Transcript Slajd 1

„Preferred Frame Quantum Mechanics; a toy model”
Toruń 2012
Jakub Rembieliński
University of Lodz
J.Rembielinski, Relativistic Ether Hypothesis, Phys. Lett. 78A, 33 (1980)
J.Rembielinski, Tachyons and the preferred frames , Int.J.Mod.Phys. A 12
1677-1710, (1997)
P. Caban and J. Rembielinski, Lorentz-covariant quantum mechanics and
preferred frame, Phys.Rev. A 59, 4187-4196 (1999)22
J.Rembielinski and K .A. Smolinski, Einstein-Podolsky-Rosen Correlations of
Spin Measurements in Two Moving Inertial Frames, Phys. Rev. A 66, 052114
(2002)23
K. Kowalski, J. Rembielinski and K .A. Smolinski Lorentz Covariant Statistical
Mechanics andThermodynamics of the Relativistic Ideal Gas and Preffered
Frame, Phys. Rev. D, 76, 045018(2007)24
K. Kowalski, J. Rembielinski and K .A. Smolinski Relativistic Ideal Fermi Gas at
Zero Temperature and Preferred Frame, Phys. Rev. D, 76, 127701 (2007)25
J. Rembielinski and K .A. Smolinski, Quantum Preferred Frame: Does It Really
Exist? EPL 2009, 10005 (2009)
J. Rembielinski and M. Wlodarczyk, „Meta” relativity: Against special relativity?
arXiv:1206.0841v1
As it is well known, it is not possible to measure one-way (open path) light velocity without
assuming a synchronization procedure (convention) of distant clocks. The issue and the
meaning of the clock synchronization was elaborated in papers by Reichenbach, Grunbaum,
Winnie, as well as in the test theories of special relativity by Robertson, Mansouri and Sexel,
Will; an accessible discussion of the synchronization question is given by Lammerzahl (C.
Lammerzahl, Special Relativity and Lorentz Invariance, Ann. Phys. 14, 71–102 (2005) ).
Consequently, the measured value of the one-way light velocity is synchronizationdependent. In particular, the Einstein synchronization procedure, assuming the pathindependent speed of light, is only one (simplest) possibility out of the variety of possibilities
which are all equivalent from the physical (operational) point of view. The relationship
between Einstein's and other synchronizations in the 1+1 D is given by the time redefinition
tEinstein = t + ε x/c
This leads to a change of the form of Minkowski metrics
while the space part of the contravariant metrics is still Euclidean
Light cone in 2+1 D:
c2 t2 + 2 t x ε + (ε 2 -1) x2 = 0
A crucial point is, how to use the synchronization freedom to solve the problem of
describing nonlocal, instantaneous influence. As was stressed above, this is equivalent to
the following question: Is it possible to realize Lorentz symmetry in a way preserving the
notion of the instant - time hyperplane by use a synchronization convention different from
the Einstein one? The answer to this question is yes!
By means of the condition of invariance of the notion of instant-time hyperplane we can
fix contravariant transformation law satisfying our requirements:
versus
This realization of the Lorentz group can by related to the standard one in the Einstein
synchronization only for velocities less or equal to c. Notice, that the time foliation of
the space-time as well as the absolute simultaneity of events is preserved by the above
transformations.
From the nonlinear transformation law of ε it follows that there
exists an inertial frame where the synchronization coefficient vanish
i.e. the Einstein convention is fulfilled. This distinguished frame we
will name as the preferred frame of reference. Putting ε'=0 we can
express the synchronization coefficient ε by the velocity of the
preferred frame as seen by an observer in the unprimed frame:
In terms of the preferred frame velocity the modified Lorentz transformations read
.
Classical free particle
A free particle of a mass m is defined by the Lagrange function derived
from the metric form
Consequently the Hamiltonian has the form
where p is the canonical (not kinematical!) momentum, i.e.
We can deduce the transformation law for momentum and Hamiltonian:
Momentum and Hamiltonian form a covariant two-vector satisfying
the invariant dispersion relation:
We can define the following invariant measures
and
Now, having the framework appropriate to description of the nonlocal phenomena
we can discuss its implementation in the quantum mechanics. To do this let us
consider a bundle of the Hilbert spaces H ε , -1< ε< 1, of the scalar square
integrable functions
with the scalar product
Under the modified Lorentz transformations the bundle forms an orbit of the Lorentz
group. As in the nonrelativistic case we quantize the system by means of the canonical
commutation relation for canonical selfadjoint observables
:
The canonical observables and the quantum Hamiltonian
transform according to the modified Lorentz transformations
We can easily verify that the Heisenberg canonical commutation relation is
covariant with respect to the above transformations, similary as the relativistic
Schroedinger equation (generalised Salpeter equation)
Realization in the coordinate representation
The above equations are covariant on the modified Lorentz group transformations in
contrast to the standard formalism of the relativistic QM
An explicit solution for m=0
Let us consider the relativistic Schroedinger equation for a massless particle
under the simplest initial condition
1
φ 𝑥, 0, 0 ≅
𝑥±𝑖
By means of the Fourier transform method we obtain two independent normalised
solutions 𝜑 𝑥, 𝑡, 𝜀 ±
𝜑 𝑥, 𝑡, 𝜀
±
±𝑖
=
(1 − 𝜀 2 ) 𝑥 +
𝜋
𝑐𝑡
1 ± −𝜀
±𝑖
We can easily calculate the proper, locally conserved and covariant
probability current (it does not exist in the standard formalism)
0
𝑗 𝑥, 𝑡, 𝜀
±
1 − 𝜀2
=
𝜋 (1 − 𝜀 2 ) 𝑥 +
1
𝑗 𝑥, 𝑡, 𝜀
±
2
𝑐𝑡
1 ± −𝜀
+1
± 1 − 𝜀2
=
𝜋 1 ± −𝜀
𝜕 0
𝑗 𝑥, 𝑡, 𝜀
𝜕𝑡
±
(1 − 𝜀 2 ) 𝑥 +
𝜕 1
+ 𝑐 𝑗 𝑥, 𝑡, 𝜀
𝜕𝑥
±
𝑐𝑡
1 ± −𝜀
= 0
2
+1
The time evelopement of the probability density distribution for the
right-handed solution (+) .
The average values of the relativistic velocity operator in the above states takes the values
<𝑽>=
±𝑐
1±(−𝜀))
So the harmonic average of 𝑐±
≡ ±𝑐±
equals to the round – trip light velocity c
THANK YOU !
Mechanika z układem wyróżnionym Szczególna Teoria Względności
c≤v≤∞
c≤v<∞
v<c
-Zachowanie przyczynowości
-Łamanie zasady względności
(układ wyróżniony)
Równoważność obu opisów
-Symetria Lorentza
-Absolutna równoczesność
- Względny czas
c→ ∞
0<v<∞
Symetria Galileusza
-Zachowanie przyczynowości
-Absolutny czas
-Łamanie przyczynowości
-Zasada względności
-Symetria Lorentza
-Względność równoczesności
-Względny czas
Phys. Lett. A 78 (1980) 33,
Int. J.Mod. Phys. A 12 (1997) 1677,
Phys. Rev. A 59 (1999) 4187,
Phys. Rev. A 66 (2002) 052114,
Phys. Rev. D 76 (2007) 045018,
Phys. Rev. D 76 (2007) 127701,
EPL 88 (2009) 10005,
Phys. Rev. A 81 (2010) 012118,
Phys. Rev. A 84 (2011) 012108.
REALIZATION OF THE LORENTZ GROUP
Einstein synchronization
Absolute synchronization
xE  xE
linear
x(u)=D(  , u )x(u)
uE  uE
linear
u=D(  , u )u
Rotations :   R ,
1 0 
R= 

