ñòð. 9 |

2

C2 = S 2 M (E + ?) ? ?p2 + p2 N 2 ? (p · N )2 ? (3.5)

(E + ?)?2 ,

?(E + ? + ?p4 )2 + (p · S)2 2?M (E + ?) ? p2

?Sab pb ? S4a (E + ? ? ?p4 )

1

G? = ?

?x4 pa ? 2P0 xa ? ,

E+?

a

2

(3.6)

S4a pa

?

K = ?P0 x4 ? ? ,

E+?

where

1

(3.7)

Sa = ?abc Sbc , Na = S4a , xk = i(?/?pk ).

2

The Casimir operator C2 (3.5) is in general the matrix which has elements depen-

ding on pk . Our second step is to diagonalise this matrix with the help of some

unitary transformation. We will look for the diagonalising operator in a form

(3.8)

U1 = exp(iS4a pa ?/p),

1/2

where p = p2 + p2 + p2 and ? is an unknown function of p, p4 .

1 2 3

With the help of the operator (3.8) one may derive from (3.4) and (3.6) a new

realisation:

?† ?†

? ? ? ?

P0 = U1 P0 U1 = P0 , Pa = U1 Pa U1 = Pa ,

(3.9)

† †

Ja = U1 Ja U1 = Ja , M = U1 M U1 = M,

?Sab pb ? S4a (E + ? + ?p4 )

†

(G+ ) = U1 G+ U1 = x4 pa ? xa M ? (3.10)

,

E+?

a a

?Sab pb ? S4a (E + ? ? ?p4 )

1

†

(G? ) = U1 G? U1 = ?

?x4 pa ? 2P0 xa ? ,

E+?

a a

2 (3.11)

† ?

= ?P0 x4 ? ?S4a pa /(E + ?),

K= U1 kU1

where

† †

xk = U1 xk U1 , Skl = U1 Skl U1 .

Reduction of the representations of the generalised Poincar? algebra

e 39

Using the Hausdorf–Campbell formula

?

1

{A, B}n ,

exp(A)B exp(?A) =

n!

n=0

{A, B}n = [A, {A, B}n?1 ], {A, B}0 = B

it is not difficult to calculate

xa = xa + pa S4b pb /p2 [??/?p ? (sin ?)/p]+

+ Sab pb /p2 (1 ? cos ?) + (1/p)S4a sin ?,

S4a = S4a cos ? + pa S4b pb /p2 (1 ? cos ?) + Sab pb (sin ?)/p, (3.12)

Sab pb = Sab pb cos ? + [(pa S4b pb /p) ? pS4a ] sin ?,

x4 = x4 + (S4b pb /p)(??/?p4 ).

Substituting (3.12) into (3.10), one obtains

pa S4b pb ?? M ?? 1 ?

(G+ ) = x4 pa ? M xa + ? ? sin ? ? sin ?+

E+?

a

p ?p4 p ?p p

E + ? + ?p4 Sab pb M ?p M

(1 ? cos ?) + ? ? (3.13)

+ +

(E + ?)p E+?

p p p

E + ? + ?p4 E + ? + ?p4

?p M

?

+ sin ? + S4a sin ? + cos ? .

E+? E+? E+?

p

The expression (3.13) for G+ is much simplified, if one puts

a

? = 2 tan?1 [p/(E + ?p4 + ?)]. (3.14)

For such a value of the parameter ?, we have:

p(E + ? + ?p4 )

1 ? cos ? = p2 /(E + p)(E + ?p4 ) ,

sin ? = ,

(E + ?)(E + ?p4 )

?? E + ?p4 ?? E + ?p4

? = ? sin ?

?

p2

?p4 p ?p

and

(G+ ) = x4 pa ? M xa . (3.15)

a

Substituting (3.9) and (3.15) into (2.6), we have

C2 = M 2 S 2 , (3.16)

where the matrix S 2 = S1 + S2 + S3 always may be chosen in the diagonal form,

2 2 2

S 2 ?s = s(s + 1)?s , |j ? ? | ? s ? j + ?.

The operators (3.9)–(3.11) are defined in a Hilbert space of square integrable

functions ?(p1 , p2 , p3 , p4 ). In order to diagonalise the operator M and (3.5) we intro-

duce in place of {p1 , p2 , p3 , p4 } the new variables {p1 , p2 , p3 , m}, where m = E + ?p4 .

Then

? p4 ? ? ? pa ?

> ?+ >

, +

?p4 E ?m ?pa ?pa E ?m

40 W.I. Fushchych, A.G. Nikitin

and the operators (3.9)–(3.11) and (3.15) take the form

2

?0 = m0 + ? p , ?

P P a = pa , M = ?m,

2m

Ja = ?i(p ? (?/?p))a + Sa , (G+ ) = ?i?m(?/?pa ), (3.17a)

a

C1 = ? 2 , C2 = m2 S 2 , (3.17b)

C3 = ?m,

K = ?im(?/?m),

(3.17c)

(G? ) = i[?pa (?/?m) ? P0 (?/?pa )] ? ?(Sab pb + S4a ?)/m,

?

a

where

? ? m < ?, m0 = ? ? 2 /2m .

