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 PDF created with pdfFactory Pro trial version www.pdffactory.com

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 2 PDF created with pdfFactory Pro trial version www.pdffactory.com

Dirac particles in static gravitational fields and uniformly accelerated frames 3 PDF created with pdfFactory Pro trial version www.pdffactory.com

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: µ γ − ψ = , µ = , , , ( i D m ) 0 0 1 2 3 µ The exact transformation of the Dirac equation for the metric = − ⋅ 2 2 0 2 2 ds V ( )( r dx ) W ( )( r d r d r ) to the Hamilton form was carried out by Obukhov: ∂ ψ 1{ V = ψ , = β + , ⋅ , = α p i H H mV F } F ∂ t 2 W n 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 4 PDF created with pdfFactory Pro trial version www.pdffactory.com

n Silenko and Teryaev used the Foldy-Wouthuysen transformation for relativistic particles in external fields and derived the relativistic Foldy-Wouthuysen Hamiltonian:     β β 2 2 m p = = βε + , − + , − (1)     H H V 1 F 1 ε ε FW FW     2 2 β m = ∇ , = ∇ − ⋅ × − ⋅ × + ∇⋅ φ Σ φ p Σ p φ φ V f F [ ( ) ( ) ] ε ε + 4 ( m ) ε = + 2 2 m p β ε + ε + ε + 3 2 2 3 m (2 2 m 2 m m ) ( + ⋅∇ ⋅ p φ p )( ) ε ε + 5 2 8 ( m ) β β ε + 2 2 [ ] ( m ) + ⋅ × − ⋅ × + ∇⋅ − ⋅∇ ⋅ . Σ Σ ( f p ) ( p f ) f ( p )( p f ) ε ε 5 4 4 5 PDF created with pdfFactory Pro trial version www.pdffactory.com

n Quantum mechanical equations of momentum and spin motion     β β 2 2 d p m p = , = − , − , φ     i H [ p ] f ε ε FW     dt 2 2 m 1 + ∇ ⋅ × − ∇ ⋅ × Π φ p Π f p ( ( )) ( ( )) ε ε + ε 2 ( ) 2 m Π d m 1 ( ) ( ) = , = × × − × × Π Σ φ p Σ [ ] f p i H ε ε + ε FW dt ( m ) 6 PDF created with pdfFactory Pro trial version www.pdffactory.com

n Semiclassical equations of momentum and spin motion 2 2 d p m p m = − − + ∇ ⋅ × φ φ p f ( P ( )) ε ε ε ε + dt 2 ( m ) 1 S − ∇ × = P × f ( ( p )), P ε 2 S d S m 1 ( ) ( ) = × × − × × φ p S S f p ε ε + ε dt ( m ) When the Foldy-Wouthuysen representation is used, the derivation of semiclassical equations consists in replacing p, Π , Σ operators with corresponding classical quantities 7 PDF created with pdfFactory Pro trial version www.pdffactory.com

Uniformly accelerated frame ⋅ ×   Π 1 ( a p ) { } = β ε + ε ⋅ + ε = + 2 2   H , a r , m p ε + FW   2 2( m ) ( ) × × Σ Π a p d p d = − βε = − a , ε + dt dt m An observer can distinguish between a gravitational field (g = – a) and a uniformly accelerated frame Helicity evolution is the same m ( ) = − = − × ω Ω o a p 2 p 8 Angular velocities of precession of spin and unit momentum vector PDF created with pdfFactory Pro trial version www.pdffactory.com

Dirac particles in stationary gravitational fields and rotating frames 9 PDF created with pdfFactory Pro trial version www.pdffactory.com

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 GM 1 ( ) = − − 2 + 2 2   ds 1 cdt dr   2 2 GM c r − 1 2 c r ( ) 4 GMa + θ + θ φ − θ φ 2 2 2 2 2 r d sin d sin d dt 2 c r 10 PDF created with pdfFactory Pro trial version www.pdffactory.com

Initial Dirac equation for the Lense-Thirring metric can be transformed to the Hamilton form by the Obukhov’s method: n Yu. N. Obukhov, Phys. Rev. Lett. 86 , 192 (2001); Fortsch. Phys. 50 , 711 (2002). ∂ ψ V = ψ , = i h H F , ∂ t W c 2 G = β + , ⋅ + ⋅ α p 2 H mc V { F } l J 2 3 2 c r h G ( )( )   + ⋅ ⋅ − ⋅ r Σ Σ J 2 3 r J r   2 5 2 c r 11 PDF created with pdfFactory Pro trial version www.pdffactory.com

