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Physical Review

Lorentz Invariant Localized States

T. O. Philips

Published Nov 9, 1964• 45 citations
Publisher

Abstract

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."

DOI
10.1103/physrev.136.b893

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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.

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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."
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.
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.
Introduction: localization and Lorentz-invariant Quantum Theory The current status of localization and related concepts, especially localized statevectors and position operators, within Lorentz-invariant Quantum Theory (LIQT) is ambiguous and controversial. Ever since the early work of Newton & Wigner (1949), and the subsequent extensions of their work, particularly by Hegerfeldt (1974, 1985), it has seemed impossible to identify localized statevectors or position operators in LIQT that were not counterintuitive – strange – in one way or another; the most striking strange property being the superluminal propagation of the localized states. The ambiguous and controversial status of these concepts arises from the varied reactions that workers have to the strange properties. Some regard them, particularly the superluminal propagation, as utterly unacceptable, and conclude that no precise concepts of localized statevectors and position operators exist in LIQT (Wigner 1973, pp. 325–7; 1983, pp. 310–13; Malament 1996). Others downplay the whole issue, on the grounds that current theoretical and experimental practice has no need of sharply formulated concepts of localization, localized states etc. (Birrell & Davies 1984, pp. 48–59; Haag 1992, p. 34). Still others, including ourselves, believe that the superluminal propagation does not lead to causal contradictions, and is not in conflict with available empirical data: so that it should not be ruled out. We also believe that the other strange properties are merely unfamiliar novelties of LIQT which we must simply learn to accept. (Like the superluminal propagation, they do not lead to causal contradictions, nor conflict with available data.) So in this essay, we aim to help remove the ambiguous and controversial status of localization concepts in LIQT.
Does the measurement of a quantum system necessarily break Lorentz invariance? We present a simple model of a detector that measures the spacetime localization of a relativistic particle in a Lorentz invariant manner. The detector does not select a preferred Lorentz frame as a Newton-Wigner measurement would do. The result indicates that there exists a Lorentz invariant notion of quantum measurement and sheds light on the issue of the localization of a relativistic particle. The framework considered is that of single-particle mechanics as opposed to field theory. The result may be taken as support for the interpretation postulate of the spacetime-states formulation of single-particle quantum theory.