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„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 x 0 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 m2 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 m2 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: ds 2 (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 dx 0 dxE0 2 if ds 2 (dx 0 u 0 u d x )2 (d x )2 2 d xE d x 0 cn velocity of light : c n n u u 0 covariant u0 in each frame ! Notice u0 1 / u 0 the same time lapse ! average : | c | C ds C C ds | c n | c