Classical and Quantum Spins in Curved Spacetimes

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Classical and Quantum Spins in Curved
Spacetimes

• Alexander J. Silenko
• Belarusian State University
• Myron Mathisson his life, work, and influence on
  current research
• Warsaw 2007

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• OUTLINE

• General properties of spin interactions with
  gravitational fields
• Classical equations of spin motion in curved
  spacetimes
• Comparison between classical and quantum
  gravitational spin effects
• Equivalence Principle and spin

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• General properties of spin interactions with
  gravitational fields
• Anomalous gravitomagnetic moment is equal to zero
• Gravitoelectric dipole moment is equal to zero
• Spin dynamics is caused only by spacetime metric!

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• Kobzarev Okun relations
• I.Yu. Kobzarev, L.B. Okun, Gravitational
  Interaction of Fermions.
• Zh. Eksp. Teor. Fiz. 43, 1904 (1962) Sov.
  Phys. JETP 16, 1343 (1963).
• These relations define form factors at zero
  momentum transfer
• gravitational and
  inertial masses are equal
• 
  anomalous gravitomagnetic moment
• is equal to zero
• gravitoelectric dipole moment is equal to zero
• Classical and quantum theories are in the best
  compliance!

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• The absence of the anomalous gravitomagnetic
  moment is experimentally checked in
• B. J. Venema, P. K. Majumder, S. K. Lamoreaux, B.
  R. Heckel, and E. N. Fortson, Phys. Rev. Lett.
  68, 135 (1992).
• see the discussion in A.J. Silenko and O.V.
  Teryaev, Phys. Rev. D 76, 061101(R) (2007).
• The generalization to arbitrary-spin particles
• O.V. Teryaev, arXivhep-ph/9904376
• The absence of the gravitoelectric dipole moment
  results in the absence of spin-gravity coupling
• see the discussion in B. Mashhoon, Lect. Notes
  Phys. 702, 112 (2006).

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• The Equivalence Principle manifests in the
  general equations of motion of classical
  particles
• and their spins
• A.A. Pomeransky and I.B. Khriplovich, Zh. Eksp.
  Teor. Fiz. 113, 1537 (1998) J. Exp. Theor. Phys.
  86, 839 (1998).

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• Classical equations of spin motion in curved
  spacetimes
• Two possible methods of obtaining classical
  equations of spin motion
• i) search for appropriate covariant equations
• Thomas-Bargmann-Mishel-Telegdi equation linear
  in spin, electromagnetic field
• Good-Nyborg equation quadratic in spin,
  electromagnetic field
• Mathisson-Papapetrou equations all orders in
  spin, gravitational field
• ii) derivation of equations with the use of some
  physical principles
• Pomeransky-Khriplovich equations linear and
  quadratic in spin, electromagnetic and
  gravitational fields

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• Good-Nyborg equation is wrong!
• The derivation based on the initial
  Proca-Corben-Schwinger equations for spin-1
  particles confirms the Pomeransky-Khriplovich
  equations
• A.J. Silenko, Zs. Eksp. Teor. Fiz. 123, 883
  (2003) J. Exp. Theor. Phys. 96, 775 (2003).

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Mathisson-Papapetrou equations
or
Myron Mathisson
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• Connection between four-momentum and
  four-velocity
• Additional force is of second order in the spin
• C. Chicone, B. Mashhoon, and B. Punsly, Phys.
  Lett. A 343, 1 (2005)

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Pole-dipole approximation
The spin dynamics given by the
Pomeransky-Khriplovich approach is the same!
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The momentum dynamics given by the
Pomeransky-Khriplovich approach results from the
spin dynamics

• S is 3-component spin
• t is world time
• H is Hamiltonian defining the momentum and spin
  dynamics
• The momentum dynamics can be deduced!

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Pomeransky-Khriplovich approach

• Tetrad equations of momentum and spin motion
• are Ricci rotation coefficients
• Similar to equations of momentum and spin motion
  of
• Dirac particle (g2) in electromagnetic field

is electromagnetic field tensor
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Pomeransky-Khriplovich approach
Tetrad variables are blue, t x0
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Pomeransky-Khriplovich approach

• Pomeransky-Khriplovich approach needs to be
  grounded
• The 3-component spin vector is defined in a
  particle rest frame. What particle rest frame
  should be used?

