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1. Majorana E., Nuovo Cimento, 1932, 9, 355;
Nambu Y., Prog. Theor. Phys. Suppl., 1966, 37-38, 368;
Fronsdal C., Phys. Rev., 1967, 156, 1665;
Barut A.O., Corrigan D., Kleinert H., Phys. Rev., 1968, 167, 1527.
2. Chodos A., Phys. Rev. D., 1970, 1, 2973 (The reader will find an extensive list of further references
in it).
3. Bakamjian B., Thomas L.H., Phys. Rev., 1953, 92, 1300;
Foldy L.L., Phys. Rev., 1961, 122, 289.
4. Fushchych W.I., Lett. Nuovo Cimento, 1974, 11, 508.
5. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1968, 7, 79; 1969, 14, 573;
Fushchych W.I., Theor. Math. Phys., 1970, 4, 360 (in Russian).
6. O’Raifeartaion L., Phys. Rev., 1965, 14, 575.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 286–300.

?
On the Poincare-invariant equations
for particles with variable spin and mass
W.I. FUSHCHYCH, A.G. NIKITIN
The Poincar?-invariant equations without redundant components, describing the motion
e
of a particle which can be in different spin and mass states are obtained. The quasi-
relativistic equation for a particle with arbitrary spin in external electromagnetic field
is found. The group-theoretical analysis of these equations in carried out.

1. Introduction
Many papers are devoted to the problem of construction of relativistic equations for
a particle which can be in different spin and mass states. There are various approaches
to this problem. The fundamentals of the equation theory describing a particle with
an infinite number of spin states have been developed by Majorana [12], and later
by Gelfand and Yaglom [8] who used infinite-dimensional unitary representations of
the homogeneous Lorentz group O(1, 3). (For the actual situation of this theory see
e.g. [10].) In other works [1, 9, 17] the wave function of such a particle is supposed
to possess some additional variables (inner degrees of freedom) besides three space
variables. On the basis of this assumption some relativistic equations for particles with
variable spin and mass (for instance, for particles of rotator type) are constructed. By
extending the four-dimensional Minkowsky space to the five-dimensional one and by
using the representations of the inhomogeneous de Sitter group P (1, 4) which includes
the Poincar? group P (1, 3) as a sub-group, equations were derived [4, 6, 7] which can
e
be interpreted as the motion equations for a particle (or for a system of two particles)
with variable discrete spin and variable continuous mass. In contrast to the above-
mentioned papers [1, 8, 17], where a particle has always an infinite number of spin
states, in the framework of the group P (1, 4) the particle has only a finite number of
spin states. This is connected with the fact that any irreducible representation of the
group O(4) which is a small group of the group P (1, 4) is decomposed into a finite
direct sum of irreducible representations of the rotation group O(3).
In the present paper, without going beyond the scope of the Poincar? group and
e
using the irreducible representations of the group O(4), we find the relativistic equati-
ons of motion in the Schr?dinger form, describing a particle which can be in finite
o
spin states. The spin s of such a particle can take the values
|j ? ? | ? s ? j + ?, (1.1)
where j and ? are the integers or half-integers labelling the irreducible representations
of the group O(4). The particle mass m can be either fixed or given by the formulas
m = a1 + b1 · s(s + 1) m2 = a2 + b2 · s(s + 1),
or (1.2)
2 2

where a1 , a2 , b1 , b2 are constants. The wave functions in the motion equation obtained
have 2(2j + 1)(2? + 1) components which corresponds to the number of the degrees
Reports on Mathematical Physics, 1975, 8, 1, P. 33–48.
Preprint ITP-121E, Institute of Theoretical Physics, Kiev, 1973, 19 p.
On the Poincar?-invariant equations for particles with variable spin and mass
e 287

of freedom of the system described. This means that the equations proposed do not
contain redundant components and hence do not lead to the well-known difficulties
[15, 18, 19].
For the construction of the equations the algebraic (nonspinor) approach developed
in [5, 11, 13, 16, 20] is used. The group-theoretical analysis of the equations is per-
formed not in terms of the Lorentz group O(1, 3) representations used traditionally but
in terms of the Poincar? group P (1, 3) ? O(1, 3) representations. This is stipulated by
e
the fact that only the invariants of the group P (1, 3) have distinct physical meaning.
2. Statement of the problem
We shall investigate the equations for a particle with variable spin and mass in
the form
??(t, x)
(2.1)
i = Hj? ?(t, x),
?t
where Hj? is the unknown operator function (the Hamiltonian of a particle) which
depends on the momenta and spin matrices, ? is the wave function which transforms
under the four-dimensional rotations and translations according to the reducible
representation of the group P (1, 3) and contains 2(2j + 1)(2? + 1) components. In the
previous paper [5] describing (up to unitary equivalence) all the Poincar?-invariant
e
equations of the form (2.1) for a particle with fixed mass m and fixed spin s we
imposed the conditions
Pµ P µ ?(t, x) = m2 ?(t, x); (2.2)
Wµ W µ ?(t, x) = m2 s(s + 1)?(t, x) (2.3)
on the solutions of equation (2,2), where Pµ is the energy-momentum operator on the
mass shell and Wµ is the Pauli–Lubansky vector. If the spin and mass of a particle
are not fixed, conditions (2.2), (2.3) should be omitted.
We resolve the problem of finding the Poincar?-invariant equations of the form (2.1)
e
using two different approaches. This is connected with the fact that the equations
obtained using the first approach may prove to be convenient in terms of quantum
mechanics and the equations derived in the second approach are useful in terms of
field theory.
In the first approach (I) the problem is reduced to the following: one has to find
I
all Hamiltonians Hj? such that the operators
?
Pa = pa = ?i
I I I
P0 = Hj? , ;
?xa
(2.4)
1
= xa pb ? xa pb + Sab , = tpa ? [xa , P0 ]+
I I I
Jab J0a
2
should satisfy the Poincar? algebra. The matrices Sab , have the following structure
e
?c
j 0 ?c
? 0
(2.5)
Sab = jc + ?c , jc = , ?c = ,
?c 0 ?c
?
0 j
where ?c and ?c are the (2j + 1)(2? + 1)-dimensional matrices satisfying the commu-
j ?
tation relations of the algebra O(4)
[?a , ?b ]? = i?c , [?a , ?b ]? = 0;
jj j [?a , ?b ]? = i?c ,
?? ? j?
288 W.I. Fushchych, A.G. Nikitin

