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? (m + E)[t3 p3 + p2 )?
t3 (p2 + p2 + p2 ) t4 (p2 + p2
? ip2 ) ? t4 p3 ]
+ t4 (p1 + ip2 ) (m + E)[t3 (p1
? 3?
W =? (m ? E)[t p .(3)
1 2 3 1 2
+ p2 )?
t5 (p2 + p2 + p2 ) t6 (p2 + p2
(m ? E)[t5 (p1 ? ip2 ) ? t6 p3 ]
+ t6 (p1 + ip2 )
53 1 2 3 1 2 3
t7 (p2 + p2 + p2 ) t8 (p2 + p2 + p2 )
(m ? E)[t7 p3 (m ? E)[t7 (p1 ? ip2 ) ? t8 p3 ]
+ t8 (p1 + ip2 ) 1 2 3 1 2 3


Здесь E = p2 + p2 + p2 + m.
1 2 3
Выбирая для матрицы (3) набор параметров аналогично предыдущему случаю
p1 ? ip2
p3 p1 + ip2
t1 = , t2 = , t3 = ,
p0 + E p0 + E p0 + E
p3
t4 = ? , t5 = t8 = 1, t6 = t 7 = 0
p0 + E
получим матрицу
? ?
p1 ? ip2
p3
1 0
? ?
m+E m+E
? ?
? ?
? ?
p1 + ip2 p3
?
? ?
0 1
? ?
m+E m+E
W0 = ? ?.
? ?
p1 ? ip2
p3
?? ?
? 1 0
? ?
m+E m+E
? ?
? p1 + ip2 ?
p3
? 0 1
m+E m+E
Матрица, обратная к W0 , имеет вид

p2 + p2 + p2 ?i pi
?1
1? 1?
1 2 3
W0 = .
2 + p2 + p2 + p2
(m + E) m+E
1 2 3

Используя формулу (2) получим общий вид обратной матрицы к матрице преобра-
зования (3).
508 В.И. Фущич, В.В. Корняк

Рассмотрим теперь уравнение Кеммера–Деффина–Петье
(?µ pµ ? m)? = 0, µ = 0, 3.
Матрицы ?µ удовлетворяют алгебре
?µ ?? ?? + ?? ?? ?µ = gµ? ?? + g?? ?µ ,
gµ? = diag (1, ?1, ?1, ?1) — метрический тензор.
Матрицы ?µ можно реализовать в виде
? ? ? ?
? ? ?
? ? ?? 0 0 0 0 ?a
00 1
?? ? ? 0? ?? 0?
?
?Sa
?
00 0 ? , ?a = ? 0 0
?0 = i ? ? ? ? ? , a = 1, 3.
?1 0 0 0? ?? 0?
? ?
0 Sa 0
??+
00 0 0 0 0 0
a

Здесь
? ? ? ? ? ?
0 ?i
00 0 00 i 0
S1 = ? 0 0 ?i ? , S2 = ? 0 0 0 ?, S3 = ? i 0 0 ?,
? ? ?
?i 0
0i0 0 00 0
?? ? ? ? ?
i 0 0
?1 = ? 0 ? , ?2 = ? i ? , ?1 = ? 0 ?,
0 0 i
? и ? — нулевые и единичные матрицы размера 3 ? 3, 0 — 3-х компонентные
0 1
нулевые столбцы или строки.
Применение программы NLP к символу оператора КДП дает следующие корни
и алгебраические кратности:
?1 = ?m — кратность =4,
?2 = ?m + P — кратность =3,
?3 = ?m ? P — кратность =3.
С помощью программы NLJW символ оператора КДП приводится к диагональ-
ному виду, т.е. алгебраические кратности совпадают с геометрическими. При этом
вычисляется 34-параметрическая матрица преобразования W , которую мы для эко-
номии места не приводим. Таким образом уравнение Кеммера–Деффина–Петье
инвариантно относительно 34-параметрической группы GL(4) ? GL(3) ? GL(3).

