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ISSN 1063-7796, Physics of Particles and Nuclei, 2018, Vol. 49, No. 1, pp. 9–10. © Pleiades Publishing, Ltd., 2018.
Dirac Particle in Riemann–Cartan Spacetimes
1
,
2
Yu. N. Obukhov
a
, A. J. Silenko
b, c,
*, and O. V. Teryaev
c, d
a
Nuclear Safety Institute, Russian Academy of Sciences, Moscow, 115191 Russia
b
Research Institute for Nuclear Problems, Belarusian State University, Minsk, 220030 Belarus
c
Joint Institute for Nuclear Research, Dubna, 141980 Russia
d
Dubna International University, Dubna, 141980 Russia
*e-mail: alsilenko@mail.ru
Abstract⎯Dynamics of a Dirac particle in general Riemann–Cartan spacetimes is considered. The Hermi-
tian Dirac Hamiltonian is derived and is transformed to the Foldy–Wouthuysen representation for an arbi-
trary spacetime geometry. The contribution of the torsion field to the Foldy–Wouthuysen Hamiltonian is
found. The new bounds on Cartan’s spacetime torsion are obtained.
DOI: 10.1134/S106377961801029X
We analyze the quantum dynamics of the Dirac
particle in the general Riemann–Cartan spacetimes.
Our notations correspond to [1–3]. The covariant
Dirac equation has the form
(1)
The Dirac matrices are defined in local Lorentz
(tetrad) frames and the spinor covariant derivatives are
given by
(2)
The general Riemann–Cartan spacetime geometry
is described by the independent tetrad and the
Lorentz connection The Dirac particle
is characterized by the electric charge q; is the
4-potential of the electromagnetic field, and
The general form of the line element of an arbitrary
gravitational field is given by
(3)
The functions V and as well as the components
of the matrix may depend arbitrarily on
The off-diagonal metric components g
0a
are related to
the effects of rotation.
In order to discuss the Dirac spinors, we need the
orthonormal frames. The preferable choice [1] is the
Schwinger gauge:
(4)
The tetrad (4) is characterized by the condition
The Hermitian Hamiltonian corresponding to
Eqs. (1)–(4) reads [2]
(5)
where
and
(6)
Hamiltonian (5) covers the general case of a Dirac
particle in an arbitrary Riemann–Cartan spacetime.
The Dirac particle in Riemannian spacetimes has been
investigated in [1–6].
In order to make the coupling of spin and torsion
explicit, we use the decomposition of the connection
into the Riemannian and post-Riemannian parts:
(7)
The tilde, as usual, denotes the Riemannian quan-
tities.
1
Talk at the International Session-Conference of the Section of
Nuclear Physics of PSD RAS, JINR, Dubna, April 12–15,
2016.
2
The article is published in the original.
α
α
γ - Ψ= α= ℏ ( ) 0, 0,1, 2,3. i D mc
α
γ
αβ
α α αβ
= =∂- + σ Γ
ℏ
, .
4
i
i i i i i
iq i
D eD D A
α
i
e
αβ βα
Γ = -Γ .
i i
i
A
( )
αβ α β β α
σ = γγ -γγ .
2
i
= -δ
× - - .
ˆ
ˆ 2 2 2 2
ˆ
ˆ
( )( )
a b
c d ab
c c d d
ds V c dt WW
dx K cdt dx K cdt
,
a
K
× 3 3
ˆ a
b
W , .
a
tx
= δ = δ - δ =
ˆ
ˆ ˆ 0 0 0
, , , 1,2,3.
a a b b
i i i b i i
e V e W cK ab
= =
ˆ
0
0, 1, 2,3.
a
e a
( ) ( )
=β + Φ+ π α +α π
+ ⋅ + ⋅ + ⋅ - ϒγ
ℏ
2
5
2
,
2 4
b a a b
b a a b
c
mc V q
c c
K π π K Ξ Σ
* ^ ^
=
ˆ
0
0
, V e = =
ˆ
0
ˆ ˆ 0
,
b b b
a a a
ee VW ^ π =- ∂- ℏ ,
i i i
i qA
ϒ=- ε Γ Ξ = ε Γ
ˆ ˆ
ˆˆ ˆ
ˆ ˆ ˆ ˆ ˆˆ ˆˆ 0
, .
abc bc
a abc abc
V
V
c
ϒ=ϒ+ , Ξ =Ξ - .
ɶ ɶ
ˆ
ˆ ˆ ˆ 0 a a a
Vc Ť VŤ