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Our task is to find the exact form of the Hamiltonian H.
The matrices ?0 and (1 ? ?0 ) are the projectors on the subspaces of upper and
lower components of the wave function ?. They satisfy the conditions
(1 ? ?0 )?a = ?a ?0 .
2
(2.8)
?0 = ?0 ,
In order to reduce equation (2.1) to the form (2.7) we first multiply (2.1) by (1 ? ?0 ).
Using (2.8), one obtains
?a pa
(1 ? ?0 )? = ? (2.9)
?0 ?,
m
or after the multiplication by p0 ,
?a pa
(1 ? ?0 )p0 ? = ? (2.10)
?0 p0 ?.
2m
On the non-relativistic motion equations in the Hamiltonian form 433

On the other hand, multiplying (2.1) by ?0 , one obtains
?0 p0 ? = ??0 ?a pa ?. (2.11)
Substituting (2.11) into (2.10) and adding the result to (2.11), we come to the equation
(?a pa )2
?
i ? = (1 ? ?0 ) ? ?a pa ? (1 ? ?0 )2m ?. (2.12)
?t 2m
Equation (2.12) with the additional condition (2.10) is completely equivalent to (2.1).
Thus we have reduced the LHH equations to the Hamiltonian form.
3. Transition to the diagonal representation
Equations (2.12), (2.10) as well as equation (2.1) are invariant with respect to
the Galilei group G. Indeed, on the set {?} of the solutions of these equations the
following representation of the algebra G is realized:
? ?
Pa = pa = ?i Jab = xa pb ? xb pa + Sab ,
P 0 = p0 = i , ,
?t ?xa
(3.1)
i
Ga = tpa ? mxa + ?a , ?a = ? ?a ?0 ,
2
where the matrices Sab realize the direct sum D(s) ? D(s) ? D(s ? 1) of the algebra
O(3) representations. One can readily see that the generators (3.1) are non-Hermitian
with respect to the usual scalar product

d3 x ?† ?2 . (3.2)
(?1 , ?2 ) = 1

The aim of this section is to transform equations (2.12), (2.10) and the operators
(3.1) to such a form that the wave function ?(t, x) has only 2s+1 non-zero components
and the generators of the Galilei group representation are Hermitian with respect to
(3.2). It is achieved by the transformation to the new wave function

?p
? > ? = V ?, V = exp ?i (3.3)
.
m

The transformed generators (3.1) take the form
Pa = V Pa V ?1 , Jab = V Jab V ?1 = Jab ,
(3.4)
P0 = V P0 V ?1 = p0 , Ga = V Ga V ?1 = tpa ? mxa .
These operators are apparently Hermitian in the scalar product (3.2). Equations (2.12),
(2.10) after the transformation (3.4) have been reduced to the diagonal form
p2 ?
a
(3.5)
? =i ?,
2m ?t
(1 ? ?0 )? = 0. (3.6)

It follows from (3.6), (2.2) that the wave function ? has only 2s + 1 non-zero
components. Thus condition (2.10) (which is equivalent to (3.6) serves to remove 4s
redundant components from the (6s + 1) component wave function ?(t, x).
434 W.I. Fushchych, A.G. Nikitin, V.A. Salogub

One can use the operator (3.3) to construct the positive definite scalar product on
the set of the solutions of equations (2.12), (2.10). Indeed, it follows from the hermitici
of the operators (3.4) with respect to (3.2) that the generators (3.4) are Hermitian
with respect to

d3 x ?† M ?2 , (3.7)
(?1 , ?2 ) = 1

where
(?† p ) · (?p )
i
† †
M = V V = 1 ? (?p ? ? p ) + (3.8)
.
m2
m
For the case s = 1/2 the transformation operator (3.3) and the metric operators
(3.8) have the form
? ?
? ? ?·p
p2
?
I 1+ a
I 0 ? ?
m2 m
M =? ?,
V =? ?·p ?, (3.9)
? ?
?·p
? I
? I
m
m
where ?a are the usual Pauli matrices.
It follows from the above that the transformation (3.3) may be considered as the
non-relativistic analog of the Foldy–Wouthuysen transformation [3].
Equation (2.10) is not the only Gallilean invariant condition which can be added
to (2.12) in order to remove the redundant components of the wave function ?. For
instance, one can use for this purpose the subsidiary condition of the form
? ??
? ? ?
2 ?1/2
? ?
? ?
(?p )2 (?p )2
1? ?
? H? ?
1? ? H ? ? ? = 0. (3.10)
+m , +m
? ?
2 4m 4m
? ?
+

Equations (2.12), (3.10), as (2.12), (2.6), are Galilean invariant and can be reduced to
the diagonal form (3.5), (3.6) by the unitary transformation
2m + (1 ? 2?0 )?a pa
? > U ?, (3.11)
U= .
4m2 + (?a pa )2
On the set of the solutions of equations (2.12), (3.10) the Galilei group generators
have the form
? ?
Pa = pa = ?i Jab = xa pb ? xb pa + Sab ,
P0 = i , ,
?t ?xa (3.12)

