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Следствие 1. Для алгебры (10)

[xa , Jµ? ]? = i(gaµ x? ? ga? xµ ), (19)
a = 1, 2, 3; µ, ? = 0, 1, 2, 3,

(20)
[x0 , Jµ? ]? = 0.

Следствие 2. Операторы (9) и (10) порождают обычные, лоренцовские, правила
преобразования для энергии и импульса, поскольку для обеих алгебр удовлетво-
ряются коммутационные соотношения

[Pµ , J?? ]? = i(gµ? P? ? gµ? P? ).

Следствие 3. Операторы (9) и (10) порождают совершенно различные правила
преобразования для пространственных и временной координат при переходе
от одной системы отсчета к другой:

= exp iJ0b ?b xa exp ?iJ0c ?c ,
xII II II
a, b, c, = 1, 2, 3,
a
(21)
?iJ0c ?c
xII II II
= exp iJ0b ?b x0 exp ;
0


= exp iJ0b ?b xa exp ?iJ0c ?c ;
xIII III III
(22)
a


= exp iJ0b ?b x0 exp ?iJ0c ?c = xIII .
xIII III III
(23)
0 0

Формула (23) является следствием соотношения (20) и указывает на то, что
время не меняется при переходе от одной системы отсчета к другой.
Суммируя все сказанное, приходим к выводу: относительно преобразований
(22), (23) релятивистское уравнение (5) инвариантно в смысле (2), хотя время не
изменяется при переходе от одной системы отсчета к другой. При этом, конечно,
нелокальные преобразования (22) не совпадают с преобразованиями Лоренца. Для
уравнений Максвелла и Дирака этот факт был обнаружен в [2].
О дополнительной инвариантности уравнения Клейна–Гордона–Фока 309

Замечание 3. Преобразование (11), осуществляющее переход от уравнения (5) к
(12), не единственно. Существует много преобразований такого типа. Вот одно из
них [1, 2]:
H H
1 1
?1
W1 = v W1 = v 1 ? ?3
1 + ?3 , .
E E
2 2
Наконец, приведем явный вид оператора координаты в представлении ?:
ipa
Xa = W ?1 xa W = xa +
? (?0 + ?3 ).
2E 2


1. Фущич В.И., Теор. и мат. физ., 1971, 7, № 1, 3; Препринт Ин-та теор. физики АН УССР,
№ 70-32, Киев, 1970.
2. Fushchych W.I., Lettere Nuovo Cimento, 1974, 11, № 10, 508.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 310–314.

On the Galilean-invariant equations
for particles with arbitrary spin
W.I. FUSHCHYCH, A.G. NIKITIN

In our preceding paper [1] the equations of motion which are invariant under the
Galilei group G have been obtained starting from the assumption that the Hamiltonian
of a nonrelativistic particle has positive eigenvalues and negative ones. These nonre-
lativistic equations as well as the relativistic Dirac equation lead to the spin-orbit
and to the Darwin interactions by the standard replacement pµ > ?µ = pµ ? eAµ .
Previously it was generally accepted the hypothesis that the spin-orbit and the Darwin
interactions are truly relativistic effects [2].
In [1] only the equations for the particles with the lowest spins s = 1 , 1, 3 have
2 2
been obtained. What puts the equations [1] in a class by themselves is that the
transformation properties of a wave function are rather complicated (nonlocal) and
it is difficult to establish their invariance under the Galilei transformations after the
replacement pµ > ?µ .
In the present note equations for arbitrary-spin particles are obtained which pos-
sess as good physical properties as the equations [1].
Moreover the wave function has simple transformation properties in the case of
the equation describing the interaction with an external field as well as in the case of
the absence of interaction.
We shall start from the assumption that under the Galilei transformation
x > x = Rx + V t + a,
(1)
t > t = t + b,
the 2(2s + 1)-component wave function ?(t, x) transforms as
?(t, x) > ? (t , x ) = exp[if (t, x)]Ds (R, V )?(t, x), (2)
where Ds (R, V ) is some numerical matrix, depending on the parameters of the trans-
formation (1), exp[if (t, x)] is the phase factor [3]
1
f (t, x) = mV · Rx + mv 2 t. (3)
2
The generators of the group G, which corresponds to the transformation (2), have the
form
? ?
Pa = pa = ?i Jab = xa pb ? xb pa + Sab ,
P0 = i , ,
?t ?xa
(4)
sab 0
Ga = tpa ? mxa + ?a , Sab = ,
0 sab
where sab are the generators of the irreducible representation D(s) of the group O3 ,
?a are some numerical matrices, which have to be such that the operators (4) satisfy
Lettere al Nuovo Cimento, 1976, 16, № 3, P. 81–85.
On the Galilean-invariant equations for particles with arbitrary spin 311

