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i
?t
(19)
?E ?i
0
H ? = 0, B = ?2 ? S,
2
?= , ?2 = .
H i 0
Equations (19) by Erikson–Beckers transformation [5]
? ?
10 0
H
1 ?0 1 0 ?.
U1 = v 1 + (?3 ? 1 ) v
3 3
(20)
, 1=
H2
2 00 1
transfer into
??1 (t, x)
= H1 ?1 (t, x),
c
i ?1 = U1 ?,
?t
(21)
10
H1 = (?3 ? 1 )E,
c 3
?3 = .
0 ?1
From (21) it is clear that the condition (12) (with the Hamiltonian H1 ) is satisfied
c

for Q ? {Q1 , Q2 }. Of course in (11) the 4 ? 4 matrix ?0 must de substituted by the
matrix ?3 ? 13 , and the 4 ? 4 spin matrices by B.

1. Fushchych W.I., Lett. Nuovo Cimento, 1973, 6, 133; Theor. Math. Phys., 1971, 7, 3 (in Russian).
2. Dirac P.A.M., The Principles of Quantum Mechanics, 4th ed., Oxford, 1958.
3. Foldy L.L., Phys. Rev., 1956, 102, 568.
4. Good R.H., Phys. Rev., 1957, l05, 1914.
5. Beckers J., Nuovo Cimento, 1965, 38, 1362.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 278–281.

On a motion equation for two particles
in relativistic quantum mechanics
W.I. FUSHCHYCH

Breit [1] was the first who proposed to describe the motion for two relativistic parti-
cles by means of a semi-relativistic Dirac-type equation. The wave function of this
equation has sixteen components. The possibility of covariant description of a system
of particles interacting m quantum mechanics was proved by Thomas and Bakamji-
an [2] and Foldy [3]. In quantum field theory the two-body problem is described
by means of the Bethe–Salpeter equation or the Logunov–Tavkhelidze–Kadyshevsky
equations [4].
The purpose of the present note is to propose, in the framework of relativistic
quantum mechanics, a new Poincar?-invariant equation for two particles with masses
e
1
m1 , m2 and spin s1 = s2 = 2 . It is a first-order linear differential equation for the
eight-component wave function. With the help of this equation the description of the
motion of two-particle systems is reduced to the description of one-particle systems
in the (1 + 6)-dimensional Minkowski space which can be in two spin states (s = 0
or s = 1).
At first we derive the equation for two noninteracting particles. To this end we
shall pass from the momenta of two particles p1 , p2 to the new canonical variables
P = (P1 , P2 , P3 ) = p1 + p2 , K = (K1 , K2 , K3 ).
The connection between the variables K and p1 , p2 is rather complicated (see, e.g.,
[5, 6]) and we do not equate it here. The total energy of the two-particle system in
the variables P and K has for our discussion a very convenient structure [5, 6]
1/2 1/2
E = (P 2 + M 2 )1/2 , M = m2 + K 2 + m2 + K 2 (1)
.
1 2

The square energy for the case when m1 = m2 ? 1 m takes the very simple form
2

pa ? Pa , pa+3 ? 2Ka ,
E 2 = p2 + p 2 + m2 , (2)
a = 1, 2, 3.
a a+3

The square root from this expression is the equation for two particles
??(t, x1 , x2 , . . . , x6 )
= H(?1 , p2 , . . . , p6 )?(t, x1 , x2 , . . . , x6 ), (3)
i p? ?
?t
where
H(?1 , p2 , . . . , p6 ) = ?0 ?a pa + ?0 ?a+3 pa+3 + ?0 m,
p? ? ? ?
(4)
? ?
pa = ?i pa+3 = ?i
? , ? ,
?xa ?xa+3
the 8 ? 8 matrices ?0 , ?a , ?a+3 obey a Clifford algebra, and has such a representation:
?0 = ?3 ? 1, ?a = 2i?2 ? sa , ?a+3 = 2i?1 ? ?a , (5)
Lettere al Nuovo Cimento, 1974, 10, 4, P. 163–167.
On a motion equation for two particles in relativistic quantum mechanics 279
? ? ? ?
010 0 0 0 10
?1 0 0 0 ? 1? 0 0 0 i ?
1
s1 = ? ?, s2 = ? ?
? 1 0 0 0 ?,
2 ? 0 0 0 ?i ? 2
0 ?i 0 0
00i 0
? ? ? ?
0 ?1 0 0
00 0 1
? 0 0 ?i 0 ? 1 ? ?1 0 0 0 ?
1
s3 = ? ?, ?1 = ? ?,
2? 0 i 0 0 ? 2? 0 0 0 ?i ?
10 0 0 0 0i0
? ? ? ?
0 ?1 0 0 0 ?1
0 0
1? 0 i? 1 ? 0 0 ?i 0 ?
0 0
?2 = ? ?, ?3 = ? ?,
2 ? ?1 0 0? 2? 0 i 0 0?
0
0 ?i 0 ?1 0 0
0 0

