References - Bogoliubov Laboratory of Theoretical Physics - JINR

a describes acceleration of the observer and ω is an angular velocity of a noninertial
reference system.
The covariant Dirac equation for spin-1/2 particles and the orthonormal tetrad
e �0
i = Vδ 0
i , e �a i = W � δ a i − K a δ 0
�
i , a,b =1, 2, 3, (5)
can be transformed to the familiar Schrödinger form with the Hermitian Hamiltonian
H ′ = βmc 2 V + c
2
[(α · p)F + F(α · p)]
+ 2G
c2 �G
J · (r × p)+
r3 2c2r3 �
3(r · J)(r · Σ)
r2 �
− J · Σ . (6)
Dirac Hamiltonian (6) contains the first part describing the static gravitational field and
the second one characterizing the contribution of rotation of the central body.
This FW transformation [2, 3] leads to the final Hamiltonian which is given by
HFW = H (1)
2G
FW + H(2)
FW , H(2)
FW =
c2 �G
J · l +
r3 2c2r3 �
3(r · J)(r · Σ)
r2 �
− J · Σ
− 3�G
�
1
8 ɛ(ɛ + mc2 ) ,
�
2{(J · l), (Σ · l)}
r5 + 1
�
(r · J)
(Σ · (p × l) − Σ · (l × p)) ,
2
r5 �
�
+ Σ · (p×(p×J)), 1
r3 ���
− 3�2c2 �
G
(5p
8
2 r −p 2 ) 2ɛ2 +ɛmc2 +m2c4 ɛ4 (ɛ + mc2 ) 2 , (J · l)
r5 �
,
(7)
where l = r×p is an angular momentum operator, and the operator p 2 r = −�2
r2 � �
∂ 2 ∂
r
∂r ∂r
is proportional to the radial part of the Laplace operator. The rotation-independent
contribution H (1)
FW has been calculated earlier [4].
The newly obtained operator of angular velocity of rotation of the spin in the static
gravitational field is equal to
Ω (2) = G
c2r3 �
3(r · J)r
r2 �
− J − 3G
�
1
4 ɛ(ɛ + mc2 ) ,
�
2{l, (J · l)}
r5 + 1
�
(r · J)
(p × l − l × p),
2
r5 � �
+ (p × (p × J)), 1
r3 ���
. (8)
The semiclassical formula corresponding to Eq.
average spin has the form
(8) and describing the motion of
Ω (2) = G
c2r3 �
3(r · J)r
r2 �
3G
− J −
r5ɛ(ɛ + mc2 [l(l · J)+(r · p)(p × (r × J))] .
)
(9)
The presented quantum mechanical and semiclassical equations are principal new results.
Description of a spin requires the introduction of a tetrad. A choice of a tetrad means
a selection of a local reference system of an observer.
There are infinitely many tetrads since a reference frame of an observer can obviously
be constructed in infinitely many ways. In particular, from a given tetrad field e α i we can
obtain a continuous family of tetrads by performing the Lorentz transformation e ′α i =
Λ α βe β
i , where the elements of the Lorentz matrix Λα β(x) are arbitrary functions of the
spacetime coordinates. In practice, there are three most widely used gauges.
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