(a, b, c) is the cycle (1, 2, 3),
?a = j(j + 1),
j2 ?2 (2.6)
?a = ? (? + 1).
In the second approach (II) the problem1 is formulated as follows: one has to find
II
all the Hamiltonians Hj? such that the operators
?
Pa = pa = ?i
II II II
P0 = Hj? , ;
?xa (2.7)
= xa pb ? xa pb + Sab , = tpa ?
II II II
Jab J0a xa P0 + i?3 S4a
should satisfy the Poincar? algebra.
e
Here
I 0
?a = ±?a , (2.8)
?3 = , S4a = ja + ?a ,
?I
0
I is the (2j + 1)(2? + 1)-dimensional unit matrix. In particular, when j = 0, ? = 1 ,
2
the operators (2.4) and (2.7) coincide, since
H0 1 = H0 1 = ?1 m + 2?3 ? · p.
I II
(2.9)
2 2


The operator (2.9) is the Dirac Hamiltonian. For other numbers j and ? , as will
be shown later, the representations (2.4) and (2.7) do not coincide. The choice of
the representation structures for the algebra P (1, 3) in the form (2.4) and (2.7) is
stipulated by the fact that on the set of solutions of the Dirac equation the Poincar?
e
algebra may be represented either in the form (2.4) or (2.7). One of the principal
differences between the operators (2.4) and (2.7) consists in the fact that all the
operators (2.4) are Hermitian with respect to the usual scalar product

d3 x ? (t, x)?2 (t, x), (2.10)
(?1 , ?2 ) = 1


while the operators (2.7) are non-Hermitian with respect to (2.10). But the operators
(2.7) are Hermitian with respect to the scalar product

d3 x ? (t, x)M ?2 (t, x),
? (2.11)
(?1 , ?2 ) = 1

?
where M (j, ?, p) is some metric operator whose form will be found later.
Equation (2.1) will obviously be Poincar?-invariant if the operators (2.4) or (2.7)
e
satisfy the Poincar? algebra, as in this case the condition
e
?
? Hj? , exp(i?Q) (2.12)
i ?(t, x) = 0
?t ?

is satisfied, where Q is an arbitrary generator and ? is the parameter of the group
P (1, 3). For the infinitisimal transformations this condition takes the form
?
? Hj? , Q (2.13)
i ?(t, x) = 0.
?t ?
1 The operators with indices I and II refer to the approaches I and II, respectively. When these indices
are omitted, the corresponding relations are true both for the approach I and the approach II.
On the Poincar?-invariant equations for particles with variable spin and mass
e 289

On the set {?} of the solutions of (2.1) we define the operators of discrete transfor-
mations
P ?(t, x) = r1 ?(t, ?x);
T ?(t, x) = r2 ?? (?t, x); (2.14)
C?(t, x) = r3 ?? (t, x),

where r1 , r2 , r3 are the matrices which can be chosen (without loss of generality) in
the form
I 0
?
I I II
or
r1 = ?1 r1 = I = , r1 = ?1 ;
0 I
(2.15)
? 0
I II I II
r2 = r2 = ?, r3 = r3 = ?2 ?, ?= ,
0 ?

where ? is the matrix satisfying the relations

??
? ?a = ??a ? , ? ?a = ??a ? . (2.16)
j j ?

The proof of existence of such a matrix is not given here.
The operators P , T , C and the generators Pµ , Jµ? must satisfy the relations

[P, P0 ]? = [P, Pa ]+ = [P, Jab ]? = [P, J0a ]+ = [C, Pa ]+ = [C, P0 ]+ = 0;
(2.17)
[C, Jab ]+ = [C, J0a ]+ = [T, P0 ]? = [T, Pa ]+ = [T, Jab ]+ = [T, J0a ]? = 0.

Let equation (2.1) be invariant under P , T , C-transformations. In this case the relati-
ons (2.17) must be added to the Poincar? algebra.
e
Thus the problem of finding all the Poincar?-invariant equations for a particle
e
with variable spin and mass is reduced to that of finding the operators Hj? which
satisfy the relations

[Hj? , Pa ]? = [Hj? , J0a ]? = 0, [Pa , J0b ]? = i?ab Hj? ;
(2.18)
[Jab , J0c ]? = i(?ac J0b ? ?bc J0a );

(2.19)
[Hj? , J0a ]? = ipa ;

[J0a , J0b ]? = ?iJab ; (2.20)

(2.21)
[P, Hj? ]? = [C, Hj? ]+ = [T, Hj? ]? = 0.

I
3. Explicit form of the operators Hj?
I
In this section we solve problem I, i.e. we find those operators Hj? which satisfy
the set of relations (2.18)–(2.21) in the case where the representation of the Poincar?
e
algebra has the structure (2.4).
The squared-mass operator for the representation (2.4) (which, generally speaking,
is not a multiple of the unit one) is of the form

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. 64
( 122 .)



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