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d3 x,
Aa (t, x) p 0 Ab (t, x) + Ab (t, x) p 0 (24б)
?ab = Aa (t, x)
2 p p

Формулы (24а) определяют спин векторного поля [15]. Мы видим, что из допол-
нительной симметрии уравнений для вектор-потенциала, описываемой теоремой 2,
следует сохранение во времени еще шести интегральных комбинаций (24б). Особо
простой вид интегралы (24) принимают в импульсном пространстве. Полагая

d4 k ? k0 ? k 2 exp(?ikx)Aµ (k),
2
Aµ (x) =

получаем

d3 k A+ (k)A? (k),
Sa = i?abc c
b
(25)
A+ (k)A? (k) A+ (k)A? (k)
?
3
?ab = 2 dk ,
a a
b b

где
?(k0 )
A± (k) = v Ac (±k).
c
2k0
В заключение обсудим физическую интерпретацию интегралов движения
(23), (24). Можно показать, что если электромагнитное поле представляет собой
плоскую волну, то формулы (23а)–(23в) задают параметры Стокса, описывающие
поляризацию этой волны. В общем случае интегралы (23а)–(23в) можно рассма-
тривать как некое обобщение этих параметров на случаи произвольных решений
уравнений Максвелла. Что же касается соотношений (24), то в случае монохрома-
тической волны они могут быть сведены к матричным элементам поляризационной
матрицы плотности для поля со спином 1.
Таким образом, нелиевская симметрия уравнений движения может использо-
ваться для описания поляризационных свойств электромагнитного поля. То же
самое можно сказать и о релятивистских уравнениях для частиц с отличной от
нуля массой и произвольным спином, например, нелиевская симметрия уравнения
Дирака [12] может быть использована при описании поляризации электрона [18].
О новых симметриях и законах сохранения для электромагнитного поля 365

1. Plybon D., Am. J. Phys., 1974, 42, 998.
2. Heaviside O., Phil. Trans. Roy. Soc. A, 1893, 183, 423.
3. Larmor I., Collected papers, London, 1928.
4. Rainich G.I., Trans. Am. Math. Soc., 1925, 27, 106.
5. Beteman H., Proc. London Math. Soc., 1909, 8, 223.
6. Cunningham E., Proc. London Math. Soc., 1909, 8, 77.
7. Ибрагимов Н.Х., ДАН СССР, 1968, 178, 566.
8. Fushchych W.I., Ргерrint IТР-70-32E, Kiev, 1970.
9. Фущич В.И., ТМФ, 1971, 7, 3.
10. Fushchych W.I., Lett. Nuovo Cimento, 1974, 11, 506.
11. Никитин А.Г., Сегеда Ю.Н., Фущич В.И., ТМФ, 1976, 29, 82.
12. Fushchych W.I., Nikitin A.G., J. Phys. A: Math. Gen., 1979, 12, 747.
13. Fushchych W.I., Nikitin A.G., Lett. Nuovo Cim., 1979, 24, 220.
14. Фущич В.И., Наконечный В.В., УМЖ, 1980, 32, 267.
15. Fushchych W.I., Nikitin A.G., Czech. J. Phys. B, 1982, 32, 470.
16. Фущич В.И., Владимиров В.А., ДАН СССР, 1981, 257, 1105.
17. Боголюбов Н.Н., Ширков Д.В., Введение в теорию квантованных полей, М., Наука, 1973.
18. Стражев В. И., Весц. АН БССР, 1981, 5, 75.
W.I. Fushchych, Scientific Works 2000, Vol. 2, 366–378.

On one- and two-particle Galilei-invariant
wave equations for any spin
W.I. FUSHCHYCH, A.G. NIKITIN
The problem of the motion of any spin charged particle in Coulomb field is solved by
using the Galilei-invariant wave equations, which have been obtained by the authors
recently. Galilei-invariant motion equations for a system of two interacting particles of
any spin are deduced.

Решается проблема движения заряженной частицы с произвольным спином в Куло-
новском поле, используя волновые уравнения, инвариантные относительно преобра-
зований Галилея, которые были получены авторами ранее. Выводятся уравнения
движения, инвариантные относительно преобразований Галилея, для системы двух
взаимодействующих частиц с произвольным спином.

