Transcript Quantum evolution of the early Universe in the quasi

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
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
2
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!
3

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
f1  g1  1
gravitational and inertial masses are equal
f2  1
anomalous gravitomagnetic moment
is equal to zero
f3  0
gravitoelectric dipole moment is equal to zero
Classical and quantum theories are in the best compliance!
4

The absence of the anomalous gravitomagnetic moment is
experimentally checked in:



The generalization to arbitrary-spin particles:


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).
O.V. Teryaev, arXiv:hep-ph/9904376
The absence of the gravitoelectric dipole moment results in
the absence of spin-gravity coupling:
W ~ g S

see the discussion in: B. Mashhoon, Lect. Notes Phys. 702, 112
(2006).
5

The Equivalence Principle manifests in the general
equations of motion of classical particles

Du
0
d

and their spins:

DS
0
d

A.A. Pomeransky and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. 113,
1537 (1998) [J. Exp. Theor. Phys. 86, 839 (1998)].
6



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
7



Good-Nyborg equation is wrong!
The derivation based on the initial ProcaCorben-Schwinger equations for spin-1
particles confirms the PomeranskyKhriplovich equations
A.J. Silenko, Zs. Eksp. Teor. Fiz. 123, 883 (2003) [J. Exp. Theor.
Phys. 96, 775 (2003)].
8
Mathisson-Papapetrou equations

Dp
1   
  R u S
d
2

DS
d

 

or

p u p u
S p  0

S u  0
Myron Mathisson
9

Connection between four-momentum and fourvelocity:


p  mu  E

E ~S


Du
d

Additional force is of second order in the spin

C. Chicone, B. Mashhoon, and B. Punsly, Phys.
Lett. A 343, 1 (2005)
10
Pole-dipole approximation

Du
1   
m
  R u S
d
2

DS
d
0
The spin dynamics given by
the Pomeransky-Khriplovich approach
is the same!
11
The momentum dynamics given by
the Pomeransky-Khriplovich approach
results from the spin dynamics
dS
 ΩS
dt
H  H0  Ω  S




S is 3-component spin
t is world time
H is Hamiltonian defining the momentum and spin
dynamics
The momentum dynamics can be deduced!
12
Pomeransky-Khriplovich approach

Tetrad equations of momentum and spin motion
dSa
b c
  abc S u
d


 abc
dua
b c
  abc u u
d
are Ricci rotation coefficients
Similar to equations of momentum and spin motion of
Dirac particle (g=2) in electromagnetic field
dS 
e

 F S
d
m
F
du
e

 F u
d
m
is electromagnetic field tensor
e
 abc u  Fab
m
c
13
Pomeransky-Khriplovich approach
e
e
Fab   E, B 
m
m
Ei   0ic u ,
c
 abcu  E, B 
c
1
c
Bi   eikl  klc u
2
dS
1
1
  0 B S  0 0
 u  E  S  Ω  S
dt
u
u  u  1
du
u B
E 0
dt
u
Tetrad variables are blue, t ≡ x0
14
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?

S u   S ua  0
a
When the metric is nonstatic, covariant
and tetrad velocities are equal to zero
(u=0 and u=0)
in different frames!
15
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.

(i  D  m)  0
  01 2 3
(i a Da  m)  0
a  01 2 3
a are the Dirac matrices but

 are not.
Definition of 3-component spin vector in the classical and quantum
theories agrees with the Pomeransky-Khriplovich approach
16
Pomeransky-Khriplovich approach
dS
1
1
  0 B S  0 0
 u  E  S  Ω  S
dt
u
u  u  1
0
du u
u B
 0 E 0
dt u
u
Pomeransky-Khriplovich gravitomagnetic field is nonzero
even for a static metric!
1
u
u
i  eikl   klc  0  0lc  0 ,
u 1
2
u
k
c
H  H0  Ω  S
17
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:
 abc
1
  hbc ,a  hac ,b  ,
2
hab  g ab  ab ,
ab  (1, 1, 1, 1)
This approximation is right for static metric but
incorrect for nonstatic metric!
Pomeransky-Khriplovich equations agree with quantum theory
resulting from the Dirac equation
18
Pomeransky-Khriplovich approach
can be verified for a rotating frame
g 
  ω  r 2   2 r 2
1 
c2

(1)

ωr

 
c2


(2)
ω

r


 

c2

(3)
ω

r





c2

ω  r 

(1)
c2
ω  r 

(2)
c2
ω  r 

c2
1
0
0
0
1
0
0
0
1
e00  1, ei0  0, eij   ij , e0i
e00  1, ei0  0, eij   ij ,
e00  1, e0i  0, ei 0
ω  r 

ω  r 
e0  
c
ω  r 

c






.






