Noninertial frames. ▫ Classical limit of relativistic quantum mechanics of Dirac
particles. ▫ Comparison of spin effects in electromagnetic and gravitational fields.
October 8-12, 2013
Spin-1/2 particles in arbitrary strong gravitational fields Yuri N. Obukhov*, Alexander J. Silenko+¤, Oleg V. Teryaev¤
*Nuclear Safety Institute, RAS, Moscow, Russia +Research Institute for Nuclear Problems, BSU, Minsk, Belarus ¤Joint Institute for Nuclear Research, Dubna, Russia XV Workshop on High Energy Spin Physics Dubna, Russia
OUTLINE
Transformation to the Foldy-Wouthuysen representation for relativistic particles Dirac particles in strong stationary gravitational fields and noninertial frames
Static metric Stationary metric The most general case Equivalence Principle for the spin and its experimental verification Mathisson-Papapetrou equations Noninertial frames
Classical limit of relativistic quantum mechanics of Dirac particles Comparison of spin effects in electromagnetic and gravitational fields Summary
2
Transformation to the FoldyWouthuysen representation for relativistic particles
3
The quantum theory is based on the covariant Dirac equation:
i bc i Da mc 0 Da ea 4 bca , i b c bc c b ( ), abc bac , a, b, c 01 2 3. 2
a
General metric:
ds V c dt ab W cW ˆˆ 2
2 2
aˆ
2
bˆ
c c d d dx K cdt dx K cdt d
Isotropic stationary metric :
ds V c dt W ab dx K cdt dx K cdt 2
2 2
2
2
a
a
b
b
Schwinger gauge:
e0 V 0 , ei W i K i 0 Tetrad indexes are blue
4
Nonunitary transformation of the initial equation brings the Dirac Hamiltonian: c c c 2 H D mc V {F α p} K p p K K Σ, 2 2 4 F V /W. Yu. N. Obukhov, A.J. Silenko and O.V. Teryaev, Phys. Rev. D 80, 064044 (2009).
General form of Dirac Hamiltonian: 1 0 H D M E O, , [ , M ] [ , E ] 0, , O 0 0 1 Relativistic Foldy-Wouthuysen transformation is performed by the unitary operator: M O U , M 2 O2 . 2 M O A.J. Silenko, Phys. Rev. A 77, 012116 (2008).
5
After the first step, the exact transformed Hamiltonian is 1 H ' E [T ,[T , F ]] [O,[O, M ]] 2T [O,[O, F ]] [ M ,[ M , F ]] 1 [ M ,[ M , O]] {O,[ M , F ]} { M ,[O, F ]] , T 2 F E i , T M O . t Then one determines the even and odd parts of this Hamiltonian: 1 1 E ' H ' H ' , O ' H ' H ' . 2 2 After the second step, one obtains the FoldyWouthuysen Hamiltonian: 1 1 H FW E ' O '2 , . 4 6
Dirac particles in strong stationary gravitational fields and noninertial frames
7
Static metric ds V c dt W ab dx dx 2
2 2
2
2
a
b
Schwarzschild field Accelerated frame
Foldy-Wouthuysen Hamiltonian: H
(1) FW
'
mc 4 4
1 [ Σ (Φ p) Σ (p Φ) Φ] 2 2 2 ' mc { ',V }
c2 1 [ Σ ( G p ) Σ ( p G ) G ] 16 ' 1 2 2 2 2 4 2 ' m c V c F , p Φ F 2V G F 2 Only ħ is a small parameter. All terms are exact. All terms of order of ħ and leading terms of order of ħ2 describing contact interaction are included Any direct spin-gravity coupling proportional to Σ·Φ, Σ·G and violating CP invariance does not appear!
8
Stationary metric
ds V c dt W ab dx K cdt dx K cdt 2
2 2
2
2
a
a
b
b
Foldy-Wouthuysen Hamiltonian: (1) (2) H FW H FW H FW
с c H K p p K Σ K 2 4 c3 1 2 ,F Σ Q 2 2 16 2 ' mc { ',V } (2) FW
Q p K p p K K p p K p p p K K p p Yu. N. Obukhov, A.J. Silenko and O.V. Teryaev, Phys. Rev. D 80, 064044 (2009).
