ? ? 2 /m2
? exp(iS0a pa ?2 /p)
and then the transition from the canonical basis to the Galilei one (see § 3). So
W = U1 V,
where V and U1 are given by equations (7.5), (3.8), (3.14).
The transformation (7.2)–(7.4) may be used to establish the connection between
the vectors in the Galilei and in the Poincar? bases. This connection is given by the
e
equations:
??
?(p, m, s, s3 ) = W Ps Ps3 Ps Ps3 ?(p, m(m, p), s, s3 ),
?
?(p, m, s, s3 ) = W ?1 Ps Ps Ps Ps ?(p, m(m, p), s, s3 ),
?? ?
3 3
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