ñòð. 64 |

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 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

ñòð. 64 |