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M 2 = Pµ P µ = (Hj? )2 ? p2 .
I
(3.1)
290 W.I. Fushchych, A.G. Nikitin

The commutativity of M 2 with the operators (2.4) and (2.14) yields the following
relations
[Pa , M 2 ]? = [xa , M 2 ]? = [(ja + ?a ), M 2 ]? = 0; (3.2)

[P, M 2 ]? = [C, M 2 ]? = [T, M 2 ]? = 0; (3.3)
I
[Hj? , M 2 ]? = 0. (3.4)

From (2.14), (2.15), (2.16) it follows that the general form of the operator M 2 satis-
fying the conditions (3.2), (3.3) is given by the formula
M 2 = a0 + a1 (j · ? ) + a2 (j · ? )2 + · · · , (3.5)
where a0 , a1 , . . . are real numbers. The series (3.5) contains only a finite number of
terms, as follows from the relation
[j · ? ? ?s ] = 0, (3.6)
|j?? |?s?j+?

where ?s is the eigenvalue of the operator j · ? . Between the numbers ?s and j, ? , s
there is the relation
2?s = ?j(j + 1) ? ? (? + 1) + s(s + 1). (3.7)
From (3.6) it is seen that the particle system capable of being in different mass
states may be compared with the representation (2.4) and hence with the equation
(2.1). As will be shown later, with a proper choice of the coefficients an in (3.5) we
can obtain the mass formula (1.2).
I
In order to find the explicit structure of the operator Hj? satisfying relations
(2.18)–(2.21) and condition (3.1) we expand it in a complete system of ortoprojectors
I
dI 3 ?3 (p) + ?1 gj3 ?3 (p) + ?2 hI 3 ?3 (p) + ?3 fj3 ?3 (p) ?j3 ??3 ,
I I
(3.8)
Hj? = j j
j3 ?3

where
jp ? j3 ?p ? ?3
?j3 = ; ??3 = ;
j3 ? j3 ?3 ? ?3
j3 =j3 ?3 =?3
(3.9)
j·p ? ·p
j3 = ?j, ?j + 1, . . . , j, ?3 = ??, ?? + 1, . . . , ?,
jp = , ?p = ,
p p
dI 3 ?3 , hI 3 ?3 , fj3 ?3 , gj3 ?3 are the unknown functions which depend on p and M .
I I
j j
It is easy to see that the operators (3.9) are the orthoprojectors on the proper
subspaces of the operators jp and ?p , i.e. they satisfy the relations
j j
?j3 ?j3 = ?j3 j3 ?j3 ; ?j3 = 1; jp = j3 ?j3 ;
j3 =?j j3 =?j
(3.10)
? ?
??3 ??3 = ??3 ?3 ??3 ; ??3 = 1; ?p = ?3 ??3 .
?3 =?? ?3 =??
On the Poincar?-invariant equations for particles with variable spin and mass
e 291

The condition (2.21) is satisfied if
?
dI 3 ?3 = hI 3 ?3 = 0; I I I I I
for
fj3 ?3 = f?j3 ??3 ; gj3 ?3 = g?j3 ??3 r1 = I,
j j
(3.11)
fj3 ?3 = ?f?j3 ??3 ;
dI 3 ?3 = hI 3 ?3 = 0; I I I I I
for
gj3 ?3 = g?j3 ??3 r1 = ?1 .
j j

In order that (3.4) be fulfilled with an arbitrary choice of the coefficients an in (3.6),
it is necessary to set
I I
(3.12)
fj3 ?3 = ?1 (j3 + ?3 ), gj3 ?3 = ?2 (j3 + ?3 ),
I I
i.e. fj3 ?3 , gj3 ?3 may depend only on the sum of the indices. If (3.12) is not fulfilled,
then the operator M 2 commutes with the Hamiltonian (the condition (3.4)) only in
the case where a1 = a2 = a3 = · · · = 0.
The condition (3.1) imposes the additional restriction
2 2
I I
= p2 + M 2 (3.13)
fj3 ?3 + gj3 ?3
I I
on the functions fj3 ?3 and gj3 ?3 . Direct verification shows that if the conditions (3.4),
(3.13) are fulfilled, the relations (2.18), (2.19), (2.21) are satisfied. Thus, it remains
to satisfy the relation (2.18) which together with (3.11)–(3.13) will determine the
I I
ultimate structure of the operator Hj? , i.e. the explicit form of the functions fj3 ?3 and
I
gj3 ?3 .
The relations (2.20) for the operator (2.4) may be reduced to the form [5]
= ?4iSab .
I I
(3.14)
[Hj? , xa ]? , [Hj? , xb ]? ?

