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. 63
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??(t, x1 , . . . , x6 ) m1 + m 2
? 0 ?a pa + v
i = ? ?0 ?a+3 pa+3 +
?
?t m1 m2

(18)
+(m1 + m2 )?0 ?(t, x1 , . . . , x6 ),

? ?
?
pa = ?i pa+3 ? K a = ?i
? , ? .
?xa ?xa+3
In this equation ? is also an eight-component function.
I wish to thank A.G. Nikitin, A.L. Grishchenko for useful comments and N.V. Hna-
tjuk for helpful rending the manuscript.

1. Breit G., Phys. Rev., 1929, 34, 553.
2. Bakamjian B., Thomas L.H., Phys. Rev., 1953, 92, 1300.
3. Foldy L.L., Phys. Rev., 1961, 122, 289.
4. Logunov A.A., Tavkhelidze A.N., Nuovo Cimento, 1963, 29, 380;
Kadyshevsky V.G.., Nucl. Phys., 1968, 36, 125.
5. Fong R., Sucher J., J. Math. Phys., 1964, 5, 456.
6. Osborn H., Phys. Rev., 1968, 176, 1514.
7. Fushchych W.I., Lett. Nuovo Cimento, 1973, 6, 133.
8. Kadyshevsky V.G., Mateev M.D., Mir-Kasimov R.M., J. Nucl. Phys., 1970, 11, 692 (in Russian).
W.I. Fushchych, Scientific Works 2000, Vol. 1, 282–285.

?
Poincare-invariant equations
with a rising mass spectrum
W.I. FUSHCHYCH

In recent years many papers have been devoted to the construction of infinite-compo-
nent wave equations to describe properly the spectrum of strongly interacting particles
[1, 2]. As a rule, the derived equations have a number of pathological properties:
the unrealistic mass spectra, the appearence of spacelike solutions (p2 < 0), the
µ
breakdown of causality etc. [2].
In this note we shall construct, in the framework of relativistic quantum mechanics,
the Poinca?e-invariant motion equations with realistic mass spectra. These equations
r
describe a system with mass spectra of the form m2 = a2 + b2 s(s + 1), where a and
b are arbitrary parameters. Such equations are obtained by a reduction of the motion
equation for two particles to a one-particle equation which describes the particle in
various mass and spin states. It we impose a certain condition on the wave function
of the derived equation, such an equation describes the free motion of a fixed-mass
particle with arbitrary (but fixed) spin s.
Let us consider the motion equation for two free particles with masses m1 = m2 =
m and spins s1 and s2 in the Thomas–Bakamjian–Foldy form [3]
??(t, x, ?)
= (Pa + M 2 )1/2 ?(t, x, ?),
2
(1)
i
?t
where
M = 2(m2 + k2 )1/2 ,
Pa = p(1) + p(2) ,
a a
(1) (2)
pa , pa are components of the momenta of the two particle, k the relative momen-
tum, x the co-ordinate of the centre of mass, ? is the relative co-ordinate.
On the manifold of solutions {?} of eq. (1) the generators of the Poincar? group
e
P1,3 have the form
?
Pa = pa = ?i
P0 = (Pa + M 2 )1/2 ,
2
, a = 1, 2, 3,
?xa
Mab = xa pb ? xb pa ,
Jab = Mab + Lab , Lab = mab + Sab , (2)
(1) (2)
mab = ?a kb ? ?b ka , Sab = sab + sab , [xa , pb ]? = i?ab ,
[?a , kb ]? = i?ab , [?a , pb ]? = 0,
(1) (2)
where sab and sab are the spin matrices satisfying the Lie algebra of the rotation
group O3 .
Equations (1) is invariant with respect to algebra (2) since the condition
?
? (Pa + M 2 )1/2 , Jµ? ? = 0,
2
(3)
i µ = 0, 1, 2, 3,
?t
Lettere al Nuovo Cimento, 1975, 14, 12, P. 435–438.
Poincar?-invariant equations with a rising mass spectrum
e 283

is satisfied. In spherical co-ordinates the operator k2 is

1? ? 12
? ? ? 2 = ?1 + ?2 + ?3 ,
k2 = ?2 2 2 2
(4)
+ m,
? 2 ab
? 2 ?? ??

where mab is the square of the angular momentum with respect to the centre of mass.
Let us impose on the function ?(t, x, ?, ?, ?) the condition

??(t, x, ?, ?)
(5)
= 0.
??
This condition means that the wave function ? constant on the sphere of radius
r0 = ? ? ? 2 with respect to internal variables ?1 , ?2 , ?3 . If we take into account the
condition (5), eq. (1) now becomes
1/2
??(t, x, ?, ?) 4
p2 + 4m + 2 m2
2
(6)
i = ?(t, x, ?, ?).
a
r0 ab
?t

Equation (6) may yield the mass spectrum only for the bosons so that mab should
be replaced by Lab . Having done this, we obtain the equation
1/2
??(t, x, ?, ?) 4
p2 + 4m + 2 L2
2
(7)
i = ?(t, x, ?, ?).
a
r0 ab
?t

Equation (7) shows that the mass operator M 2 = P0 ?Pa has on the set {?(t, x, ?, ?)}
2 2

the discrete mass spectrum of the form

42 4
M 2? = 4m2 + 4m2 + (8)
2L ?= 2 s(s + 1) ?,
r0 ab r0

where

Lab = mab = ?a kb ? ?b ka ,
if (9)
s = 0, 1, 2, . . .

