Classical and Quantum Spins in Curved Spacetimes

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!

1 1 1f g

2 1f

3 0f

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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, arXiv:hep-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).

W ~ g S

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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)].

0Du

d

0DS

d

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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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1

2

DpR u S

d

Mathisson-Papapetrou equations

or

DSp u p u

d

0S u 0S p

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)

p mu E

DuE S

d

~

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Pole-dipole approximation

1

2

Dum R u Sd

0

DS

d

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!

d

dt

SΩ S

0 H H Ω S

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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 (g=2) in electromagnetic field

b caabc

dSS u

d

b ca

abc

duu u

dabc

dS e

F Sd m

du e

F ud m

is electromagnetic field tensorF cabc ab

eu F

m

!

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, ,cab abc

e eF u

m m E B E B

0 0 0

1 1

1

d

d u u ut

S

S u S Ω SB E

0

1,

2c c

i ic i ikl klcu e u E B

0d ut

d

u u BE Tetrad variables are blue, t ≡ x0

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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?

0aaS u S u

When the metric is nonstatic, covariant and tetrad velocities are equal to zero(u=0 and u=0) in different frames!

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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

( ) 0 0 1 2 3i D m

( ) 0 0 1 2 3aa m ai D

a are the Dirac matrices but are not.

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0 0 0

1 1

1

d

dt u u u

SS u S SΩB E

0

0

0

d u

dt u u

u u BE

00 0

1,

2 1

k c

i ikl klc lc

u ue

u u

Pomeransky-Khriplovich gravitomagnetic field is nonzeroeven for a static metric!

0 H H Ω S

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, ,

1, ,

2abc bc a ac b ab ab abh h h g

In the referenceA.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:

( 1, 1, 1, 1)ab This approximation is right for static metric but incorrect for nonstatic metric!

Pomeransky-Khriplovich equations agree with quantum theory resulting from the Dirac equation

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Pomeransky-Khriplovich approachcan be verified for a rotating frame

2 (1) (2) (3)2 2

2 2 2 2

(1)

2

(2)

2

(3)

2

1

1 0 0.

0 1 0

0 0 1

r

c c c c

cg

c

c

ω r ω r ω r ω r

ω r

ω r

ω r

( )

0 0

( )

0 0 0

0 0

0

0 0

( )

0 0

1, 0, , ,

1, 0, , ,

1, 0, , .

i

ii j j

i

i ij

i i

i j i

i

i

i j iij

e e e ec

e e e ec

e e e ec

ω r

ω r

ω r

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Pomeransky-Khriplovich approachresults in the Gorbatsevich-Mashhoon equation

0i

ij

kjke

c

equationE sxa :

a d

c

n

t

d

dt Ω

Sω ω S

A. Gorbatsevich, Exp. Tech. Phys. 27, 529 (1979);B. Mashhoon, Phys. Rev. Lett. 61, 2639 (1988).

A. J. Silenko (unpublished).

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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)].

3 0 0 0

is the gravitational radius

2 1

2 1 / 1 1 /

g

g g

g

r

mr u u r r u r r

r

Ω L

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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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:

( ) 0 0 1 2 3i D m

2 2 0 2 2( ) ( ) ( ) ( )ds V dx W d d r r r r

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1{ }

2

Vi H H mV F Ft W

α p

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

2 2

1 12 2

[ ( ) ( ) ]4 ( )

FW

m pH V F

m

m

Σ φ p Σ p φ φ

3 2 2 3

5 2

2 2

5

(2 2 2 )( )( )

8 ( )

( )( ) ( ) ( )( )

4 4

m m m m

m

m

p p φ

Σ f p Σ p f f p p f

2 2

V F

m p

φ f

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Quantum mechanical equations of momentum and spin motion

2 2

[ ]2 2FW

d m pi H

dt

pp φ f

1( ( )) ( ( ))

2 ( ) 2

m

m

Π φ p Π f p

1[ ]

( )FW

d mi H

dt m

Π

Π Σ φ p Σ f p

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Semiclassical equations of momentum and spin motion

2 2

( ( ))2 ( )

1( ( )),

2

d m p m

dt m

S

pφ f P φ p

SP× f p P

1

( )

d m

dt m

S

S φ p S f p

Pomeransky-Khriplovich equations give the same result!

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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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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).

H m α p ω J

,2

Σ

J L S L r p S

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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:

2 2FWH m p ω J

d

dt

Sω S

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The particle motion is characterized by the

operators of velocity and acceleration:

For the particle in the rotating frame

00

[ ]i

i idxv i H x x t

dx

0[ ] [ ]

ii i idvw i H v H H x

dx

2 2m p

p

v ω r

2 ( ) 2 ( )

p ω

w ω ω r v ω ω ω r

w is the sum of the Coriolis and centrifugal accelerations

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The classical and quantum approaches are in the best

agreement

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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 spin2 2

( ( ))2 ( )

1( ( )),

2

d m p m

dt m

pφ f P φ p

P× f p

1

( )

d m

dt m

S

S φ p S f p

φ depends only on but f is a function of both and

00gijg00g

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Dynamics of unit momentum vector n=p/p:2

( ) ( )d m p

dt p

nω n ω φ n f n

( )m

p o Ω ω φ n

Difference of angular velocities of rotation of spin and momentum depends only on :00g

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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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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, arXiv:hep-ph/9904376

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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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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

Classical and Quantum Spins in Curved Spacetimes

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