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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Lorentz-Invariant Localization for Elementary Systems. III. Zero-Mass Systems

Andrés J. Kálnay

Published May 15, 1971• 15 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.3.2357

Related papers

Lorentz-Invariant Localization for Elementary Systems. II

Andrés J. Kálnay

The purpose of this paper is to find to what extent the notion of position in the relativistic quantum mechanics of free elementary systems of nonzero mass is defined (or precluded) by the requirement that the descriptions of a localized state as seen from two observers in different frames of reference should be physically consistent. Neither the existence of a time component of the position operator nor the compatibility of the space components of position is assumed. Hermiticity with respect to the quantum-relativistic scalar product in state-vector space is neither assumed nor rejected. There are three possibilities: (i) The position is represented as usual by a Hermitian operator; this is the point-type position. (ii) Non-Hermitian operators are accepted, and the $k$ component of position can represent segments of the $k$ axis with length of the order of $\frac{1}{m}$; when the operator is not only non-Hermitian, but also non-normal, this is the extended-type position. (iii) Non-Hermitian operators are accepted, but the segments are reduced to points; this is the limiting case of the extended-type position. Case (ii) is not studied in this paper. General conditions are found for any spin as regards cases (i) and (iii), and their consequences are obtained for spin 0 and spin \textonehalf{}. The results are the following: There is no position operator in case (i), but in case (iii) the position operator is unique for spin 0 and unique up to a parameter for spin \textonehalf{}. For both spins, in spite of the lack of relativistic Hermiticity, the eigenvalues of position are real, and the velocity is the expected one. The eigenstates of one component of the position are found. The components of the position are compatible with each other for spin 0, and incompatible for spin \textonehalf{}. In the former case, the simultaneous eigenstates of all components are Philip's class-I localized states. Related work is discussed.

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Journal of Physics Conference Series•Apr 22, 2013•4 citations
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.
The purpose of this paper is to find to what extent the notion of position in the relativistic quantum mechanics of free elementary systems of nonzero mass is defined (or precluded) by the requirement that the descriptions of a localized state as seen from two observers in different frames of reference should be physically consistent. Neither the existence of a time component of the position operator nor the compatibility of the space components of position is assumed. Hermiticity with respect to the quantum-relativistic scalar product in state-vector space is neither assumed nor rejected. There are three possibilities: (i) The position is represented as usual by a Hermitian operator; this is the point-type position. (ii) Non-Hermitian operators are accepted, and the kkk component of position can represent segments of the kkk axis with length of the order of 1m\frac{1}{m}m1​; when the operator is not only non-Hermitian, but also non-normal, this is the extended-type position. (iii) Non-Hermitian operators are accepted, but the segments are reduced to points; this is the limiting case of the extended-type position. Case (ii) is not studied in this paper. General conditions are found for any spin as regards cases (i) and (iii), and their consequences are obtained for spin 0 and spin \textonehalf{}. The results are the following: There is no position operator in case (i), but in case (iii) the position operator is unique for spin 0 and unique up to a parameter for spin \textonehalf{}. For both spins, in spite of the lack of relativistic Hermiticity, the eigenvalues of position are real, and the velocity is the expected one. The eigenstates of one component of the position are found. The components of the position are compatible with each other for spin 0, and incompatible for spin \textonehalf{}. In the former case, the simultaneous eigenstates of all components are Philip's class-I localized states. Related work is discussed.
In parts II and III of the present series of papers, the localization problem was studied for the relativistic nonzero-mass (spin 0 and \textonehalf{}) and zero-mass (spin 0, \textonehalf{}, and 1) cases. With a set of postulates of which the basic one was the imposition of Lorentz invariance of localization (which is a self-consistency requirement), it was possible to define the position operator uniquely (except for a constant for spin \textonehalf{}, nonzero mass). However, the position operator has unusual properties, e.g., non-Hermiticity. This does not preclude its having physical meaning, but is such a strange result that in the present paper we test the validity of the postulates in the nonrelativistic case (spin 0 and \textonehalf{}) where the position operator is well known. We use the same set of postulates, except that here we impose the Galilean invariance of localization. The result of the test is positive; the position operator found is the usual one. Moreover, with our set of postulates, we restate the notion of localization in a more physical basis, giving a precise definition of it without imposing avoidable requirements.
In parts II and III of the present series of papers, the localization problem was studied for the cases of relativistic nonzero mass (spin 0, 12\frac{1}{2}21​) and zero mass (spin 0, 12\frac{1}{2}21​, and 1). With a set of postulates, of which the basic one is the imposition of Lorentz invariance of localization (LIL) (which is a self-consistency requirement), it was possible to define the position operator uniquely (except for a constant in the spin-12\frac{1}{2}21​, nonzero-mass case). For the determination of the position operators it was enough to consider only very special cases of homogeneous Lorentz transformations. One of the purposes of the present paper is to consider the effect of arbitrary inhomogeneous continuous Lorentz transformations in order to obtain general Poincar\'e-invariant localization (GPIL). Our main result will be that localization is LIL if and only if it is GPIL. In II and III, LIL was obtained in spite of the fact that the position operators were not 4-vector operators but only 3-vector operators. In this paper, following Fleming, we construct formally covariant position operators from the 3-vector operators. We prove that their eigenfunctions are just the localized states defined in the GPIL sense.
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.
The problem of localization of relativistic particle in quantum physics originates from the fact that position variable is non-Hermitian in nature. Kalney1 in a series of papers, has tried to construct a position operator for a massive as well as massless particle in a unified way incorporating Lorentz invariance. His observation of non-Hermitian and non-normal nature of position operator suggests that it gives rise to the possibility of getting more information from complex eigenvalues. Kalney and Toledo2 have considered that a relativistic particle may not be a point particle and position be extended to certain region. Non-Hermitian nature of position variable is suitable for description of such extended body as each coordinate is associated with a pair of variables.
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.