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of particles with variable spin and mass.
II
By analogy with the previous section, we seek Hj? in the form
II II II
(4.1)
Hj? = (?1 gj3 ?3 + ?3 fj3 ?3 )?j3 ??3 ,
j3 ?3

II II
where the unknown functions gj3 ?3 , fj3 ?3 depending only on p, M have the following
properties

fj3 ?3 = ?f?j3 ??3 ;
II II II II
(4.2)
gj3 ?3 = g?j3 ??3 ;
2 2
II II
= p2 + M 2 . (4.3)
fj3 ?3 + gj3 ?3

We can verify directly that the relations (2.18), (2.20), (2.21) are fulfilled, provided
(4.1), (4.3), (2.19) are satisfied. Using (2.7), we reduce (2.19) to the form

?[Hj? , xa ]? Hj? + i(ja + ?a )[Hj? , ?3 ]? + i[Hj? , (ja + ?a )]? ?3 = ipa .
II II II II
(4.4)

Substituting (4.1) into (4.4), using the values of the commutators ?j3 , ??3 with xa ,
ja , ?a , (A.1) and equating linearly independent terms, we obtain the following set of
equations

gj3 ?3 gj3 +1?3 + fj3 ?3 fj3 +1?3 = E 2 + p fj3 +1?3 ? fj3 ?3 ;
II II II II II II

gj3 ?3 gj3 ?3 +1 + fj3 ?3 fj3 ?3 +1 = E 2 + ?p fj3 ?3 +1 ? fj3 ?3 ;
II II II II II II

gj3 ?3 fj3 +1?3 + p = gj3 +1?3 fj3 ?3 ? p ;
II II II II
(4.5)
? ?p ;
II II II II
gj3 ?3 fj3 ?3 +1 + ?p = gj3 ?3 +1 fj3 ?3
II II
?gj3 ?3 ?fj3 ?3 ?3
? fj3 ?3 = ±1.
II II
gj3 ?3 = 2p(j3 + ?3 ), ?=
?p ?p ?3
In the case j = 0, the set (4.5) coincides with the set of equations for the coefficient
functions obtained in [13, 16].
294 W.I. Fushchych, A.G. Nikitin

Omitting rather cumbersome calculations, we give the solution of this system
p
fj3 ?3 = E th 2(j3 + ?3 )?II ,
II
?II = arcth ,
E (4.6)
II II
= E sech 2(j3 + ?3 )?
gj3 ?3 .

By means of (4.6) and (4.1) we obtain the following explicit form of the Hamiltonians
II
Hj? for the representation (2.7)
II
?1 sech 2(j3 + ?3 )?II + ?3 th 2(j3 + ?3 )?II (4.7)
Hj? = E ?j3 ??3 .
j3 ?3

Formula (4.7) gives the solution to the problem II. Let us write out the Hamilto-
nians Hj? for j, ? ? 1. According to (4.7), (3.12), we have
II

H II 1 = H II0 ?1 H0 1 E ?1 ;
II
1 1
22 2 2

H II1 = H II0 ?1 H01 E ?1 ;
II
(4.8)
1 1
2 2

H11 = H10 ?1 H01 E ?1 ,
II II II

where
H II0 = ?1 M + 2?3 (j · p); H0 1 = ?1 M + 2?3 (? · p);
II
1
2 2

H01 = H00 ? 2E(? · p)[?1 (? · p) ? ?3 E](E 2 + p2 )?1 ; (4.9)
II II II
H00 = ?1 E;
H10 = H00 ? 2E(j · p)[?1 (j · p) ? ?3 E](E 2 + p2 )?1
II II

are the Hamiltonians of the particles with fixed spin and mass obtained in [13, 16, 20].
5. Transition to the canonical representation
To give an unambiguous answer to the question what particles are described by
the equation (2.1) with the Hamiltonians obtained in Sections 3, 4 it is necessary to
find the explicit form of the Cazimir operators Wµ W µ and Pµ P µ of the group P (1, 3).
These operators prove to be of the simplest form in the canonical representation of
the Foldy–Shirokov type. Let us pass from the representations (2.4) and (2.7) to the
canonical one. Such a transition for the representation (2.4) is performed by means
of the operator
? ?
1
Uj? = exp ?i?2 ?j ? ? j ? ? ? .
I
(5.1)
2 33 3 3
j? 33

For the Hamiltonian (3.20a) the operator (5.1) has the simple form
I
E + ?1 Hj?
I
(5.2)
Uj? = ,
2E(E + M )
for the Hamiltonians (3.20b) the operator (5.1) is of the form

(j + ?) I
I
(5.3)
Uj? = exp i?2 p? .
p
On the Poincar?-invariant equations for particles with variable spin and mass
e 295

