Dynamics of Dirac particle spins in arbitrary stationary - - PowerPoint PPT Presentation

Dynamics of Dirac particle spins in arbitrary stationary gravitational fields

Yuri N. Obukhov (UCL, London, UK) Alexander J. Silenko (INP BSU, Minsk, Belarus) Oleg V. Teryaev (JINR, Dubna, Russia)

International Workshop “Bogoliubov readings” Dubna 2010

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OUTLINE

n Dirac particles in static gravitational

fields and uniformly accelerated frames

n Dirac particles in stationary

gravitational fields and rotating frames

n Comparison between classical and

quantum equations of spin motion

n Dirac particles in arbitrary strong

stationary gravitational fields

n Summary

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Dirac particles in static gravitational fields and uniformly accelerated frames

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This problem has been solved in the work:

n A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71,

064016 (2005).

The quantum theory is based on the Dirac equation: The exact transformation of the Dirac equation for the metric to the Hamilton form was carried out by Obukhov:

( ) 0 1 2 3 i D m

µ µ

γ ψ µ − = , = , , ,

2 2 2 2

( )( ) ( )( ) ds V dx W d d = − ⋅ r r r r

1{ } 2 V i H H mV F F t W ψ ψ β ∂ = , = + , ⋅ , = ∂ α p

• nYu. 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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n Silenko and Teryaev used the Foldy-Wouthuysen

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

2 2 (1)

1 1 2 2 [ ( ) ( ) ] 4 ( )

FW FW

m p H H V F m m β β βε ε ε β ε ε     = = + , − + , −         − ⋅ × − ⋅ × + ∇⋅ + 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 β ε ε ε ε ε β β ε ε ε + + + + ⋅∇ ⋅ + + + ⋅ × − ⋅ × + ∇⋅ − ⋅∇ ⋅ . p p φ Σ f p Σ p f f p p f

2 2

V F m p ε = ∇ , = ∇ = + φ f

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n Quantum mechanical equations of

momentum and spin motion

2 2

[ ] 2 2

FW

d m p i H dt β β ε ε     = , = − , − ,         p p φ f 1 ( ( )) ( ( )) 2 ( ) 2 m m ε ε ε + ∇ ⋅ × − ∇ ⋅ × + Π φ p Π f p

( ) ( )

1 [ ] ( )

FW

d m i H dt m ε ε ε = , = × × − × × + Π Π Σ φ p Σ f p

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n Semiclassical equations of momentum

and spin motion

2 2

( ( )) 2 ( ) 1 ( ( )), 2 d m p m dt m S ε ε ε ε ε = − − + ∇ ⋅ × + − ∇ × = p φ f P φ p S P× f p P

( ) ( )

1 ( ) d m dt m ε ε ε = × × − × × + S S φ p S f p

When the Foldy-Wouthuysen representation is used, the derivation

• f semiclassical equations consists in replacing p, Π, Σ operators

with corresponding classical quantities

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Uniformly accelerated frame

{ }

2 2

1 ( ) , , 2 2( )

FW

H m m Π a p a r p β ε ε ε ε ⋅ ×   = + ⋅ + = +   +  

, d dt p a βε = −

( )

d dt m Σ a p Π ε × × = − +

An observer can distinguish between a gravitational field (g = – a) and a uniformly accelerated frame Helicity evolution is the same

( )

2

m

• ω

Ω a p p = − = − ×

Angular velocities of precession of spin and unit momentum vector

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Dirac particles in stationary gravitational fields and rotating frames

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Spin motion in the Lense-Thirring metric

Albert Einstein’s theory of general relativity predicts that rotating bodies drag spacetime around themselves (frame dragging or the Lense- Thirring effect)

Lense-Thirring metric (an example of a stationary spacetime):

( )

( )

2 2 2 2 2 2 2 2 2 2 2

2 1 1 2 1 4 sin sin GM ds cdt dr GM c r c r GMa r d d d dt c r θ θ φ θ φ   = − − +     − + + −

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Initial Dirac equation for the Lense-Thirring metric can be transformed to the Hamilton form by the Obukhov’s method:

( )( )

2 2 3 2 2 5

, 2 { } 2 3 2 V i H F t W c G H mc V F c r G r c r ψ ψ β ∂ = , = ∂ = + , ⋅ + ⋅   + ⋅ ⋅ − ⋅   α p l J r Σ r J Σ J h h

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

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( )( ) ( ) ( )

{ }

[ ] [ ]

( )

[ ]

( )

(2) 2 2 3 2 5 2 5 5 3 2 2 2 2 2 2 4 2 5

2 3 2 2 , 3 1 , 8 ( ) 1 1 , , 2 3 2 5 , . 8 ( )

FW r

G G H r c r c r G c m r r r m m p c m r ε ε ε ε ε ε   = ⋅ + ⋅ ⋅ − ⋅     ⋅ ⋅  −    +     ⋅      + ⋅ × − ⋅ × + ⋅ × ×                 + + ⋅ − −   +   l J r Σ r J Σ J Σ l l J r J Σ p l Σ l p Σ p p J l J p h h h