Substituting (3.8) into (3.14), using the commutation relations (A.1) and taking into
account the linear independence of the vectors (A.2), we obtain the following equati-
I I
ons for fj3 ?3 , gj3 ?3

gj3 ?3 gj3 +1?3 + fj3 ?3 fj3 +1?3 = gj3 ?3 gj3 ?3 +1 + fj3 ?3 fj3 ?3 +1 = M 2 ? p2 .
I I I I I I I I
(3.15)
I I
From (3.13) it is seen that the functions gj3 ?3 , fj3 ?3 can be represented in the form
I I
p2 + M 2 . (3.16)
fj3 ?3 = E sin ?j3 ?3 , gj3 ?3 = E cos ?j3 ?3 , E=
Inserting (3.16) into (3.15), we obtain for ?j3 ?3 the following recurrence formulas
P
?j3 +1?3 = ?j3 ?3 ± 2?I , ?j3 ?3 +1 = ?j3 ?3 ± 2?I , ?I = arctg (3.17)
.
M
I I
By means of (3.16), (3.17) we can define all the coefficients gj3 ?3 , fj3 ?3 of the operator
I I
(3.8) if at least one of the functions of the set fj3 ?3 (or gj3 ?3 ) is known. This initial
function can be found from the relations (3.17), (3.11) which, taking into account
(3.16), can be written in the form:
I
for
??j3 ??3 r1 = I;
(3.18)
?j3 ?3 =
???j3 ??3 I
for r1 = ?1 .

Finally, we are led to the following result: the operator (3.8) with coefficient functions
(3.16), (3.17) satisfies the relations (2.18)–(2.21) and this means that the problem I is
292 W.I. Fushchych, A.G. Nikitin

I
completely solved. The equation (2.1) with such Hj? will be invariant with respect to
the full Poincar? group P (1, 3).
e
Let as present the simplest solutions for the system of the recurrence relations
(3.17), (3.18) (for the details of the solutions for equations such as (3.15) see [5])
? 1
?(?1)j3 +?3 + 2 ?I , j + ? — half-integers,
?
?j3 ?3 = (?1)j3 +?3 ?I , (3.19)
j + ? — integers,
?
?
2(j3 + ?3 )?I , ?3 = ±?3 , j + ? — arbitrary numbers.
Substituting (3.19), (3.11), (3.16) into (3.8), we obtain
? 1
??1 M + ?3 p(?1)j3 +?3 + 2 ?j3 ??3 , j + ? — half-integers, (3.20a)
?
?
?
?
? j3 ?3
?
?
?
?? M + ?
?1 p(?1)j3 +?3 ?j3 ??3 ,
? j + ? — integers, (3.20b)
3
I
Hj? = j3 ?3
?
?
?E
? ?1 cos 2(j3 + ?3 )?I +
?
?
? j?
? 33
?
?
?
?
+?3 sin 2(j3 + ?3 )?I ?j3 ??3 , j + ? — arbitrary numbers.
Choosing other solutions of the system (3.17), (3.18) we arrive at Hamiltonians which
are unitarily equivalent to (3.20) but differ from them in form.
Let us write the explicit expressions for the operators (3.20) in terms of j · p, ? · p
for j, ? ? 1. Using (3.9), (3.20a), we find
H I 1 = ?1 M + 2?3 (? · p)(j · p)p?1 ;
1
22

H I 1 = H I 0 ? 4?3 (j · p)(? · p)2 p?2 ; (3.21)
1 1
2 2

H11 = ?H00 + H10 + H01 + 2?3 (j · p)2 (? · p)2 p?3 ,
I I I I

where
H I 0 = ?1 M + 2?3 (j · p);
I
H00 = ?1 M + ?3 p; 1
2
(3.22)
H01 = H00 ? 2?3 (? · p)2 p?1 ; H10 = H00 ? 2?3 (j · p)2 p?1
I I I I

are the Hamiltonians of particles with the fixed spin found earlier in [5].
Substituting (3.10) into (3.20b), we obtain
H I 1 = H I 0 ?3 H0 1 E ?1 ;
I
1 1
22 2 2

H 1 0 ?3 H0 1 E ?1 ;
I I I
(3.23)
H11 =
2 2 2

H11 = H10 ?3 H01 E ?1 ,
I I I

where
H0 1 = ?1 M + 2?3 (? · p); H I 0 = ?1 M + 2?3 (j · p);
I
1
2 2

H01 = ?1 E + 2(? · p)[?3 M ? ?1 (? · p)]E ?1 ; (3.24)
I

H10 = ?1 E + 2(j · p)[?3 M ? ?1 (j · p)]E ?1
I
On the Poincar?-invariant equations for particles with variable spin and mass
e 293

also coincide with the Hamiltonians obtained in [5]. The operator H I 0 is the Dirac
1
2
Hamiltonian.
The equation (2.1) together with the Hamiltonians H I 1 , H I 1 , H11 describes the
I
1 1
22 2
particles with spins 0 and 1, 1 and 3 , 0, 1 and 2, respectively. As it is seen from
2 2
I I I
(3.21)–(3.24), the operators Hj? can be expressed by Hj?1? , Hj? ?1 . Thus, the form
of the Hamiltonian for arbitrary j and ? is completely defined by the Hamiltonians
for j, ? = 0, 1 .
2
II
4. Explicit form of the operators Hj?
II
In this section we solve the problem II, i.e. find all the operators Hj? satisfying the
system (2.18)–(2.21), when the representation of the algebra P (1, 3) has the structure
(2.7).
Using the representation (2.7) for the special case when j = 0, ? is an arbitrary
number, Weaver, Hammer, and Good, and then, for a more general statement of the
problem, Mathews [13, 16], found equations of the type (2.1) for a particle with fixed
spin and mass. The results given below are a generalization of [13, 16, 20] to the case
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