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Physical Review

Lorentz-Invariant Localization for Elementary Systems

Juan Carlos Gallardo, Andrés J. Kálnay, Salomon Risemberg

Published Jun 25, 1967• 21 citations
Publisher

Abstract

Philips has axiomatically defined sets of localized states which are Lorentz-invariant (while Newton-Wigner sets of localized states are not) and has divided these sets into three classes. Philips's conjecture concerning spin-zero localized states, namely, that the postulates define the sets of localized states in a unique way, is proved to be incorrect by finding a class-III set that satisfies Philips's postulates. Philips's work is discussed and his calculations (spin-zero case) are repeated without using some (explicit and implicit) unnecessary hypotheses. These calculations are also extended to the spin-\textonehalf{} case, for which it is proved that there are only class-III sets of localized states. The results are discussed. Incidentally, an explicit form of the effects induced by a Lorentz transformation on representation space is found for both the spin-zero and spin-\textonehalf{} cases.

DOI
10.1103/physrev.158.1484

Related papers

Lorentz Invariant Localized States

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Newton and Wigner have previously discussed the definition of "localized states" in terms of invariance conditions. However, their localization conditions are not Lorentz covariant. The present work presents a modified set of invariance postulates which includes a Lorentz invariance condition. It is shown that for spinless systems there do exist states satisfying the modified set of postulates; these states are calculated explicitly. The procedure appears to preclude the existence of Lorentz covariant "position operators."

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Physics of Particles and Nuclei Letters•Oct 1, 2006•5 citations
Philips has axiomatically defined sets of localized states which are Lorentz-invariant (while Newton-Wigner sets of localized states are not) and has divided these sets into three classes. Philips's conjecture concerning spin-zero localized states, namely, that the postulates define the sets of localized states in a unique way, is proved to be incorrect by finding a class-III set that satisfies Philips's postulates. Philips's work is discussed and his calculations (spin-zero case) are repeated without using some (explicit and implicit) unnecessary hypotheses. These calculations are also extended to the spin-\textonehalf{} case, for which it is proved that there are only class-III sets of localized states. The results are discussed. Incidentally, an explicit form of the effects induced by a Lorentz transformation on representation space is found for both the spin-zero and spin-\textonehalf{} cases.
Newton and Wigner have previously discussed the definition of "localized states" in terms of invariance conditions. However, their localization conditions are not Lorentz covariant. The present work presents a modified set of invariance postulates which includes a Lorentz invariance condition. It is shown that for spinless systems there do exist states satisfying the modified set of postulates; these states are calculated explicitly. The procedure appears to preclude the existence of Lorentz covariant "position operators."
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 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.
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.