After the Foldy-Wouthuysen transformation, the Hamiltonian takes the form = + , (1) (2) H H H FW FW FW 2 G h G ( )( )   = ⋅ + ⋅ ⋅ − ⋅ r Σ Σ J (2) 2 l J 3 r J H r   FW 2 3 2 5 c r 2 c r { } ( ) ( )   ⋅ ⋅ Σ l  2 , l J 3 h G 1 −   ,  ε ε + 2 5 8 c  ( m ) r    ⋅      ( ) 1 r J 1 [ ] [ ] [ ] + ⋅ × − ⋅ × + ⋅ × ×   Σ Σ Σ      p l l p , p p J ,        5 3 2 r r   ε + ε + ⋅ 2 2 2 ( ) l J 3 h 2 m m − −  2 2  p 5 p , . ε ε + r 2 4 2 5   8 c ( m ) r 12 PDF created with pdfFactory Pro trial version www.pdffactory.com

Quantum mechanical equations of momentum and spin motion Force operator:     ∂ µ   i i dp 1 dp g = = − + i  j   i , , , F v p µ ∂  j    dt dt 4  x  ∂   H = = − FW   agrees with the known p i h , p µ ∂ µ   x c nonrelativistic classical result с ( ) = × − × + F curl K p p curl K F s 2 ( )  ⋅  3 r r J 2 G = −   K J curl 2 3 2   c r r 13 PDF created with pdfFactory Pro trial version www.pdffactory.com

Spin-dependent part  h G ( )( )   = ∇ ⋅ ⋅ − ⋅ r Σ Σ J 2  F 3 r J r   s  2 5 2 c r { } ( ) ( )   ⋅ ⋅ Σ l  2 , l J 3 h G 1 −   ,  ε ε + 2 5 8  ( mc ) r    ⋅      ( ) r J 1 [ ] [ ] [ ] 1 + ⋅ × − ⋅ × + ⋅  × ×  Σ p l Σ Σ       l p , p p J ,       5 3 2 r r    ε + ε + ⋅ 2 ( ) 2 2 3 h 2 m m l J − −  2 2   5 p , p ε ε + r 2 4 2 5   8 c ( m ) r agrees with the previously obtained nonrelativistic classical result: R. Wald, Phys. Rev. D 6 , 406 (1972); B.M. Barker and R. F. O’Connell, Gen. Relativ. Gravit. 11 , 149 (1979). 14 PDF created with pdfFactory Pro trial version www.pdffactory.com

Operator equation of spin motion Π d dt = × + × Ω Σ Ω Π (1) (2) Term depending on ( ) the static part of the metric  ⋅  3 r r J G = − Ω (2)   J 2 3 2   c r r { } ( )   ⋅  2 l l J , 3 G 1 −   ,  ε ε + 2 5 4 ( mc ) r     ⋅      1 r J 1 [ ] ( ) + × − × + × ×       p l l p , p p J ,     5 3 2 r r 15 PDF created with pdfFactory Pro trial version www.pdffactory.com

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 3 ( r r J ) r J = Ω G LT 2 5 c r ( ) 3 G ( ) ( ) ( )   − ⋅ + ⋅ × × l l J r p p r J .   ( ) γ γ + 2 2 5 m 1 c r 16 PDF created with pdfFactory Pro trial version www.pdffactory.com

Spin motion in the rotating frame n The simplest example of nonstatic spacetimes n The exact Dirac Hamiltonian was obtained by Hehl and Ni: = β + ⋅ − ⋅ , α p ω J H m = Σ = + , = × J L S L r p , S 2 n F. W. Hehl and W. T. Ni, Phys. Rev. D 42, 2045 (1990). 17 PDF created with pdfFactory Pro trial version www.pdffactory.com

n The result of the exact Foldy-Wouthuysen transformation is given by = β + − ⋅ . ω J 2 2 H m p FW A.J. Silenko and O.V. Teryaev, Phys. Rev. D 76 , 061101(R) n (2007). The equation of spin motion coincides with the n Gorbatsevich-Mashhoon equation: S d dt = − × ω S 18 PDF created with pdfFactory Pro trial version www.pdffactory.com