135
Lab = ?a kb ? ?b ka + Sab ,
if (10)
s= , , ,...
222

Sab = ?c /2, ?c are the 2 ? 2 Pauli matrices.
In the case (9) the operator M 2 has a simple spectrum. In the case (10) the
spectrum of M 2 is twofold degenerated. In the general case the measure of the
degeneracy depends on the dimension of the matrices Sab realizing representations of
the group O3 .
If we suppose that the energy operator P0 can have both the positive and negative
spectrum, then for fermions (the spectum (10)) we find the equation
1/2
4
p2 + 4m + 2 L2
2
p0 ?(t, x, ?, ?) = ?0 ?(t, x, ?, ?),
a
r0 ab
(11)
? 10
p0 = i , ?0 = ,
0 ?1
?t
284 W.I. Fushchych

where ? is the four-component wave function. The integro-differential equation (11)
may be written in the symmetrical form with respect to the operators p0 , pa if the
transformation [4] is carried out on it

?0 H
1 1/2
U=v 1+ v H = ?0 ?c pc + ?0 ?4 a2 + b2 L2
, , c, d = 1, 2, 3,(12)
cd
H2
2
where ?0 , ?c , ?4 are the 4 ? 4 Dirac matrices, a2 = 4m2 , b2 = 4/r0 . After the
2

transformation (12), eq. (11) takes the form
1/2
p0 ?(t, x, ?, ?) = ?0 ?c pc + ?0 ?4 a2 + b2 L2 ?(t, x, ?, ?),
cd
(13)
? = U?.

We now summarize that eq. (7) describes a boson system with increasiftg mass
spectrum if the operator Lab has the form (9). Equation (13) (or eq. (7)) describes a
fermion system with increasing mass spectrum if the operator Lab has the form (10).
The four-component eq. (13) (or (7)) may be used for describing the free motion
of a particle of nonzero mass with arbitrary half-integer spin s. Indeed, to do this it is
sufficient to impose the Poincar?-invariant condition on the wave function ?, picking
e
up a fixed spin from the whole discrete spectrum (10).
This condition has the form
1
Wµ W µ ?(t, x, ?, ?) = L2 ?(t, x, ?, ?) = s(s + 1)?, (14)
ab
2
M
where
1
?µ??? P ? J ?? , (15)
Wµ =
2
s is an arbitrary but fixed number from the set (10).
Equations (7), (13) may be obtained in another way. Let us consider the equation

??(t, x1 , x2 , . . . , x6 ) 1/2
= p2 + p2 + · · · + p2 + ? 2 (16)
i ?(t, x1 , x2 , . . . , x6 ),
1 2 6
?t
where pk = ?i(?/?xk ), k = 1, 2, . . . , 6, ? is a constant. The equation is invariant
under the generalized Poincar? group P1,6 [5].
e
P1,6 is the group of rotations and translations in (1 + 6)-dimensional Minkowski
space. Equation (16) is invariant with respect to the algebra [5]
? ?
Pk = pk = ?i
P 0 = p0 = i , , k = 1, 2, . . . , 6,
?t ?xk (17)
Jµ? = xµ p? ? x? pµ + Sµ? , µ, ? = 0, 1, 2, . . . , 6.

Equation (16), together with the supplementary condition of the type (5), is equi-
valent to eq. (7). This may be shown by passing from the variables x4 , x5 , x6 to
the new variables ?, ?, ?. It is to be emphasized, however, that the supplementary
condition of the type (5) breaks down the invariance with respect to the whole group
P1,6 but conserves the invariance relative to its subgroup P1,3 ? P1,6 .
Poincar?-invariant equations with a rising mass spectrum
e 285

Note 1. On the set {?} besides the representations of the Poincar? algebra P1,3 (the
e
external algebra), we may construct one more algebra of Poincar? K1,3 (the internal
e
algebra). The representation of the algebra K1,3 has the following form:
1 ?
Ka = ka = ?i
K0 = M, , Lab = mab + Sab ,
2 ??a
(18)
1 Sab kb
mab = ?a kb ? ?b ka , = ? (?a K0 + K0 ?a ) ?
L0a .
2 K0 + m
This algebra describes an intrinsic relative motion of the two-particle system with
respect to the centre of mass. The algebra P1,3 describes a motion of the centre of
mass. Equations (7), (13) are not invariant in respect to the whole algebra K1,3 .
Note 2. We note that the results obtained do not contradict the O’Raifeartaigh’s
theorem [6] since the operators (2) of the algebra P1,3 together with the operators
(18) of the algebra K1,3 form the infinite-dimensional Lie algebra.
Note 3. Equation (13) jointly with tin condition (14) for the case s = 1 is equivalent
2
to the ordinary four-component Dirac equation for the particle with the spin s = 1 .
2


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