1. Фущич В.И., О дополнительной инвариантности релятивистских уравнений движения, Теор. и
мат. физика, 1971, 7, № 1, 3–12.
2. Фущич В.И., О новом методе исследования групповых свойств уравнений математической фи-
зики, ДАН СССР, 1979, 246, № 4, 846–850.
3. Фущич В.И., О новом методе исследования групповых свойств систем дифференциальных урав-
нений в частных производных, В кн.: Теоретико-групповые методы в математической физике,
Киев, Ин-т математики АН УССР, 1978, 5–44.
4. Фущич В.И., Никитин А.Г., Симметрия уравнений Максвелла, Киев, Наукова думка, 1983,
200 c.
5. Tobey R., Baker J., Crews R. et al., PL/I–FORMAC symbolic mathematics interpreter, IBM, Proj.
No 360-03.3.004, Contributing Program Library, IBM, 1969.
6. Воеводин В.В., Кузнецов Ю.А., Матрицы и вычисления, М., Наука, 1984, 320 с.
7. Демидович Б.П., Марон И.А., Основы вычислительной математики, М., Наука, 1970, 404 c.
В.I. Фущич, Науковi працi 1999, т.2, 509–516.

Relativistic particle of arbitrary spin
in the Coulomb and magnetic-monopole field
W.I. FUSHCHYCH, A.G. NIKITIN, W.M. SUSLOPAROW
Exact solutions of relativistic wave equations for any spin charged particle in the
Coulomb and magnetic-monopole fields are found.

1. Introduction
The description of interaction of charged spinning particle with external field is
important problem of quantum mechanics. The interest in such problems is stimulated
by the research of quark models with effective potential (see, e.q., [1]).
The experimental discovery of the relatively stable resonances with spins s > 1 2
and searches of the exotic atoms, in which these resonances play the role of an orbital
particles [2, 3] lead to the necessity of description of high-spin particle motion in
an external field. At the same time relativistic wave equations for such particles lead
to contradictions of principle — such as the absence of stable solutions in Coulomb
field [4], the causality violation [5], etc. (see, e.q., [6]).
In papers [7, 8] Poincar?-invariant wave equations for particles of arbitrary spin
e
are proposed which allow us to avoid many of these difficulties. By using these
equations the solutions of many problems connected with any spin particle motion
in an external field have been found for homogeneous magnetic field, Coulomb one
and also for Redmond field — i.e. the combination of plane wave and homogeneous
magnetic, field [9, 10]. In [11] the alternative possibility, of describing the spinning
particle in the Coulomb field is considered, one that makes use of Galilei-invariant
wave equations.
In present paper the problem of interaction of any spin relativistic particle with
magnetic-monopole field is solved, using the equations proposed in [7, 8]. Such a
problem for spinless particle was first considered by Dirac [12] and Tamm [13].
Harish-Chandra [14] obtained the exact solution of Dirac equation for electron in-
teracting with magnetic-monopole field. A number of publications, devoted to the
description of the motion of a charge in monopole field has appeared last time (see,
e.g., [15–18]), but the case of a particle of any spin was not yet considered.
Besides we obtain the exact solutions of Poincar?-invariant equations for particles
e
with arbitrary spin, interacting with the combination of the Coulomb and magnetic-
pole fields.

2. Poincar?-invariant equations for particles of arbitrary spin
e
We will start from the following equations, describing the relativistic particle of
spin s in an external electromagnetic field [7, 8]:
e 1
?µ ? µ ? m + (1 ? i?4 ) Sµ? ? i?µ ?? F µ? ? = 0,
4m s (2.1)
(?µ ? µ + m)(1 ? i?4 )[Sµ? S µ? ? 2s(s ? 1)]? = 16ms?,
Nuovo Cimento A, 1985, 87, № 4, P. 415–424.
510 W.I. Fushchych, A.G. Nikitin, W.M. Susloparow