Ga = tpa ? mXa , Xa = xa + [U , xa ]? U.
The generators (3.12) are Hermitian with respect to the usual scalar product (3.2)
but, in contrast to (3.1), are non-local (integral) operators.
4. The Hamiltonian equations without redundant components
In this section we obtain new (different from (2.1)) equations for arbitrary spin
particles, which are invariant under the Galilei group G. The main property of these
equations is that the wave function of a particle with spin s has 2(2s + 1) components.
On the non-relativistic motion equations in the Hamiltonian form 435

It allows to establish the direct connection between our equations and the relativistic
equations without redundant components.
We shall start from the following representation for the generators of the Galilei
group
? ? ?
Pa = pa = ?i Jab = xa pb ? xb pa + Sab ,
P0 = i , ,
?t ?xa
(4.1)
?
Sc 0
? ?
Ga = tpa ? mxa + ?a , Sab = , (a, b, c) = (1, 2, 3),
?c
0 S
?
where the matrices Sc realize the irreducible representation D(s) of the group O(3),
?
and ?a are arbitrary numerical matrices which have to be such that the generators
(4.1) satisfy the algebra G. It can be shown that the most general (up to equivalence)
?
form of the matrices ?a , satisfying such a requirement, is
1
? ? (4.2)
?a = k(?1 + i?2 )Sa , Sa = ?abc Sbc ,
2
?
where ?1 , ?2 are the 2(2s + 1)-dimensional Pauli matrices which commute with Sab ,
k is an arbitrary constant.
To obtain the Galilean invariant equations in the form
(4.3)
Ls ?(t, x) = 0
we must find the operators Ls satisfying the conditions
(4.4)
[Pµ , Ls ]? = [Jab , Ls ]? = [Ga , Ls ]? = 0.
Thus our problem has been reduced to the solution of the commutation relations (4.4).
In order to solve relations (4.4) we reduce the generators (4.4) to the diagonal
representation
?
P0 = V P0 V ?1 = i Pa = V pa V ?1 = pa ,
,
?t (4.5)
?1 ?1
?
= xa pb ? xb pa + Sab , = tpa ? mxa .
Jab = V Jab V Ga = V Ga V
The transition operator V has the form

?p
(4.6)
V = exp i .
m

We require that the wave function of the spin-s particle has, in the diagonal
representation (4.5), 2s + 1 non-zero components. This requirement may be written
in the form of the Galilean invariant condition
(4.7)
(1 + ?3 )? = 0.
Another natural assumption is that each component of ? satisfies the non-relativistic
Schr?dinger equation
o
p2
?
? = a?. (4.8)
i
?t 2m
436 W.I. Fushchych, A.G. Nikitin, V.A. Salogub

One can write (4.7) and (4.8) in the form of the single equation
p2
1 ? 1
(?1 + i?2 ) i ? a + (?1 ? i?2 )2m ? = 0. (4.9)
Ls ? =
2 ?t 2m 2
Equation (4.9) is Galilean invariant inasmuch as the following relations are satisfied
(4.10)
[Ls , Pµ ]? = [Ls , Jab ]? = [Ls , Ga ]? = 0.
To obtain equation (4.9) in the representation (4.1) it is sufficient to use the transition
operator (4.6). Making the transformation
? > ? = V ?1 ? , Ls > Ls = V ?1 Ls V, (4.11)
one obtains equation (4.3), where

p2 2
1 ? 2 (Sp )
Ls = (?1 + i?2 ) i ? + (?1 ? i?2 )m + ?3 k(Sp ).
a
(4.12)
+k
2 ?t 2m 2m

Thus we have found the Galilean invariant equation (4.3), (4.12) for the 2(2s + 1)-
component wave function. For s = 1/2, k = 1/s ((4.3), (4.12)) coincide with the
Levi–Leblond equation [16].
Equations (4.3), (4.12), as well as equation (2.1), may be reduced to the Hamil-
tonian form. Indeed, multiplying (4.12) by i?2 , one obtains from (4.3), (4.12) the
following expression:
? ?? ?
?0 B ? ?a pa + (1 ? ?0 )2m ? = 0, (4.13)

where
p2 k 2 (Sp )2
? 1
? ?
B=i ? ?0 = ?k?1 Sa .
a
(4.14)
+ , ?0 = (1 + ?3 ),
?t 2m 2m 2
??
The matrices ?0 , ?a satisfy thereby relations (2.8) as the ?0 , ?a . Repeating the
computations (2.9)–(2.12) one easily obtains from (4.13) the equations
p2
? ? ? ? (pa pb + pb pa ) + ?a pa ? (1 ? ?0 )2m, (4.15)
? ?
H = a ? ?0 ?a ?b
i ? = H?,
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