After the Foldy-Wouthuysen transformation, the Hamiltonian takes the form

(1) (2) FW FW FW

H H H = + ,

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Quantum mechanical equations of momentum and spin motion

( )

curl curl 2

s

с = × − × + F K p p K F

1 , , , 4 ,

i i i j i j FW

dp dp g F v p dt dt x H p i x c

µ µ µ µ

    ∂   = = − +    ∂       ∂   = = −   ∂   p h

Force operator:

( )

2 3 2

3 2 curl G c r r ⋅   = −     r r J K J

agrees with the known nonrelativistic classical result

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( )( ) ( ) ( )

{ }

[ ] [ ]

( )

[ ]

( )

2 2 5 2 5 5 3 2 2 2 2 2 2 4 2 5

3 2 2 , 3 1 , 8 ( ) 1 1 , , 2 3 2 5 , 8 ( )

s r

G r c r G mc r r r m m p c m r ε ε ε ε ε ε    = ∇ ⋅ ⋅ − ⋅       ⋅ ⋅  −    +     ⋅        + ⋅ × − ⋅ × + ⋅ × ×                + + ⋅ − −    +   F r Σ r J Σ J Σ l l J r J Σ p l Σ l p Σ p p J l J p h h h

Spin-dependent part

• R. Wald, Phys. Rev. D 6, 406 (1972); B.M. Barker and
• R. F. O’Connell, Gen. Relativ. Gravit. 11, 149 (1979).

agrees with the previously obtained nonrelativistic classical result:

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Operator equation of spin motion

( ) ( )

{ }

( )

[ ]

(2) 2 3 2 2 5 5 3

3 2 , 3 1 , 4 ( ) 1 1 , , 2 G c r r G mc r r r ε ε ⋅   = −       ⋅  −    +     ⋅      + × − × + × ×           r r J Ω J l l J r J p l l p p p J

(1) (2)

d dt = × + × Π Ω Σ Ω Π

Term depending on the static part of the metric

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Semiclassical limit of quantum mechanical equations of momentum and spin motion can be found Relativistic formula for the angular velocity of the Lense-Thirring spin precession:

( ) ( ) ( ) ( )

( )

2 2 5 2 2 5

3 ( ) 3 . 1

LT

r G c r G m c r γ γ ⋅ − =   − ⋅ + ⋅ × ×   + r r J J Ω l l J r p p r J

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Spin motion in the rotating frame

n The simplest example of nonstatic spacetimes

n The exact Dirac Hamiltonian was obtained by Hehl

and Ni:

n F. W. Hehl and W. T. Ni, Phys. Rev. D 42, 2045 (1990).

H m β = + ⋅ − ⋅ , α p ω J

, 2 = + , = × = Σ J L S L r p S

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n The result of the exact Foldy-Wouthuysen

transformation is given by

n

A.J. Silenko and O.V. Teryaev, Phys. Rev. D 76, 061101(R) (2007).

n

The equation of spin motion coincides with the Gorbatsevich-Mashhoon equation:

2 2 FW

H m p β = + − ⋅ . ω J

d dt = − × S ω S

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n The particle motion is characterized by the

• perators of velocity and acceleration:

n For the particle in the rotating frame

[ ]

i i i

dx v i H x x t dx ≡ = , , ≡ ,

[ ] [ ]

i i i i

dv w i H v H H x dx   ≡ = , = − , , .  

2 2

m p β ε ε = − × , = + , p v ω r 2 ( ) 2 ( ) β ε × = + × × = × − × × . 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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Comparison between classical and quantum equations

• f spin motion

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n The Equivalence Principle manifests in the general

equations of motion of classical particles

n and their spins:

n

A.A. Pomeransky and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. 113, 1537 (1998) [J. Exp. Theor. Phys. 86, 839 (1998)].

n When one neglects a non-geodesic motion of

spinning particles, Pomeransky-Khriplovich and Mathisson-Papapetrou approaches bring the same results

Du d

µ

τ = DS d

µ

τ =

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Pomeransky-Khriplovich equation for the three-component spin

d t d = × S Ω S

1 . 2 1

k c i ikl klc lc

u u ce u u γ γ Ω = + +      

Tetrad variables are blue, t ≡ x0

For nonstatic metric, the Pomeransky-Khriplovich equation of spin motion depends on a choice of a tetrad!

