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e
may be written down as

[(m + ?µ pµ )[Sµ? S µ? ? 2s2 ](1 ? i?4 ) ? 4ms]? = 0. (5.20)

The system of equations (1) with the Hamiltonian (5.19) and (5.20) describes the free
motion of a particle with the fixed spin s. When the interaction is included into these
equations by the standard substitution pµ > ?µ = pµ ? eAµ , no difficulties typical
On the Poincar?-invariant equations for particles with variable spin and mass
e 297

for the equations of Bargman–Wigner, Rarita–Schwinger and other equations [18, 19]
arise. This question will be discussed in the next paper.
6. The equation for a charged particle in an external electromagnetic field
The generalization of equation (2.1) to the case of a charged particle in an external
electromagnetic field proves to be a difficult problem owing to the complicated depen-
dence of the Hamiltonians Hj? on momenta. In this section the problem is solved, the
assumption being that the particle momenta are small compared with the particles
masses which are considered to be fixed. With the help of successive unitary trans-
formations we found the equation for the positive energy states of the particle with
arbitrary spin and fixed mass just as it had been done by Foldy and Wouthuysen [3]
for s = 1 .
2
For p m the Hamiltonians Hj? have been represented as series in powers of
1/m (Compton wavelength). Restricting ourselves to the constituents of power 1/m2
and using the relation
l
S4a pa
(j2 ? ?3 ) ?j3 ??3 = S4a = ja ? ?a ,
l
(6.1)
, l = 0, 1, . . . ,
p
a
j3 ?3

we write the operators (3.20b), (4.7) in the form
? ?
1
Hj? = ?1 ?m + dab (pa pb ? pb pa )? +
?
m
a,b
(6.2)
1? 1
+?3 ba p a + h (p) + o ,
m2 m3
a

where
1
?ab ? S4a S4b ,
? = I, II; dab = ba = 2S4a ,
4
2
hI (p) = ?2hII (p) = S4a dbc pa pb pc ; a, b, c = 1, 2, 3.
3
It is seen from (6.2) that the Hamiltonians (3.20b), (4,7) coincide in the approxi-
mation to terms of power 1/m and are polinomials in pa . Equation (2.1) with the
Hamiltonian (6.2) describes the free motion of a particle without any spin. In order
to describe the motion of a charged particle in an external electromagnetic field we
make in (2.1), (6.2) the usual replacement pµ > ?µ = pµ ? cAµ . The result is
?
?
(6.3)
Hj? (?)?(t, x) = i ?(t, x);
?t

S·H
?2 (S4a ?a )2
?2 ?e
?
Hj? (?) = ?1 m + +
2m m m
a

(6.4)
1? 1
+eA0 + ?3 2 S4a ?a + h (?) + o ;
m2 m3
a

H = curl A, S = j + ?.
298 W.I. Fushchych, A.G. Nikitin

One can verify directly that the Hamiltonian (6.4) has positive energy eigenvalues
as well as negative ones. We obtain from (6,3) the equation for the positive energy
states. It is achieved by the unitary transformation
?U †
Hj? (?) > Hj? (?) = U Hj? (?)U † ? i
? > ? = U ?, I ?
(6.5)
U,
?t
where
U ? = exp(iS3 ) exp(iS2 ) exp(iS1 );
?

S4a ?a S4a Ea
S1 = ??2 , S2 = ? 3 e ;
2m2
m
a a

(6.6)
?2 4
S3 = ? (S4a ?a )3 ?
?
h? (?) + [? 2 , S4a ?a ]+ +
2m3 3 a a

e? ?Aa ?A0
[S · H, S4a ?a ]+ Ea = ? ?
+ S4a Ea + e , .
2 ?t ?t ?xa
a a

From (6.5), (6.6) one obtains

S·H
?2 e
?e (6.7)
Hj? (?) = ?1 m+ + eA0 + [S4a Ea , S4b ?b ]? .
2m2
2m m
a,b

The operator (6.7) commutes with ?1 . On the set of functions ?+ which satisfy to
the condition

?1 ?+ = ?+ (6.8)

the Hamiltonian (6.7) is positive definite and equals

S·H
?2
?e
+
Hj? (?)? = m+ + eA0 +
2m m
(6.9)
e ?
[S4a Ea , S4b ?b ]? ?+ = i ?+ .
+2
2m ?t
a,b

Formula (6.9) should be considered as a generalization of the Pauli equation for
a particle of spin 1 to the case of a particle with an arbitrary (in general, variable)
2
spin.
To inquire into the physical sense of the constituents which are included in Hj? (?),
we consider in detail the special case of the equation (6.9) when j = 0, S4a = Sa = ?a .
According to (1.1) it corresponds to the particle with a fixed spin. Using the identity

