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AIP conference proceedings

Localization problem in quantum mechanics and preferred frame

Paweł Caban, Jakub Rembieliński

Published Jan 1, 1998• 1 citation
Publisher

Abstract

In this paper the localization problem in the relativistic quantum mechanics is considered in the framework of a nonstandard synchronization scheme for clocks. Such a synchronization preserves Poincaré covariance but (at least formally) distinguishes an inertial frame. Our analysis has been focused mainly on the problem of existence of a proper position operator for massive particles. We have found the explicit form of the position operator and have demonstrated that in the preferred frame our operator coincides with the Newton–Wigner one. Moreover, full algebra of observables consisting of position operators and fourmomentum operators is manifestly Poincaré covariant in this framework. Our results support expectations of other authors (Bell [1], Eberhard [3]) that a consistent formulation of quantum mechanics demands existence of a preferred frame.

DOI
10.1063/1.57125

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In this paper the localization problem in the relativistic quantum mechanics is considered in the framework of a nonstandard synchronization scheme for clocks. Such a synchronization preserves Poincaré covariance but (at least formally) distinguishes an inertial frame. Our analysis has been focused mainly on the problem of existence of a proper position operator for massive particles. We have found the explicit form of the position operator and have demonstrated that in the preferred frame our operator coincides with the Newton–Wigner one. Moreover, full algebra of observables consisting of position operators and fourmomentum operators is manifestly Poincaré covariant in this framework. Our results support expectations of other authors (Bell [1], Eberhard [3]) that a consistent formulation of quantum mechanics demands existence of a preferred frame.
In this paper the relativistic quantum mechanics (QM) is considered in the framework of the nonstandard synchronization scheme, which preserves the Poincar\'e covariance but (at least formally) distinguishes an inertial frame. This enables one to avoid the problem with a strong formulation of local causality related to the breaking of Bell's inequalities in QM. Our analysis has been focused mainly on the problem of the existence of a proper position operator for massive particles. We have proved that in our framework such an operator exists for particles with arbitrary spin. This operator is Hermitian and covariant, it has commuting components, and its eigenvectors (localized states) are covariant too. We have found an explicit form of the position operator and demonstrated that it coincides with the Newton-Wigner one in the preferred frame. We have also defined a covariant spin operator and constructed an invariant spin squared operator. Moreover, full algebra of observables consisting of position, four-momentum, and spin operators is manifestly Poincar\'e covariant in this framework. Our results support expectations of other authors [J. S. Bell, in Quantum Gravity, edited by C. J. Isham, R. Penrose, and D. W. Sciama (Oxford University Press, Oxford, 1981), p. 611; P. H. Eberhard, Nuovo Cimento B 46, 392 (1978)] that a consistent formulation of quantum mechanics demands the existence of a preferred frame.
A review is given of a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. It is formulated in global non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories and then restricted to the intrinsic inertial rest frame of the isolated system. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare' algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. This implies that the Hilbert space must be the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one.
The present series of papers is devoted to the localization problem in relativistic quantum mechanics. The consequences of imposing the condition that the descriptions of a localized state seen from two different frames of reference should be physically consistent were investigated in Paper II. Only instantaneous localization was studied. Considering elementary systems of nonzero mass and spin 0 and \textonehalf{}, the following results were found. (a) As regards the localization problem, one cannot avoid accepting at least one of the following strong departures from the usual ideas: (i) Position has no meaning; (ii) it violates the physical equivalence of inertial frames; (iii) it is the only quantum variable which cannot be represented by an operator; (iv) it is non-Hermitian; (v) some unusual interaction effects do not disappear when the interaction is switched off. (b) If a component of position is to be precisely measurable as a point, then for spin 0 there is only one possible position operator; for spin \textonehalf{} the operator is unique up to a parameter; in both cases the operators are non-Hermitian with respect to the relativistically invariant scalar product, but in spite of this the eigenvalues of all components are real. The components of position are compatible with each other for spin 0 and incompatible for spin \textonehalf{}. The communication relations of position with linear momentum are the standard ones. The velocity operator, which is Hermitian, has the expected form. Several authors have stated that the localization problem can have a solution for mass greater than zero, but for the zero-mass case a solution does not exist. In this paper the localization problem is studied for zero-mass elementary systems by only imposing, as in Paper II, the above requirement of physical consistency. We consider systems with spin 0, \textonehalf{}, and 1. We prove that the consequences are essentially the same as those obtained for systems of mass greater than zero. The parameter is no longer arbitrary.
Although the concept of localization in non-relativistic quantum mechanics is mathematically well-defined there is no obvious and unambiguous way to generalize it to relativistic quantum theory. After a brief review of localization in quantum mechanics the Newton-Wigner localization scheme is introduced and it is shown by the example of a massive spinless system how it leads to a position operator with seemingly unphysical properties. This motivated the development of several theorems which claim to rule out the existence of localizable particles. A particularly important one of them is presented and its validity discussed. In the last section however it is shown that if localization is considered with respect to spacelike hyperplanes, the properties of the Newton-Wigner position operators are not necessarily unphysical and localizable particles can exist.
The idea of the preferred frame as a remedy for difficulties of the relativistic quantum mechanics in description of the non-local quantum phenomena was undertaken by physicists such as J. S. Bell and D. Bohm. The possibility of the existence of a preferred frame was also seriously treated by P. A. M. Dirac. In this paper, we propose an Einstein-Podolsky-Rosen–type experiment for testing the possible existence of quantum preferred frame. Our analysis suggests that to verify whether a preferred frame of reference in the quantum world exists, it is enough to perform an EPR-type experiment with a pair of observers staying in the same inertial frame and with use of the massive EPR pair of spin–one-half or spin-one particles.
A new formulation of relativistic quantum mechanics is proposed in the framework of the rest-frame instant form of dynamics, where the world-lines of the particles are parametrized in terms of the Fokker-Pryce center of inertia and of Wigner-covariant relative 3-coordinates inside the instantaneous Wigner 3-spaces, and where there is a decoupled (non-covariant and non-local) canonical relativistic center of mass. This approach: (a) allows us to make a consistent quantization in every inertial frame; (b) leads to a description of both bound and scattering states; (c) offers new insights on the relativistic localization problem; (d) leads to a non-relativistic limit with a Hamilton-Jacobi treatment of the Newton center of mass; (e) clarifies non-local aspects (spatial non-separability) of relativistic entanglement connected with Lorentz signature and not present in its non-relativistic treatment.