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0,

i i

e e = = For t leads he rotating frame, on to the exact equatio ly the Schwin n of spin m ger gaug

• tion

e

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Lense-Thirring effect in general relativity

Lense-Thirring effect for spin:

L.I. Schiff, Am. J. Phys. 28, 340 1960;

• Proc. Nat. Acad. Sci. 46, 871 (1960);
• Phys. Rev. Lett. 4, 215 (1960).
• L. Schiff has described the effect of precession of

gyroscope (classical spin) caused by a rotation

• f a central body which angular momentum is J

2 2 5

3 ( ) , . d r G dt c r ⋅ − = × = S r r J J Ω S Ω

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( ) ( ) ( ) ( )

( )

2 2 5 2 2 5

3 ( ) 3 . 1

LT

r G c r G m c r γ γ ⋅ − =   − ⋅ + ⋅ × ×   + r r J J Ω l l J r p p r J

Our relativistic formula for the Lense-Thirring spin precession: Pomeransky-Khriplovich formula:

( )

( ) ( ) ( )

( )

2 2 5 2 2 2 5

3 ( ) 3 . 1

LT

r G c r G r m c r γ γ ⋅ − =   − ⋅ + × ×   +

PK

r r J J Ω l l J p p J

• A. A. Pomeransky and I. B.

Khriplovich, Zh. Eksp. Teor.

• Fiz. 113, 1537 (1998) [Sov.
• Phys. JETP 86, 839 (1998)].

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n When the symmetric tetrad is used, the

Pomeransky-Khriplovich equation should be added by the correction for the gravitoelectric field including the correction for the Thomas precession:

( ) ( ) ( )

( )

( )

( )

2 2 5 2

3 1 . G m c r r Ω r p p r J p p J δ γ γ  = ⋅ × ×  +  − × × 

Relativistic formula for the Lense-Thirring spin precession is our new result

Semiclassical equation of spin motion does not agree with the Pomeransky-Khriplovich result

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The classical and quantum approaches are in the best agreement

The classical and quantum equations

• f spin motion agree

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Dirac particles in arbitrary strong stationary gravitational fields

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Any stationary metric can be brought to the form diagonal in spatial coordinates and then to the isotropic form by an appropriate transformation of spatial coordinates:

( )( )

2 2 2 2 2 i i j j ij

ds V c dt W dx K cdt dx K cdt δ = − − −

Schwinger gauge:

( )

,

i i i

e V e W K

µ µ µ µ µ

δ δ δ = = −

Dirac Hamiltonian:

( )( )

2 2 3 2 2 5

2 { } 2 3 , 2

D

c G H mc V F c r G V r F c r W β = + , ⋅ + ⋅   + ⋅ ⋅ − ⋅ =   α p l J r Σ r J Σ J h

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Foldy-Wouthuysen Hamiltonian:

(1) (2) FW FW FW

H H H = + ,

(1) 2

1 ' [ ( ) ( ) ] 4 2 ' { ', } 1 [ ( ) ( ) ] 16 '

FW

m H m V β βε ε ε β ε   = − , ⋅ × − ⋅ × + ∇⋅   +     + , ⋅ × − ⋅ × + ∇⋅     Σ Φ p Σ p Φ Φ Σ p Σ p h h h h G G G

{ } ( )

2 4 2 2 2 2 2

1 ' , 2 m c V c F F V F ε

2

= + , = ∇ , = ∇ p Φ G

( ) ( )

{ }

(2) 2 2

2 4 1 , 16 2 ' { ', }

FW

с c H c F m V ε ε = ⋅ + ⋅ + ⋅ ∇×   − , ⋅   +   K p p K Σ K Σ Q h h

( ) ( ) ( )

( )

( )

( )

= ×∇ ⋅ + ⋅ −∇ ⋅ + ⋅ × − × × ∇× − ∇× × × Q p K p p K K p p K p p p K K p p

Only ħ is a small parameter. All terms are exact. All terms

• f order of ħ and leading

terms of order of ħ2 describing contact interaction are included

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Operator equation of spin motion

(1) (2)

d dt = × + × Π Ω Σ Ω Π

Term depending on the static part of the metric

(1) 2

1 1 1 ( ) ( ) 2 2 ' { ', } 8 ' m m V ε ε ε     = − , × − × + , × − ×     +     Ω Φ p p Φ p p G G

( )

{ }

(2) 2 2

1 , 2 8 2 ' { ', } c c F m V ε ε   = ∇× − ,   +   Ω K Q This is the general solution of the problem

Semiclassical equation of spin motion

d dt = × S Ω S

( )

( )

2 2 2

1 ' ' 2 ' 2 4 ' ' m c c F m V m V ε ε ε ε ε = − × + × + ∇× − + + Ω Φ p p K Q G

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Summary

n Foldy-Wouthuysen Hamiltonians and operator and

semiclassical equations of spin motion are derived for Dirac particles in static gravitational fields and uniformly accelerated frames. An observer can distinguish between a static gravitational field and a uniformly accelerated frame

n Foldy-Wouthuysen Hamiltonians and operator and

semiclassical equations of spin motion are obtained for Dirac particles in the Lense-Thirring metric and rotating frames

n Behavior of classical and quantum spins in stationary

spacetimes is the same and any important quantum effects do not appear. The classical and quantum approaches are in the best agreement

n Foldy-Wouthuysen Hamiltonians and operator and

semiclassical equations of spin motion are derived for Dirac particles in arbitrary strong stationary gravitational fields

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