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Operator equation of spin motion
dΠ i (1) (2) [ H FW , Π] Ω Σ Ω Π dt
Term depending on the static part of the metric
c2 1 mc 4 1 Ω ( Φ p p Φ ) ( G p p G ) 2 2 2 2 ' mc { ',V } 8 ' (1)
Ω(2)
c c3 1 2 K 2 ,F Q 2 2 8 2 ' mc { ',V }
Semiclassical equation of spin motion
dS ΩS dt 2
mc 4 c c cF 2Q Ω 2 Φp G p K 2 ' mc 'V 2 ' 2 4 '2 mc 2 'V This is the general solution of the problem
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The most general case ds V c dt ab W cW ˆˆ 2
2 2
2
aˆ
bˆ
c c d d dx K cdt dx K cdt d
Dirac Hamiltonian:
Yu. N. Obukhov, A.J. Silenko and O.V. Teryaev, Phys. Rev. D 84, 024025 (2011); arXiv:1308.4552 [gr-qc] (2013). 11
Foldy-Wouthuysen Hamiltonian:
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In the general case, the momentum and spin dynamics in gravitational fields and noninertial frames can be perfectly described with gravitoelectric and gravitomagnetic fields defined by the Pomeransky-Khriplovich relations: i
c
0icu
c i
c eikl 2
c u klc
A.A. Pomeransky and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. 113, 1537 (1998) [J. Exp. Theor. Phys. 86, 839 (1998)].
The gyrogravitomagnetic ratio is exactly the same for any particles: ggrav=2
This is the manifestation of the post-Einsteinian equivalence principle (equivalence principle for spin; no difference between the classical gravity and the classical limit of quantum mechanics) 13
Angular velocity of spin rotation:
The classical limit is
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Equivalence Principle for the spin and its experimental verification
Kobzarev – Okun relations define form factors at zero momentum transfer
I.Yu. Kobzarev, L.B. Okun, Gravitational Interaction of Fermions. Zh. Eksp. Teor. Fiz. 43, 1904 (1962) [Sov. Phys. JETP 16, 1343 (1963)].
Kobzarev – Okun relations result in the absence of the anomalous gravitomagnetic moment and the gravitoelectric dipole one Motion of gyroscopes and particle spins in curved spacetimes is the same. Classical and quantum theories are in the best compliance! This is a manifestation of the post-Einsteinian equivalence principle for the spin!
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The Equivalence Principle manifests in the general equations of motion of classical particles
and their spins:
Du 0 d
(Einsteinian)
DS 0 d
(post-Einsteinian)
see, e.g., A.A. Pomeransky and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. 113, 1537 (1998) [J. Exp. Theor. Phys. 86, 839 (1998)].
The Equivalence Principle is not exactly satisfied for spinning particles (Mathisson-Papapetrou equations) 16
Mathisson-Papapetrou equations
Dp 1 R u S d 2
DS d
S
p u p u
p 0
or
S u 0
Myron Mathisson 17
Connection between four-momentum and fourvelocity:
p mu E Du E ~S d
Additional term in the equation of spin motion is of the second order in the spin!
C. Chicone, B. Mashhoon, and B. Punsly, Phys. Lett. A 343, 1 (2005)
For Dirac particles, terms of the second order in the spin are reduced
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In this case
Du 1 m R u S d 2
DS d
0
The spin dynamics given by the Pomeransky-Khriplovich approach is the same! 19
Yu. N. Obukhov, A.J. Silenko and O.V. Teryaev, arXiv:1308.4552 [gr-qc] (2013).
These equations can be obtained in the framework of classical spin physics
Mathisson force
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The
restriction on the anomalous gravitomagnetic moment may be obtained from experimental data: B.
J. Venema, P. K. Majumder, S. K. Lamoreaux, B. R. Heckel, and E. N. Fortson, Phys. Rev. Lett. 68, 135 (1992). the analysis in A.J. Silenko and O.V. Teryaev, Phys. Rev. D 76, 061101(R) (2007). by
( 201 Hg) 0.369 ( 199 Hg) 0.042 (95% C.L.)
21
before correction
after correction
We have extracted the new restriction of about 0.9% from the experimental data presented in C. Gemmel et al, Eur. Phys. J. D 57, 303 (2010).
(He) G (Xe) 0.009,
gHe G g Xe
The absence of the gravitoelectric dipole moment results in the absence of direct spin-gravity coupling
W ~ g S
see the discussion in B. Mashhoon, Lect. Notes Phys. 702, 112 (2006).
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Noninertial frames Uniformly accelerated frame 1 Π (a p) H FW , a r , m2 p 2 2 2( m) A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71, 064016 (2005).
dp a, dt
Σ a p dΠ dt m
An observer can distinguish between a gravitational field (g = – a) and a uniformly accelerated frame Helicity evolution is the same
m o ω Ω 2 a p p Angular velocities of precession of spin and unit momentum vector
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The exact Foldy-Wouthuysen transformation for the rotating frame A.J. Silenko and O.V. Teryaev, Phys. Rev. D 76, 061101(R) (2007).