1. Introduction
The description of motion of the a charged spinning particle in a central field is one
of the important problems of quantum mechanics. But the formulation of this problem
for particles with spin s > 1 is confronted with principle difficulties because of such
2
fundamental relativistic equations as Kemmer–Duffin, Proca ones and others lead to
contradictions when one tries to depict the interaction of the spin-one particle with the
Coulomb field. Among them are the particle fall on centrum, the absence of stable
solutions, nonrenormalizability and many others [1, 2]. The well-known paradoxes
which arise by relativistic description of the interaction of highest-spin particles with
an electromagnetic field are connected with the breakdown of causality (see, e.g.,
[3]).
In the present paper the problem of a the motion of charged particle with any spin
in Coulomb field is solved by using Galilei-invariant wave equations (GIWE). The
interest for such equations has been awaked by the paper of Levi-Leblond [4], who
has obtained the GJWE for a particle of spin 1 . The Levi-Leblond equation аs well
2
as Dirac one gives the correct description of Pauli interaction of particle spin with a
magnetic field. Unfortunately neither Levi-Leblond equation nor its generalization for
any spin, obtained by Hagen and Hurley [5, 6], take into account such an important
physical effect as spin-orbit coupling.
In papers [7–11] the GIWE for any spin particles are found which describe the
spin-orbit interaction. These equations do not have pretensions to give a complete
description of charged-particle interaction with an electromagnetic field, but they
design adequately the physical situation in the cases in which the particle energy
is too small to be enough for the pair creation, — i.e. when the one-particle Dirac
equation is applicable. In spite of the absence of relativistic invariance, the equations
found in [7–11] describe correctly the spin-orbit, Darwin and quadrupole couplings
of any spin particle with an external field. It means specifically that the mentioned
couplings are not to be interpreted as a relativistic corrections without fail, but may
be described consistently in the frame of the Galilean-invariant approach.
Nuovo Cimento A, 1984, 81, № 3, P. 644–660.
On one- and two-particle Galilei-invariant wave equations for any spin 367

In this paper the explicit solutions of GIWE [9, 10] are found for the case of
interaction of any spin particle with Coulomb field. The analog of Sommerfeld formula
for any spin is obtained. It is demonstrated that by the solution of GIWE the diffi-
culties do not arise, which characterize the relativistic equations, but at the same
time the fine structure of Galilean particle spectrum contains the contribution from
spin-orbit coupling.
Besides the problem of the description of particle interaction with an external field
the two-body quantum-mechanical problem is of great interest for physics. In the last
years such an interest is additionally stimulated by the successes in meson masses
description in the frame of quark models.
In present paper, starting from one-particle equations [7, 9, 10] two-body GIWE
are derived for particles of any spin. For the case in which the particle spins are equal
to 1 the equation is obtained, which leads to the same fine and hyperfine spectrum
2
structure as the Breit one [12] and is explicitly invariant under the Galilei group,
whereas the Breit equation is invariant neitlier under Galilean group nor under the
Poincar? one.
e

2. GIWE of first order
Here we consider the systems of partial differential equations of a from
L? ? (?µ pµ + ?5 m) ? = 0, (2.1)
µ = 0, 1, 2, 3,
where p0 = i(?/?t), pa = i(?/?xa ), ?µ and ?5 are (n ? n)-dimensional square matri-
ces, ? is n-component function, m is c-number.
A great deal of papers are devoted to the description of relativistic equations of
type (2.1), but Galilean-invariant equations of first order remain almost nonstudied
ones.
A wide class of GIWE of type (2.1) is obtained in papers [9, 10], the main result
of which are used here.
Equation (2.1) is invariant under Galilei transformations
x > x = Rx + vt + a, t>t =t+b (2.2)
if a set of (n ? n)-dimensional matrices Sa and ?a (a = 1, 2, 3) exists, which satisfies
the relations [9, 10]
(2.3)
[Sa , Sb ] = i?abc Sc , [Sa , ?b ] = i?abc ?c , [?a , ?b ] = 0;
?† ?0 ? ?0 ?a = 0, ?† ?5 ? ?5 ?a = i?a ,
a a
(2.4)

?a ?b ? ?b ?a = ?i?ab ?0 , [Sa , ?5 ] = [Sa , ?0 ] = 0.
If eq.(2.1) admits the Lagrangian formulation, eqs.(2.3), (2.4) give necessary and
sufficient conditions of its Galilean invariance [9]. The sufficiency of these conditions
is rather obvious, as soon as the following relations may be obtained from (2.3), (2.4):
[L, Ga ] = ?† ? ?a L, (2.5)
[L, Pµ ] = [L, Ja ] = 0, a

where Pµ , Ja , Ga are the Galilei group generators
? ?
Pa = pa = ?i
P0 = i , ,
?t ?xa (2.6)
Ga = tpa ? mxa + ?a .
Ja = ?abc xb pc + Sa ,
368 W.I. Fushchych, A.G. Nikitin