the commutation relations of the algebra G. It can be shown that the most general
(up to equivalence) form of the matrices ?a satisfying this requirement is
1
(5)
?a = k(?3 + i?2 )Sa , Sa = ?abc Sbc ,
2
where ?2 , ?3 are the 2(2s + 1)-dimensional Pauli matrices which commute with Sab ,
k is an arbitrary constant.
To find the motion equations for arbitrary-spin particles
?
(6)
i ?(t, x) = Hs (p, s)?(t, x)
?t
it is sufficient to construct such an operator (Hamiltonian) Hs (p, s) that eq. (6) be
invariant under the Galilei group G. Equation (6) will be invariant relative to G, if
the following conditions are satisfied:
[Hs (p, s), Ga ]? = ?iPa . (7)
[Hs (p, s), Pa ]? = 0, [Hs (p, s), Jab ]? = 0,
Thus our problem has been reduced to the solution of the commutation relations (7)1 .
In order to solve relation (7) we expand Hs in a complete system of the orthopro-
jectors and Pauli matrices
?µ aµ ?r , (8)
Hs (p, s) = µ = 0, 1, 2, 3,
r
µ,r

where
s · p/p ? r
r, r = ?s, ?s + 1, . . . , s,
?r = ,
r?r
r=r

and ?0 is the 2(2s + 1)-dimensional unit matrix, aµ (p) are unknown coefficient func-
r
tions. Substituting (8) into (7), using the relations [4]
pa S · p
Sab pb i
(2?r ? ?r+1 ? ?r?1 ) + Sa ? (?r+1 ? ?r?1 ),
[?r , xa ] =
2p2 2p pp (9)
[?r , Sab ] = pa [?r , xb ] ? pb [?r , xa ],
and taking into account the completeness and the orthogonality of the orthoprojeotors,
we have found that, up to equivalence, the general form of the Hamiltonian Hs (p, s),
satisfying (7), is given by the formula
2 (S · p)
p2 2
? ?1 2i?hS · p ? (?3 + i?2 )?k (10)
Hs = m0 + ?3 ?m + ,
2m m
where ? is an arbitrary constant.
Formula (10) gives the free nonrelativistic Hamiltonian for a particle with an
arbitrary spin. Equation (6) with the Hamiltonian (10) is invariant under the group G.
For the spin 1 particle (when s = 1 , k = ?i, ? = 1) equation (6) may be written in
2 2
the compact form
p2
(?µ p + m)? = (1 + ?4 ? ?0 )
µ
(11)
?,
2m
where ?µ are the Dirac matrices.
1 The analogous problem has been eolved in the relativistic case in [4]. Lately the method of the work
[4] has been further developed in works of R.F. Guertin [5].
312 W.I. Fushchych, A.G. Nikitin

The Hamiltonian (10) and the generators (4) are non-Hermitian under the usual
scalar product. They are, however, Herinitian under

d3 x ?† M ?2 , (12)
(?1 , ?2 ) = 1


where M is the positive-definite metric operator
2
S·p S·p
? ?
M = 1 + [i(k ? k )?3 ? (k + k )?2 ] + 2|k|2 (1 + ?1 ) (13)
.
m m
Besides, if ?, k satisfy the condition ?k = (?k)? , the Hamiltonians are Hermitian also
in the indefinite metric

d3 x ?† ??2 , (14)
(?1 , ?2 ) = 1


where
if ? ? = ?, k ? = k,
?3 ,
?=
if ? ? = ??, k ? = ?k.
?2 ,

With the help of the transformation
?·p
Hs > Hs = V Hs V ?1 , (15)
V = exp i ,
m
the Hamiltonian (10) may be reduced to the diagonal form

p2
(16)
Hs = m0 + ?3 ?m + .
2m
It is interesting to note that the condition of Galilei invariance admits the pos-
sibility to introduce two masses: the rest mass, or the rest energy (?1 = m0 + ?m,
?2 = m0 ? ?m) and the kinetic mass (the coefficient of p2 ). Below we consider the
case when m0 = 0, ? = 1, i.e. the rest mass is equal to the kinetic mass.
To describe the motion of the charged particle in external electromagnetic fields
we make in (6) and (10) the replacement pµ > ?µ (symmetrizing preliminarily the
Hamiltonian in pa [1]). This leads to the equation
?
(17)
i ?(t, x) = Hs (?)?(t, x),
?t

?2 2k 2 1
+ ?1 2ik(S · p) + (?3 + i?2 ) (S · ?)2 + (S · M ) ,(18)
Hs (?) = ?3 m +
2m m 2

where Ha = i?abc [?b , ?c ]? are components of the magnitude of the magnetic field.
It is important to note that eq. (17) as before is invariant with respect to the
Galilei transformations (1) and (2), if the vector potential is transformed according
to [2]

A > A = RA, A0 > A0 = A0 + V RA. (19)
On the Galilean-invariant equations for particles with arbitrary spin 313

To prove this statement it is sufficient to use the exact form of the matrix Ds (R, V )
in (2)

Ds (R) 0
D (R, V ) = (1 + i? · V ) ·
s
(20)
,
s
0 D (R)

where Ds (R) the matrices from the representation D(s) of the group O3 .
As in the case of the Dirac equation [6] the Hamiltonian (18) cannot be diagonali-
zed exactly. We shall make the approximate diagonalization of the operator (18) up to
the terms of the power 1/m2 with the help of the operator
s s s
(21)
V (?) = exp[iB3 ] exp[iB2 ] exp[iB1 ],

where

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