When the metric is nonstatic, covariant and
tetrad velocities are equal to zero (u0 and u0)
in different frames!
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Pomeransky-Khriplovich approach

• Local flat Lorentz frame is a natural choice of
  particle rest frame.
• Only the definition of the 3-component spin
  vector in a flat tetrad frame is consistent with
  the quantum theory.
• Definition of 3-component spin vector in the
  classical and quantum theories agrees with the
  Pomeransky-Khriplovich approach

are the Dirac matrices but
are not.
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Pomeransky-Khriplovich approach
Pomeransky-Khriplovich gravitomagnetic field is
nonzero even for a static metric!
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Pomeransky-Khriplovich approach
In the reference A.A. Pomeransky and I.B.
Khriplovich, Zh. Eksp. Teor. Fiz. 113, 1537
(1998) J. Exp. Theor. Phys. 86, 839 (1998) the
following weak-field approximation was used
This approximation is right for static metric but
incorrect for nonstatic metric! Pomeransky-Khrip
lovich equations agree with quantum theory
resulting from the Dirac equation
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Pomeransky-Khriplovich approach

• can be verified for a rotating frame

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Pomeransky-Khriplovich approachresults in the
Gorbatsevich-Mashhoon equation

1. Gorbatsevich, Exp. Tech. Phys. 27, 529 (1979)
2. Mashhoon, Phys. Rev. Lett. 61, 2639 (1988).

A. J. Silenko (unpublished).
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Pomeransky-Khriplovich approach

• Another exact solution was obtained for a
  Schwarzschild metric
• A. A. Pomeransky, R. A. Senkov, and
  I. B. Khriplovich, Usp. Fiz. Nauk 43, 1129 (2000)
  Phys. Usp. 43, 1055 (2000).

However, Pomeransky-Khriplovich and
Mathisson-Papapetrou equations of particle
motion does not agree with each other!
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Comparison of classical and quantum gravitational
spin effects

• Classical and quantum effects should be
  similar due to the correspondence
  principle

Niels Bohr
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Comparison of classical and quantum gravitational
spin effects

• A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71,
  064016 (2005).
• Silenko and Teryaev establish full agreement
  between quantum theory based on the Dirac
  equation and the classical theory
• The exact transformation of the Dirac equation
  for the metric
• to the Hamilton form was carried out by Obukhov

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Comparison of classical and quantum gravitational
spin effects

• Yu. N. Obukhov, Phys. Rev. Lett. 86, 192 (2001)
• Fortsch. Phys. 50, 711 (2002).

This Hamiltonian covers the cases of a weak
Schwarzschild field and a uniformly accelerated
frame
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Comparison of classical and quantum gravitational
spin effects

• Silenko and Teryaev used the Foldy-Wouthuysen
  transformation for relativistic particles in
  external fields and derived the relativistic
  Foldy-Wouthuysen Hamiltonian

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Comparison of classical and quantum gravitational
spin effects

• Quantum mechanical equations of momentum and spin
  motion

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Comparison of classical and quantum gravitational
spin effects

• Semiclassical equations of momentum and spin
  motion

Pomeransky-Khriplovich equations give the same
result!
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Comparison of classical and quantum gravitational
spin effects

• These formulae agree with the results obtained
  for some particular cases with classical and
  quantum approaches
• A. P. Lightman, W. H. Press, R. H. Price, and S.
  A. Teukolsky, Problem book in relativity and
  gravitation (Princeton Univ. Press, Princeton,
  1975).
• F. W. Hehl and W. T. Ni, Phys. Rev. D 42, 2045
  (1990).
• These formulae perfectly describe a deflection of
  massive and massless particles by the
  Schwarzschild field.

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Comparison of classical and quantum gravitational
spin effects

• Spinning particle in a rotating frame
• The exact Dirac Hamiltonian was obtained by Hehl
  and Ni
• F. W. Hehl and W. T. Ni, Phys. Rev. D 42, 2045
  (1990).