0



Boosts:
nonlinear !
D(R,u)=R
, T   I
 WE0


 WE


linear
Lorentz factors:


T
WE  WE 
I
1  1  WE2 
 WET
c =1
WE Fourvelocity of the primed frame with respect to the unprimed one W 
x' 0 
1 0
x
0
W
x 0  constant
is a covariant notion!
D(Λ,u) triangular !!!
Consequences:
time does not mix with spatial coordinates !!!
x' 0   1 x0
x0  constant hyperplane
x' 0  constant hyperplane
Consequently there exists a covariant time foliation of the Minkowski
space- time!!! This fact has extremely important implications for time developement
of physical systems (covariance).
Cauchy conditions consistent with an instantaneous (nonlocal) influence too !
Velocity transformations without singularities also for superluminal signals!
Solution of the dispersion relation p p   m 2 in terms of the
covariant momenta p  pi  on the upper momenta hyperboloid :
p 0  u 0 m 2  p   u p 2
2

p0  u0  u p  m 2  p   u p 2
2

invariant measures :
d  p,m  =  p 0 )  ( p 2  m 2   d 4 p  d 3 p /  2 p 0 
d  p,m   2up  d  p,m   u0 d 3 p
d  x   u d x if dx  0 (invariant condition)
0
3
d  x  ~ u d 
0
absent in the
standard SR
Einstein’s ( subscript E) versus absolute
synchronization
Relationship:
xE0  x 0  u 0 ux
,
xE = x ,
(u )  u  1
,
uE  u ,
0
E
2
2
E
consequently :
Preferred frame: u=0, u0 =1
uE0  1 / u 0
Minkowski space-time:
ds2  (dxE0 )2  (d xE )2 ,
 1

u 0 uT
g(u)   0
0 2
T
u
u

I

(u
)
u

u


1 0 


0 I 
dl  (d xE )  (d x )
2
dx0  dxE0
2
ds2  (dx0  u0 u d x )2  (d x )2
2
if d xE  d x  0
cn
velocity of light : c n  
 n u u 0
covariant
u0
in each frame !
Notice u0  1 / u 0
the same time lapse !
average : | c | C 
 ds
C

C ds | cn |
c