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„ Preferred Frame Quantum Mechanics ; a toy model ” Toruń 2012 Jakub Rembieliński

„ 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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„ Preferred Frame Quantum Mechanics ; a toy model ” Toruń 2012 Jakub Rembieliński

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  1. „PreferredFrame Quantum Mechanics; a toy model” Toruń 2012 Jakub Rembieliński University of Lodz

  2. 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

  3. 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

  4. Lightconein 2+1 D: c2 t2 + 2 t x ε + (ε 2 -1) x2 = 0

  5. 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.

  6. 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:

  7. In terms of the preferredframevelocity the modified Lorentz transformationsread

  8. . 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.

  9. 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

  10. 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 :

  11. 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)

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

  13. 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

  14. We caneasilycalculate the proper, locallyconservedand covariant probabilitycurrent (itdoes not exist in the standard formalism)

  15. The timeevelopement of the probabilitydensitydistribution for the right-handedsolution(+) .

  16. The averagevalues of the relativisticvelocity operator in the abovestatestakes the values So the harmonicaverage of equals to the round – triplightvelocityc THANK YOU !

  17. 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.

  18. 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 !!!

  19. 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

  20. 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 :

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