The ?a are the Pauli matrices.
The two-particle equation (3) will be defined completely in that case if we deter-
mine both the Hamiltonian and the Poincar? generators [7]. The generators of the
e
P1,3 group on {?} have such a form:
P0 = H(?1 , . . . , p6 ) = ?0 ?A pA + ?0 m,
p ? ? Pa = p a , A = 1, 2, . . . , 6,
Jab = Mab + mab + Sab , a, b = 1, 2, 3,
(6)
(2)
H (Sab + mab )pb
1
J0a = tpa ? (xa H + Hxa ) ? v v ,
H2 H2 + M
2
where
(1) (2)
Mab ? xa pb ? xb pa , mab ? xa+3 pb+3 ? xb+3 pa+3 ,
?? ?? ? ? ? ? Sab = Sab + Sab ,
i i
(1) (2)
(?a ?b ? ?b ?a ), Sab = (?a+3 ?b+3 ? ?b+3 ?a+3 ),
Sab =
4 4 (7)
[?a , pb ]? = i?ab ,
x? [?a+3 , pb+3 ]? = i?ab ,
x ?
[?a , xb ]? = [?a , xa+3 ]? = [?a+3 , xb+3 ]? = 0,
x? x? x ? [?a , pb+3 ]? = [?a+3 , pb ]? = 0.
x? x ?
It can be immediately verified that the operators (6) satisfy the Poincar? algebra. It
e
follows that eq. (3) is Poincar? invariant. If we perform the unitary transformation
e
(E + M + ?c pc )(M + m + ?c+3 pc+3 )
(8)
U=
2{M E(E + m)(M + m)}1/2
on the operators (6), then we obtain
P0 = U P0 U = ?0 E, Jab = U Jab U = Jab ,
c c c
P a = pa ,
(9)
1 mab pb + Sab pb
= tpa ? (xa P0 + Po xa ) ? ?0
c c c
J0a .
2 E+M
The transformed generators (9) have canonical form [2, 3]. The position operators Xa
and Xa+3 on a set {?} look like
(1)
Sab pb ?a p a ?c p c m + ?c+3 pc+3

?
Xa = U xa U = xa + +i ,(10)
2E 2 (E + M )
E(E + M ) 2E M
280 W.I. Fushchych

(2)
S pb+3 i?a+3

= U xa+3 U = xa+3 + a+3 b+3 ?
Xa+3 +
M (M + m) 2M
(11)
pa+3 ?c+3 pc+3 pa+3
?i ? i 2 2 ?c pc (m + ?c+3 pc+3 ).
2M 2 (M + m) 2E M

An interaction Hamiltonian for two particles, in the absence of external fields, can
have the form

H = ?0 ?A pA + ?0 {m2 + V (r)}1/2 , (12)

where V (r) is an arbitrary function depending on r ? x2 . In the special case
c+3
when V (r) = e4 /r2 the interaction Hamiltonian can be written as

e2 (16) (16)
(16) (16)
H = ?0 (13)
?A p A + ? ?7 + ?0 m,
r0
(16) (16) (16)
where the 16 ? 16 matrices ?0 , ?A , ?7 satisfy a Cifford algebra. An external
electromagnetic field is introduced in eq. (3) in the following way:

pa > ?a = pa ? eAa (t, x1 , x2 , x3 ), pa+3 > ?a+3 = pa+3 ? eAa+3 (t, x4 , x5 , x6 ).

An extraction of the positive solutions from eq. (3) is realized by means of the
subsidiary condition
? ?
H µ
?1 ? ?µ p ? ? = 0,
1? v or
?=0 µ = 0, 1, 2, . . . , 6.
H2 2


It is evident that these conditions are invariant under the Poincar? group.
e
It should be noted that the function V (r) may be of arbitrary form, therefore the
relative velocity Va+3 ,
?
Va+3 ? ? ?i[Xa+3 , H]? = Va+3 ?, (14)

with respect to the centre-of-mass may be arbitrary. To do Va+3 smaller than the
photon velocity it is necessary to impose the condition

Va+3 = V4 + V5 + V6 < 1.
2 2 2 2


These questions will be considered in more detail in another paper.
Finally we shall find the equation for two particles with mass m1 = m2 . Let us,
with Kadyshevsky et al. [8], represent M in such a form
m1 + m2 2
M= v (m1 m2 + K )1/2 , (15)
m1 m2

where
2
m1 m 2
2
= ?m1 m2 + 2 2
m2 m2 (16)
K +K + +K .
1 2
(m1 + m2 )2
On a motion equation for two particles in relativistic quantum mechanics 281

In the variables P and K formula (2) can be rewritten as
(m1 + m2 )2 2
2
2
+ (m1 + m2 )2 . (17)
E =P + K
m1 m2
It follows that the equation of motion for the two particles is

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