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.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.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsFeb 15, 197027 citations

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

Andrés J. Kálnay

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 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.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMay 15, 197115 citations

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.