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„ Preferred Frame Quantum Mechanics ; a toy model ” Toruń 2012 Jakub Rembieliński University of Lodz. J. Rembielinski , Relativistic Ether Hypothesi s, Phys. Lett . 78A, 33 (1980) J . Rembielinski , Tachyons and the preferred frames , Int.J.Mod.Phys . A 12 1677-1710, (1997)

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slide1

„PreferredFrame Quantum Mechanics; a toy model”

Toruń 2012

Jakub Rembieliński

University of Lodz

slide2

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.Rembielinskiand 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. Rembielinskiand K .A. SmolinskiLorentz Covariant Statistical MechanicsandThermodynamicsof the RelativisticIdealGas and PrefferedFrame,Phys. Rev. D, 76, 045018(2007)24

K. Kowalski, J. Rembielinskiand K .A. SmolinskiRelativisticIdeal Fermi Gasat Zero Temperatureand PreferredFrame,Phys. Rev. D, 76, 127701 (2007)25

J. Rembielinskiand K .A. Smolinski, Quantum Preferred Frame: Does It Really Exist?EPL2009, 10005 (2009)

J. Rembielinski and M. Wlodarczyk, „Meta” relativity: Against special relativity?

arXiv:1206.0841v1

slide4

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 synchronization-dependent. In particular, the Einstein synchronization procedure, assuming the path-independent 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 inthe 1+1 D is given by the time redefinition

tEinstein = t + ε x/c

Thisleads to a change of the form of Minkowski metrics

while the space part of the contravariantmetricsisstillEuclidean

slide6

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 canfix contravariant transformation law satisfying our requirements:

versus

Thisrealization of the Lorentz group can by related to the standard one inthe Einstein

synchronizationonly for velocities less orequal to c. Notice, that the timefoliation of

the space-time as well as the absolutesimultaneity of eventsispreserved by the above

transformations.

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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 bythe velocity of the preferred frame as seen by an observer in the unprimedframe:

slide9

.

Classical free particleA free particle of a mass m is defined by the Lagrange function derived fromthemetric form

Consequently the Hamiltonian has the form

where p is the canonical (notkinematical!) momentum, i.e.

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We candeduce the transformation law for momentum and Hamiltonian:

Momentum and Hamiltonian form a covarianttwo-vectorsatisfying

the invariantdispersionrelation:

We can define the following invariant measures

and

slide11

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 functionswith the scalar product

Under themodified Lorentz transformationsthe 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 :

slide12

The canonicalobservables and the quantum Hamiltonian

transform according to themodified Lorentz transformations

We caneasilyverifythat the Heisenberg canonicalcommutationrelationis

covariant with respect to the abovetransformations, similary as the relativistic

Schroedinger equation (generalisedSalpeterequation)

slide13

Realization in the coordinaterepresentation

The above equations are covariant on themodified Lorentz group transformations in contrast to the standard formalism of the relativistic QM

slide14

Anexplicitsolution for m=0

Letusconsider the relativistic Schroedinger equation for a masslessparticle

under the simplestinitialcondition

By means of the Fourier transformmethod we obtaintwo independent normalised

solutions

slide15

We caneasilycalculate the proper, locallyconservedand covariant

probabilitycurrent (itdoes not exist in the standard formalism)

slide17

The averagevalues of the relativisticvelocity operator in the abovestatestakes the values

So the harmonicaverage of equals to the round – triplightvelocityc

THANK YOU !

slide19

Phys. Lett. A 78(1980) 33,

Int.J.Mod. Phys. A12(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,

EPL88 (2009) 10005,

Phys. Rev. A 81 (2010) 012118,

Phys. Rev. A 84 (2011) 012108.

slide20

REALIZATION OF THE LORENTZ GROUP

Einstein synchronization

Absolute synchronization

linear

linear

linear

nonlinear !

Lorentz factors:

c=1

Boosts:

Fourvelocity of the primed frame with respect to the unprimed one

D(Λ,u)triangular !!!

slide21

Consequences:

time does not mix with spatial coordinates !!!

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!

absent in the

standard SR

slide22

in each frame ! Notice

covariant

Einstein’s( subscriptE) versusabsolutesynchronization

Relationship:

Preferred frame:u=0, u0 =1

Minkowski space-time:

the same time lapse !

velocity oflight :

average :