(i )
c
i
(3)
,
(i )
,
(i )
, eij   i j .
19
Pomeransky-Khriplovich approach
results in the Gorbatsevich-Mashhoon equation
A. Gorbatsevich, Exp. Tech. Phys. 27, 529 (1979);
B. Mashhoon, Phys. Rev. Lett. 61, 2639 (1988).
k
ijk
ij 0


e 
c
Exact equations:
dS
Ω  ω and
 ω  S
dt
A. J. Silenko (unpublished).
20
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)].
rg 
2
1
 0
ΩL

3
0
2mr  u  u 1  rg / r 1  u 0 1  rg / r

rg is the gravitational radius




However, Pomeransky-Khriplovich and Mathisson-Papapetrou
equations of particle motion does not agree with each other!
21
Comparison of classical and
quantum gravitational spin effects
Classical and quantum effects
should be similar due to the
correspondence principle
Niels Bohr
22
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

(i  D  m)  0
  01 2 3
The exact transformation of the Dirac equation for the metric
ds2  V 2 (r)(dx0 )2  W 2 (r)(dr  dr)
to the Hamilton form was carried out by Obukhov:
23
Comparison of classical and quantum gravitational spin effects

1
V
i
 H  H   mV  {F  α  p} F 
t
2
W
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
24
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:
   p2

  m2
H FW     V  1    F  1
2
 2

m

[ Σ  (φ  p)  Σ  (p  φ)  φ]
4 (  m)
 m(2 3  2 2 m  2 m2  m3 )

(p )(p φ)
5
2
8 (  m)
φ  V  f  F
  m2  p 2

 ( 2  m2 )
  Σ  (f  p)  Σ  (p  f )  f  
(p )(p f )
5
4
4
25
Comparison of classical and quantum gravitational spin effects

Quantum mechanical equations of
momentum and spin motion
2
2



dp
 m
 p 
 i[ H FW  p]     φ     f 
dt
2
 2 
m
1

(Π  (φ  p))  (Π  (f  p))
2 (  m)
2
dΠ
m
1
 i[ H FW  Π] 
Σ  φ  p   Σ  f  p 
dt
 (  m)

26
Comparison of classical and quantum gravitational spin effects

Semiclassical equations of momentum
and spin motion
dp
m2
p2
m

φ f 
(P  (φ  p))
dt


2 (  m)
1
S
 (P × (f  p)),
P
2
S
dS
m
1

S  φ  p   S  f  p 
dt  (  m)

Pomeransky-Khriplovich equations give the same result!
27
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.
28
Comparison of classical and quantum gravitational spin effects

Spinning particle in a rotating frame

The exact Dirac Hamiltonian was obtained by Hehl
and Ni:
H   m  α  p  ω  J
Σ
J  L  S L  r  p , S 
2

F. W. Hehl and W. T. Ni, Phys. Rev. D 42, 2045 (1990).
29
Comparison of classical and quantum gravitational spin effects

The result of the exact Foldy-Wouthuysen
transformation is given by
H FW   m  p  ω  J
2


2
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:
dS
 ω  S
dt
30
Comparison of classical and quantum gravitational spin effects

The particle motion is characterized by the
operators of velocity and acceleration:
i
dx
i
i
0
v  0  i[ H  x ] x  t
dx
i
dv
wi  0  i[ H  vi ]    H  [ H  xi ] 
dx

For the particle in the rotating frame
v
p
 ω  r   m  p 
2
2

pω
w  2
 ω  (ω  r )  2 v  ω  ω  (ω  r )

w is the sum of the Coriolis and centrifugal accelerations
31
Comparison of classical and quantum gravitational spin effects
The classical and quantum
approaches are in the best
agreement
32
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
33
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!
34
Equivalence Principle and spin
dp
m2
p2
m

φ f 
(P  (φ  p))
dt


2 (  m)
1
 (P × (f  p)),
2
dS
m
1

S  φ  p   S  f  p 
dt  (  m)

φ depends only on g 00 but f is a function of
both g 00 and gij
35
Equivalence Principle and spin

Dynamics of unit momentum vector n=p/p:
2
dn
m
p
 ω  n ω 
(φ  n)  (f  n)
dt
p

Difference of angular velocities of rotation of
spin and momentum depends only on g 00 :
m
o  Ω  ω   (φ  n)
p
36
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)
37
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, arXiv:hep-ph/9904376
38
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)
39
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!
40
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
41
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
42
Thank you for attention
43