The transition from the representation (2.7) to the canonical one is performed by the
isometric operator

E
ch (j3 + ?3 )?II ? i?2 sh (j3 + ?3 )?II ?
II
Uj? =
M j3 ?3

?sech 2(j3 + ?3 )?II ?j3 ??3 ; (5.4)
E
?1
II
ch (j3 + ?3 )?II + i?2 sh (j3 + ?3 )?II
Uj? = ?j3 ??3 .
M j3 ?3

The operators (2.4) and (2.7) under the transformations (5.1) and (5.4) take the form
?
Pa = pa = ?i Jab = xa pb ? xb pb + Sab ;
,
?xa
(5.5)
1 Sab Pb
= tpa ? ?1 [xa , E]+ ? ?1
P0 = ?1 E, J0a .
2 E+M
The operators (5.5) are Hermitian operators with respect to the usual scalar product

d3 x ?† (, x)?2 (t, x), (5.6)
(?1 , ?2 ) = 1

where the functions ? are related to the solutions of the equation (2.1) by
?(t, x) = Uj? ?I (t, x) = Uj? ?II (t, x).
I II
(5.7)
In the representation (5.5) the Casimir operators are expressed by the matrices of
spin (2.5)

Wµ W µ = M 2 (j + ? )2 = M 2 j(j + 1) + ? (? + 1) + 2(j, ? ) , (5.8)

Pµ P µ = M 2 , (5.9)

where the squared-mass operator in the general case is given by formula (3.6). It
follows from (5.8) that equation (2.1) in the case of arbitrary j and ? describes a
particle whose spin can have the values |j ? ? | ? s ? j + ? . Equation (2.1) describes
a particle with the fixed spin S0 , if one imposes additional relativistically-invariant
conditions on the solutions ?(t, x). These conditions have the form

(j · ? )Uj? ?I (t, x) = ?s ?I (t, x);
I I
(5.10)
Uj?
?1
(j · ? )Uj? ?II (t, x) = ?s ?II (t, x).
II II
(5.11)
Uj?

The solutions of equations (2.1) and (5.10), (2,1) and (5.11) are eigenfunctions of the
operators Wµ W µ and Pµ P µ
Wµ W µ ?(t, x) = m2 s(s + 1)?(t, x), Pµ P µ ?(t, x) = m2 ?(t, x), (5.12)
s s

where
(5.13)
ms = a + bs(s + 1),
296 W.I. Fushchych, A.G. Nikitin

if we set in (3.6)
a0 = (a + b?)2 , a2 = 4b2 ,
a1 = 4b(a + b?),
(5.14)
a3 = a4 = · · · = 0, ? = j(j + 1) + ? (? + 1),
or

m2 = a2 + b2 s(s + 1) (5.15)
s

if in (3.6)

a2 = a3 = · · · = 0. (5.16)
a0 = a + b?, a1 = 2b,

Choosing the other values for the coefficients an in (3.6), one obtains another depen-
dence of the mass on spin.
At the conclusion of this section we present the explicit form of the metric operator
for the representation (2.7)
E

? II II
sech 2(j2 + ?3 )?II ?j3 ??3 . (5.17)
Mj? = Uj? Uj? =
M j3 ?3

For the problem of external motion of a charged particle in an external electromagnetic
field to be solved, it is more convenient to use the equation (2.1) with the Hamiltonian
I II
Hj? , and for the second quantization the equation (2.1) with the Hamiltonian Hj? is
more preferable. The first problem is considered in the next section, the second one
is solved in [14] for the case j = 0.
Notation. When formulating the problems I and II, we restricted ourselves to the
case where the matrices Skl in (2.4), (2.7) form the irreducible representation D(j, ? )
j+?
?D(s, 0) of group O(4). Using the reducible representations
or the direct sum
s=|j?? |
with more complex structure, we obtain qualitatively new equations of the type (1).
Let, for instance, the matrices Skl in (2.7) realize the representation D(0, s ? 1 )
2
?[D( 1 , 0) ? D(0, 1 )] of the group O(4). This means that these matrices have been
2 2
represented in the form
i
? ?k ?l + ?l ?k = ?2?kl , (5.18)
Skl = Skl + ?k ?l , [Skl , ?? ]? = 0,
2
?
where Skl are the generators of the representation D(0, s ? 1 ). In this case the
2
Hamiltonian H, which satisfies the conditions (2.17)–(2.21), has the form

(5.19)
H = ?0 ?a pa + ?0 m, ?0 = ?3 ?4 ,

and the Poincar?-invariant supplementary condition (5.11), which selects the spin S,
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