One concludes from (2.5) that the Lie algebra of Galilei group is realized on the set
of eq.(2.1) solutions.
So, to describe all GIWE in the form (2.1), it is necessary to solve the system of
matrix relations (2.3), (2.4). The simplest (i.e. realized by the matrices of minimal
dimensions) solutions of these relations are [10]
ann ? I bnn ? I
0 0
?0 = , ?5 = 2 ,
0n?1 n?1 ? ? cn?1 n?1 ? ?
0 1 0 1

?
dnn ? Sa en n?1 ? Ka
i
?a = ,
(en n?1 )† ? Ka 0n?1 n?1 ? ?
1
s
(2.7)
?
Inn ? Sa 0
Sa = ,
?
In?1 n?1 ? Sa
0

?
fn ? Sa gn n?1 ? Ka
1
?a = ,
hn?1 n ? Ka 0n?1 n?1 ? ?
1
2s
? ?
where Sa and Sa are the matrices, which realize irreducible representations D(s) and
D(s?1) of O3 algebra, Ka are the (2s?1)?(2s+1)-dimensional matrices, determined
by the relations

? ?
Ka Sb ? Sb Ka = i?abc Kc , Sa Sb + Ka Kb = is?abc Sc + s2 ?ab , (2.8)

I and ? are unit matrices of dimension (2s + 1) ? (2s + 1) and (2s ? 1) ? (2s ? 1). The
1
symbols Anl signify (n ? l)-dimensional matrices (n = 2, 3), whose nonzero matrix
elements are
(a22 )11 = (b22 )22 = c2 c11 = (d22 )12 = ?(d22 )21 = c(e21 )11 =
(2.9)
= (f22 )21 = c?1 (h12 )11 = (I22 )jj = 1, j = 1, 2,

(a33 )12 = (a33 )21 = (b33 )23 = a(b33 )13 = a(b33 )22 = a(b33 )31 =
= 2a2 (b33 )12 = 2a2 (b33 )21 = ?a(c22 )11 = ?(d33 )31 = (d33 )13 =
= (f33 )21 = (f33 )32 = c?1 (h23 )11 = c?1 (h23 )22 = ?c?1 (g32 )31 = 1, (2.10)
(c22 )12 = (c22 )21 = c?1 (e32 )12 = s ? 1, (e32 )21 = cs,
(I33 )jj = 1, j = 1, 2, 3,

where c = 2s ? 1, a is an arbitrary parameter.
The formulae (2.1), (2,7), (2.9) determine the GIWE, which are equivalent to
Hagen–Hurley ones [5, 6]. These equations may be interpreted as Galilean-invariant
motion equations of a free particle with spin s [6].
Equations (2.1) (2.7), (2.10) also describe a free Galilean particle of spin s and mass
m, but in contrast to (2.1), (2.7), (2.9) these equations after minimal substitution

pµ > ?µ = pµ ? eAµ (2.11)

describe the spin-orbit coupling of a particle with a field. They are just the equations
to be used to solve the problem of any spin particle motion in Coulomb field.
On one- and two-particle Galilei-invariant wave equations for any spin 369

3. Equations in the Schr?dinqer form
o
Consider together with (2.1) GIWE for spin-s particle in the Schr?dinger form
o
?
(3.1)
i ? = Hs (p)?,
?t
where Hs (p) is second-order differential operator, ? is 2(2s + 1)-component wave
function. Such equations will be used as a basis for a construction of two-particle
GIWE.
Equation (3.1) is invariant under Galilei transformations (2.2), if the Hamiltonian
Hs , satisfy the following commutation relations:
(3.2)
[Hs , Pa ] = [Hs , Ja ] = 0, [Hs , Ga ] = iPa ,
where Pa , Ga , Ja are the Galilei group generators (2.6). Without loss of generality
the matrices Sa and ?a from (2.6) may be taken in the form
? 0 0
Sa 0
(3.3)
Sa = , ?a = ,
?
?a Sa 0
0 S
?
where Sa are the generators of irreducible representation D(s) of O3 group, 0 is the
(2s + 1)-row zero matrix.
The generators (2.6), (3.3) form (together with Hs ) the Lie algebra of extended
Galilei group, satisfying the following commutation relations:
[Pµ , P? ] = [Ga , Gb ] = 0, [Pa , Gb ] = i?ab m,
(3.4)

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