On Newton-Wigner localization and superluminal propagation speeds

J. E. Ingall

R. M. Mir‐Kasimov

Nicola Cufaro Petroni, Jean Pierre Vigier

Daniele Colosi, Robert Oeckl

Abstract The Newton-Wigner states and operator are widely accepted to provide an adequate notion of spatial localization of a particle in quantum field theory on a spacelike hypersurface. Replacing the spacelike with a timelike hypersurface, we construct one-particle states of massive Klein-Gordon theory that are localized on the hypersurface in the temporal as well as two spatial directions. This addresses the longstanding problem of a “time operator" in quantum theory. It is made possible by recent advances in quantization on timelike hypersurfaces and the introduction of evanescent particles. As a first application of time-localized states, we consider the time-of-arrival problem. Our results are in accordance with semiclassical expectations of causal propagation of massless and massive particles. As in the Newton-Wigner case, localization is not perfect, but apparent superluminal propagation is exponentially suppressed.

Gordon N. Fleming, Jeremy Butterfield

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.

Francis S. G. Von Zuben

Two related problems in relativistic quantum mechanics, the apparent superluminal propagation of initially localized particles and dependence of spatial localization on the motion of the observer, are analyzed in the context of Dirac’s theory of constraints. A parametrization invariant formulation is obtained by introducing time and energy operators for the relativistic particle and then treating the Klein–Gordon equation as a constraint. The standard, physical Hilbert space is recovered, via integration over proper time, from an augmented Hilbert space wherein time and energy are dynamical variables. It is shown that the Newton–Wigner position operator, being in this description a constant of motion, acts on states in the augmented space. States with strictly positive energy are nonlocal in time; consequently, position measurements receive contributions from states representing the particle’s position at many times. Apparent superluminal propagation is explained by noting that, as the particle is potentially in the past (or future) of the assumed initial place and time of localization, it has time to propagate to distant regions without exceeding the speed of light. An inequality is proven showing the Hegerfeldt paradox to be completely accounted for by the hypotheses of subluminal propagation from a set of initial space–time points determined by the quantum time distribution arising from the positivity of the system’s energy. Spatial localization can nevertheless occur through quantum interference between states representing the particle at different times. The nonlocality of the same system to a moving observer is due to Lorentz rotation of spatial axes out of the interference minimum.