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Comparison of classical and quantum gravitational
spin effects

• The result of the exact Foldy-Wouthuysen
  transformation is given by
• A.J. Silenko and O.V. Teryaev, Phys. Rev. D 76,
  061101(R) (2007).
• The equation of spin motion coincides with the
  Gorbatsevich-Mashhoon equation

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Comparison of classical and quantum gravitational
spin effects

• The particle motion is characterized by the
  operators of velocity and acceleration
• For the particle in the rotating frame

w is the sum of the Coriolis and centrifugal
accelerations
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Comparison of classical and quantum gravitational
spin effects

• The classical and quantum approaches are in the
  best agreement

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Equivalence Principle and spin

• Gravity is geometrodynamics!
• The Einstein Equivalence Principle predicts the
  equivalence of gravitational and inertial effects
  and states that the result of a local
  non-gravitational experiment in an inertial frame
  of reference is independent of the velocity or
  location of the experiment

Albert Einstein
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Equivalence Principle and spin

• The absence of the anomalous gravitomagnetic and
  gravitoelectric dipole moments is a manifestation
  of the Equivalence Principle
• Another manifestation of the Equivalence
  Principle was shown in Ref.
• A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71,
  064016 (2005).
• Motion of momentum and spin differs in a static
  gravitational field and a uniformly accelerated
  frame but the helicity evolution coincides!

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Equivalence Principle and spin
f depends only on but f is a function of
both and
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Equivalence Principle and spin

• Dynamics of unit momentum vector np/p

Difference of angular velocities of rotation of
spin and momentum depends only on
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Equivalence Principle and spin

• Pomeransky-Khriplovich equations assert the exact
  validity of this statement in strong static
  gravitational and inertial fields
• The unit vectors of momentum and velocity rotate
  with the same mean frequency in strong static
  gravitational and inertial fields but
  instantaneous angular velocities of their
  rotation can differ
• A.J. Silenko and O.V. Teryaev (unpublished)

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Equivalence Principle and spin

• Gravitomagnetic field
• Equivalence Principle predicts the following
  properties
• Gravitomagnetic field making the velocity rotate
  twice faster than the spin changes the helicity
• Newertheless, the helicity of a scattered massive
  particle is not influenced by the rotation of an
  astrophysical object
• O.V. Teryaev, arXivhep-ph/9904376

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Equivalence Principle and spin

• Gravitomagnetic field
• Analysis of Pomeransky-Khriplovich equations
  gives the same results
• Gravitomagnetic field making the velocity rotate
  twice faster than the spin changes the helicity
• Newertheless, the tetrad momentum and the spin
  rotate with the same angular velocity
• Directions of the tetrad momentum and the
  velocity coincide at infinity
• As a result, the helicity of a scattered massive
  particle is not influenced by the rotation of an
  astrophysical object
• A.J. Silenko and O.V. Teryaev (unpublished)

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Equivalence Principle and spin

• Gravitomagnetic field
• Alternative conclusions about the helicity
  evolution made in several other works
• Y.Q. Cai, G. Papini, Phys. Rev. Lett. 66, 1259
  (1991)
• D. Singh, N. Mobed, G. Papini, J. Phys. A 3, 8329
  (2004)
• D. Singh, N. Mobed, G. Papini, Phys. Lett. A 351,
  373 (2006)
• are not correct!

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Summary

• Spin dynamics is defined by the Equivalence
  Principle
• Mathisson-Papapetrou and Pomeransky-Khriplovich
  equations predict the same spin dynamics
• Anomalous gravitomagnetic and gravitoelectric
  dipole moments of classical and quantum particles
  are equal to zero
• Pomeransky-Khriplovich equations define
  gravitoelectric and gravitomagnetic fields
  dependent on the particle four-momentum
• Behavior of classical and quantum spins in curved
  spacetimes is the same and any quantum effects
  cannot appear

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Summary

• The helicity evolution in gravitational fields
  and corresponding accelerated frames coincides,
  being the manifestation of the Equivalence
  Principle
• Massless particles passing throughout
  gravitational fields of astrophysical objects
  does not change the helicity
• The evolution of helicity of massive particles
  passing throughout gravitational fields of
  astrophysical objects is not affected by their
  rotation
• The classical and quantum approaches are in the
  best agreement

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• Thank you for attention