where ? = ?(x) is the 8s-component wave function, x = (x0 , x1 , x2 , x3 ), ?µ =
?i(?/?xµ ) ? eAµ , Aµ is the vector potential, F µ? is the electromagnetic-field tensor,
?µ are (8s ? 8s)-dimensional matrices satisfying the Clifford algebra
(2.2)
?µ ?? + ?? ?µ = 2gµ? ,
?4 = ?0 ?1 ?2 ?3 , Sµ? are the generators of the representation D 1 , 0 ? D 0, 1 ?
2 2
D s ? 1 , 0 of Lorentz group.
2
In the case s = 1 the system (2.1) is reduced to Dirac equation for electron. If
2
s is arbitrary integer or half-integer, this system describes the causal motion of the
charged particle of spin s in an external electromagnetic field [7, 8].
To solve the above-mentioned problems it is convenient to pass from eqs.(2.1) to
the system of second-order equations. Multiplying eqs.(2.1) from the left by 1 (1±i?4 )
2
and expressing ?+ = 1 (1 + i?4 )? via ?? = 1 (1 ? i?4 )? we obtain
2 2
e
?µ ? µ ? m2 ? Sµ? F µ? ?? = 0, (2.3a)
2s
[Sµ? S µ? ? 4s(s + 1)]?? = 0, (2.3b)

1
?µ ? µ ?? . (2.3c)
?+ =
m
According to (2.3), solving of eqs.(2.1) is reduced to finding of the function ?? ,
satisfying (2.3a), (2.3b) inasmuch as general solution of eqs.(2.1) may be presented
as ? ? ?+ + ?? , and ?? is expressed via ?+ in accordance with (2.3c).
It follows from (2.3b) that the function ?? has only 2s + 1 nonzero components
and is spinor from the space of D(s, 0) representation of the Lorentz group. On the
set of such functions the matrices Sµ? are reduced to (2s + 1) ? (2s + 1)-dimensional
generators of the irreducible representation D(s) of O3 group (indicated below as
S = (S1 , S2 , S3 )), and eq.(2.3a) comes to the following form:
e
?µ ? µ ? m2 ? S(H ? iE) ?s = 0, (2.4)
m
where ?s is (2s + 1)-component function (including nonzero components of ?? ), E
and H are the vectors of electric and magnetic fields, respectively. For s = 1 eq.(2.4)
2
coincides with well-known Zaiteev–Feynman–Gell-Mann equation [19, 20].

3. Arbitrary-spin particle in the magnetic-monopole field
In the spherical co-ordinates the vector potential and the corresponding vectors of
the electric and magnetic-field strength created by the magnetic monopole are [14]
n
(1 ? cos ?),
A0 = Ar = A? = 0, A? =
2c
(3.1)
nr
E = 0, H= ·,
2e r3
where n is integer, r = (r sin ? cos ?, r sin ? sin ?, r cos ?).
Writing eqs.(2.4), (3.1) in the spherical co-ordinates one obtains
n S·r
1 ? 2? 1
? + 2 ?? ? + (?2 ? m2 )? = (3.2)
r ?,
r2 ?r ?r 2s r3
r
Relativistic particle of arbitrary spin 511

where ? is the stationary-state energy,

n2 1 ? cos ?
1 ?2
1? ? in ?
?? = ?
sin ? + + .
sin2 ? ??2
sin ? ?? ?? 1 + cos ? ?? 4 1 + cos ?

Equation (3.2) may be solved by separation of variables for any value of spin s.
With the help of the unitary transformation ? > ? = V ?, where
?
V = exp[?iL3 ?] exp[i(S2 cos ? ? S1 sin ?)?], L3 = ?i (3.3)
,
??

eq.(3.2) comet to such a form, in which the matrix S · r/r3 on the r.h.s. is diagonal
and equal to S3 /r2 :
1 ? 2? 1 n1
? + 2 ? ? ? + (?2 ? m2 )? = (3.4)
r S3 ?.
r2 ?r ?r 2s r2

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