The generators (3.17) are Hermitian with respect to the scalar product (3.3).

So we reach the following result:

Theorem. The Hilbert space of the IR D? (?, j, ? ) of the P (1, 4) algebra, correspondi-

ng to P 2 = ? 2 > 0, is expanded into the direct integral of the subspaces, which

correspond to the IR of the G(3) algebra with the following values of the invariant

operators: C1 = ? 2 , C2 = m2 s(s + 1), C3 = ?m, |?| ? m < ?, |j ? ? | ? s ? j + ? .

The explicit form of the P (1, 4) group generators in the Galilei basis and that of the

transition operator, which connects the canonical and the G(3) bases, are given by

the formulae (3.8), (3.14) and (3.17).

To conclude this section we consider the IR of the P (1, 4) algebra, corresponding

to P 2 = 0. The realisations of such an IR have been obtained in the form (Fushchych

and Krivsky [9, 10]):

1/2

P0 = ?E0 ? ? p2 + p2 , Pa = p a , P4 = p 4 ,

4

? Sab pb ?

J0a = ?i?E0 ?? J04 = ?i?E0

, ,

?pa E0 + p 4 ?p4

? ? Sab pb

? p4

J4a = i pa +? ,

?p4 ?pa E0 + p 4

where Sab are the generators of the IR D(s) of the SO(3) group. Substituting (3.18)

into (2.3), one obtains

1 ?

?

P0 = (?E0 ? p4 ), J1 = ?i p ?

M = ?E0 + p4 , + Sa ,

2 ?p a

? ? ? ?

? p4 K = ?i?E0

G+ = i pa (3.18)

+ i?E0 , ,

a

?p4 ?pa ?pa ?p4

1 ? ?? Sab pb

G? = ?ipa ? iP0 ?? .

a

2 ?p4 ?pa E0 + ?p4

It is not difficult to see that replacement of the variables {p, p4 } > {p, m}, where

m = E0 + ?p4 , reduces the generators (3.18) to the form (3.17), where, however,

? = 0, 0 ? m < ? and s has the fixed value, which characterises the IR of the SO(3)

Reduction of the representations of the generalised Poincar? algebra

e 41

group. So we have established the explicit form of the generators of the P (1, 4) group,

corresponding to P 2 = 0, in the Galilei basis.

4. The representations with P 2 < 0

We now use the IR of the P (1, 4) group, which corresponds to P 2 = ?? 2 < 0.

The generators of such representations have been obtained in the form (Fushchych

and Krivsky [9, 10, 11])

1/2

P4 = ? p 2 + ? 2 ? p 2

P 0 = p0 , Pa = p a , ,

0 a

? ?

? p? ? = ±1,

J?? = i p? + S?? , (4.1)

?p? ?p?

S?? p?

?

= ?iP4 ??

J4? , ?, ? = 0, 1, 2, 3,

|P4 | + ?

?p?

where S?? are the matrices which realise IR of the Lie algebra of the SO(1, 4) group.

Reducing the representation (4.1) by the representations of the Lie algebra of the

Galilei group, the mass operator M = P0 + P4 may take the zero value. Let us impose

the G(3)-invariant condition of turning into zero in the hyperspace, corresponding to

zero eigenvalues of the operator M , on the functions from the space of the IR (4.1)

(this hyperspace is the five-dimensional half-cylinder p2 = ? 2 , ?p0 < 0).

Using the transformation operator on the generators (4.1)

? = 2 tanh?1 [p/(? + |P4 | + ?p0 )] (4.2)

U2 = exp(iS0a pa ?/p),

and using the relations

1 ?? ?

?1

U2 x0 U2 = x0 + S0a pa , xµ = i ,

p ?p0 ?pµ

pa S0b pb ?? 1 1 Sab pb

?1

? sinh ? + S0a sinh ? + (1 ? cosh ?),

U2 xa U2 = xa +

p2

pp ?p p p

?1

U2 S0a U2 = S0a cosh ? ? (1/p)Sab pb sinh ? + (pa /p)(S0b pb /p)(1 ? cosh ?),

?1

U2 Sab pb U2 = Sab pb cosh ? + [(pa S0b pb /p) ? pS0a ] sinh ?,

p(?p0 + |P4 | + ?) ?? p

sinh ? = , = ,

(?p0 + |P4 |)(|P4 | + ?) |P4 |(|P4 | + ?)

?p0

?p2 |P4 |(?p0 + ?) + p2 + ? 2

??

1 ? cosh ? = 0

, = ,

(|P4 | + ?)(?p0 + |P4 |) |P4 |(|P4 | + ?)(|P4 | + ?p0 )

?p

one comes to the realisation

1/2

P4 = ? p 2 + ? 2 ? p 2

P 0 = p0 , Pa = p a , ,

0

? ?

? pa

Jab = i pb + Sab ,

?pa ?pb

(4.3)

? ? Sab pb + Sa0 ?

? p0 ?

J0a = i pa ,

|P4 | + ?p0

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