H FW m2 p2 ω J, J l s
The particle motion is characterized by the operators of velocity and acceleration: i dx vi 0 i[ H xi ] x 0 t dx i dv i i i w 0 i[ H v ] H [ H x ] dx
For the particle in the rotating frame
p
v ω r m2 p 2 pω w 2 ω (ω r) 2v ω ω (ω r )
w is the sum of the Coriolis and centrifugal accelerations
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General noninertial frame The unit vector of the world velocity V=cU/U0 rotates with the angular velocity
a 2V ω r N ar ωV 2ω a 1 2 ω ω r ω r , 2 V c c ar N V /V
The exact semiclassical expression for the angular velocity of spin rotation can be written in terms of V:
a V ω r
Ω c2 a r
c2 a r c2 V ω r 2
ω 2
The spin rotates with respect to the momentum direction with the angular velocity o Ωω . V
When ω=0 (an uniformly accelerated frame), the helicity of a particle with a negligible mass remains unchanged 25
When a=0 (a purely rotating frame), the exact solution is given by
o ω
V ω ω r 2
.
V We can conclude that the motion in the general noninertial (arbitrarily rotating and accelerating) frame leads to the change of the helicity even for the ultrarelativistic particle with a negligible mass. A similar effect takes place in a gravitational field when a particle trajectory is finite.
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Infinite trajectory of a particle Helicity: ζ=(s/s)(U/U) U=(U1,U2,U3) contravariant vector
At infinitely distant parts of the particle trajectory, ζ= ζ´. ζ´=(s/s)(p/p), p=(p1,p2,p3) covariant vector
n= p/p We can redefine the helicity with the covariant momentum vector
dn 1 dp ωn nn , dt p dt
1 dp ω n , p dt
c2 v c ω nΦ n G K 2 FV v 2F 2 c n n K n n K . 2
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Therefore, the vector of spin rotates with respect to the momentum direction, and the angular velocity of such rotation is
c2 c o Ωω nΦ n n K n n K . FV v 2 The helicity of a particle with a negligible mass moving on an infinite trajectory in a gravitational field remains unchanged It can be shown that this conclusion is valid for an arbitrary (even nonisotropic) stationary metric
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Classical limit of relativistic quantum mechanics of Dirac particles
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Phys. Part. Nuclei Lett. 10, 91 (2013).
We can use Wentzel–Kramers–Brillouin method at condition that a de Broglie wavelength is much smaller than the characteristic size of the inhomogeneity region of the external field l:
l ,
d 1. dx
one-dimensional problem 30
When the condition of the Wentzel– Kramers–Brillouin approximation is satisfied, the transition to the classical limit can be done by replacing the operators in the Hamiltonian and equations of motion by the respective classical quantities.
31
Pomeransky-Khriplovich equation for the three-component spin
dS ΩS dt 1 u u i ceikl klc 0 0lc 0 . u 1 2 u k
c
Tetrad variables are blue, t ≡ x0
For nonstatic metric, the Pomeransky-Khriplovich equation of spin motion depends on a choice of a tetrad! 32
With the Schwinger gauge, 1 1 c Ω v Φ v G K FV 1 2F 2
v v K v v K . 2c 1 2 p mW v, Since ' mc V ,
the classical and quantum mechanical equations of spin motion agree The classical Hamiltonian H class ' cK p s Ω agrees with the quantum one
The classical and quantum approaches are in the best agreement 33
Comparison of spin effects in electromagnetic and gravitational fields
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(i D m) 0
01 2 3
The squared Dirac equation is
( 2 g D D 2 [ ]D D m2c 2 ) 0.
Explicit computation yields 2 q ab Fab m ab R 2 c 4
2 ab ie ab cd 5 m c 0, 16 where Fab ea eb F are the tensorlike electromagnetic 2
ab
abcd
2 2
1 , ab . 2m coincides with the spin (and momentum) transport
field coefficients and ab ab
matrix in a gravitational field.
The equation shows the interactions of the spin with electromagnetic and gravitational fields are perfectly 35 analogous
Summary
Relativistic Foldy-Wouthuysen transformation allows to derive Foldy-Wouthuysen Hamiltonians and operator and semiclassical equations of spin motion for Dirac particles in strong stationary gravitational fields and noninertial frames Validity of the post-Einsteinian equivalence principle (for the spin) can be experimentally checked The classical and quantum equations of spin motion in strong gravitational fields are in the best agreement Interactions of the spin with electromagnetic and gravitational fields are perfectly analogous 36
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