(S4a )2 div E
e
e i ?Ea a
[S4a Ea , S4b ?b ]? ? ? ? ?
Qab
j?
2m2 12m2 6m2
?xb
(6.10)
a,b
e
? S · (E ? p ? p ? E); Qab = 3[S4a , S4b ]+ ? 2 ?ab (S4c )2 ,
j?
4m2 c
On the Poincar?-invariant equations for particles with variable spin and mass
e 299

we write the equation (6.9) for j = 0 as

S·H
?2 s(s + 1)
+ eA0 ? e ?e
H0? (?)?+ = div E?
m+
6m2
2m m
(6.11)
i ?Ea e ?
? S · (E ? p ? p ? E) ?+ = i ?+ ,
Qab +
0?
12m2 4m2
?xb ?t
Qab = 3[Sa , Sb ]+ ? 2?ab S(S + 1).
0?

Equation (6.11) describes in the quasi-relativistic approximation the movement of a
charged particle with an arbitrary fixed spin in an external electromagnetic field.
The Hamiltonian H0? (?) ? includes the constituents which corresponds to the dipole
?
e 1 ?Ea ? e
S · H , quadrupole ??
? ? S · (p ? E?
Qab , spin-orbital
0?
12m2 4m2
m ?xb
a,b

1
E ? p) ?
and Darwin s(s + 1) div E interactions. By substituting (6.10) into
6m2
(6.9) one verifies that similar constituents are included in the Hamiltonian Hj? (?) of
a particle with variable spin.
Thus using the equations for a free particle with an arbitrary spin obtained in
Sections 1–4 we found the quasi-relativistic equations (6.9), (6.11) for a charged parti-
cle in an external electromagnetic field. We established that in the approximation of
1/m2 both Hamiltonians (3.20b) and (4.7) are formally equivalent to (6.7). However,
II
the operator Hj? is determined in the Hilbert space where the scalar product has
I
the complicated structure (2.11), (5.17), and that’s why the Hamiltonian Hj? is more
convenient for the description of the motion of a charged particle in an external
electromagnetic field.
In the case s = ? = 1 (6.11) coincides with the equation obtained by Foldy and
2
Wouthuysen [3]. For s = ? = 1 (6.11) has the structure analogous to the equation
obtained in [2], but in addition it takes into account the quadrupole interaction of the
particle with a field.
Appendix
Here we present, without proof, some relations used in the paper. In [5] it is
shown that for the projectors of the type (3.9) the following formulas hold:

i 1 p
p ? j(?j3 ?1 ? ?j3 +1 ) ? j ? jp (?j3 ?1 + ?j3 +1 ? 2?j3 );
[j, ?j3 ] =
2p 2 p
(A.1)
1 i p
[x, ?j3 ] = 2 p ? j(?j3 ?1 + ?j3 +1 ? 2?j3 ) ? j ? jp (?j3 ?1 ? ?j3 +1 ).
2p 2p p

The corresponding relations for ??3 can be obtained from (A.1) by making replace-
ment j > ? , j3 > ?3 , jp > ?p . These relations are used in solving equations (3.14)
and (4.4). Besides, the following fact is taken into account: the linear combination

p?j p?? p
(A.2)
L= b1 + b2 + b3 j + b 4 ? + bs ?j3 ??3
p p p
300 W.I. Fushchych, A.G. Nikitin

equals zero if and only if either j3 = ±j3 , or ?3 = ±? , and then

p?j p
?i + j ? j3 j3 = ±j,
?j3 ??3 = 0,
p p
(A.3)
p?? p
?i + ? ? ?3 ?j3 ??3 = 0, ?3 = ±?
p p
is either fulfilled, or all numbers bn equal zero.

1. Blokincev D.E., Zh. Exp. Teor. Fiz., 1947, 17, 273.
2. Carrido L.M., Oliver L., Nuovo Cim. A, 1967, 52, 588.
3. Foldy L.L., Wouthuysen S.A., Phys. Rev., 1950, 78, 29.
4. Fushchych W.I., Teor. Mat. Fiz., 1970, 4, 360.
5. Fushchych W.I., Grishchenko A.L., Nikitin A.G., Teor. Mat. Fiz., 1971, 8, 192.
6. Fushchych W.I., Krivsky I.Yu, Nucl. Phys. B, 1968, 7, 79.
7. Fushchych W.I., Krivsky I.Yu, Nucl. Phys. B, 1969, 14, 754.
8. Gelfand I.M., Yaglom A.M., Zh. Exp. Teor. Fiz., 1948, 18, 703.
9. Ginsburg V.L., Tamm I.E., Zh. Exp. Teor. Fiz., 1947, 17, 273.
10. Gyuk J., Uniezava H., Phys. Rev. D, 1971, 3, 898.
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