The author has earlier defined the quantum two-time localization problem as the minimizing of a particle's position spread about specified points at two distinct times. In the present article optimum localization is found for relativistic massive free particles. For short time intervals, spreading necessarily occurs at the speed of light while for long times the previously found diffusion-like behaviour is recovered. In defining relativistic localization, use is made of the work of Newton and Wigner (1949); in particular, their restriction to the positive energy hyperboloid is found to be necessary to recover the nonrelativistic limit of wavepacket spreading.
The author has earlier defined the quantum two-time localization problem as the minimizing of a particle's position spread about specified points at two distinct times. In the present article optimum localization is found for relativistic massive free particles. For short time intervals, spreading necessarily occurs at the speed of light while for long times the previously found diffusion-like behaviour is recovered. In defining relativistic localization, use is made of the work of Newton and Wigner (1949); in particular, their restriction to the positive energy hyperboloid is found to be necessary to recover the nonrelativistic limit of wavepacket spreading.
Although the concept of localization in non-relativistic quantum mechanics is mathematically well-defined there is no obvious and unambiguous way to generalize it to relativistic quantum theory. After a brief review of localization in quantum mechanics the Newton-Wigner localization scheme is introduced and it is shown by the example of a massive spinless system how it leads to a position operator with seemingly unphysical properties. This motivated the development of several theorems which claim to rule out the existence of localizable particles. A particularly important one of them is presented and its validity discussed. In the last section however it is shown that if localization is considered with respect to spacelike hyperplanes, the properties of the Newton-Wigner position operators are not necessarily unphysical and localizable particles can exist.
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.
Constructing observables that describe the localization of relativistic particles is an important foundational problem in relativistic quantum field theory (QFT). The description of localization in terms of single-time observables leads to conflict with the requirement of causality. In this paper, we describe particle localization in terms of time-of-arrival observables, defined in terms of the interaction between a quantum field and a measuring apparatus. The resulting probabilities are linear functionals of QFT correlation functions. Specializing to the case of a scalar field, we identify several time-of-arrival observables differing on the way that the apparatus localizes particle-detection records. Maximum localization is obtained for a unique observable that is related to the Newton-Wigner position operator. Finally, we define a measure of localizability for relativistic particles in terms of a novel time-energy uncertainty relation for the variance of the time of arrival.
Abstract We challenge the widespread belief, originated by Newton and Wigner (1949 Rev. Mod. Phys. 21 400) that the incorporation of special relativity into quantum mechanics implies that a massive particle cannot be localized within an arbitrarily small spatial extent, that there is a minimum width approximately equal to the Compton wavelength. Our argument is in four parts. First, the scalar function used by Newton and Wigner as a measure of localization is not a position probability amplitude. The correct relativistic position probability amplitude becomes a delta function for a state vector localized according to the criteria of Newton and Wigner. Second, the possibility of Lorentz contraction as observed from a boosted frame means that the wavepacket width in the boost direction can take arbitrarily small values. Third, we refer to the work of Almeida and Jabs (1984 Am. J. Phys. 52 921) who show that the long time wavepacket spreading rate for relativistic position probability amplitudes is always less than the speed of light no matter how small the initial width of the wavepacket. Lastly, we show that it is a simple matter to construct scalar amplitudes with spatial widths smaller than the supposed minimum.
Two related problems in relativistic quantum mechanics, the apparent superluminal propagation of initially localized particles, and the dependence of their localization on the motion of the observer, are analyzed in the context of the theory of constraints. Time and energy operators are introduced for the free relativistic particle, and a parametrization invariant formulation is obtained through Dirac constraint theory. The resulting description is of a system constrained in momentum and energy, but not in position or time, for which observables are constants of the motion. The Klein-Gordon equation is recovered on a physical Hilbert space, constructed via integration over the proper time from an augmented Hilbert space, wherein time and energy are dynamical variables...
The concept of position is a basic one in our construction of a model of thenphysical world: yet the theory of localisation in quantum mechanics is fraughtnwith difficulties, especially in the relativistic case. A strictly localised state forna nonrelativistic particle cannot be described by an ordinary function. In thenrelativistic case the problems are manifold. For a free electron, using the Diracnposition operator x, we cannot construct positive-energy states which localisenthe particle within a compact region. Furthermore it has been suggested that anconsistent one-particle description of localisation on a scale small compared tonthe Compton wavelength of the particle is ruled out because pair-productionnwould take place. These matters are dicussed at greater length in Chaptern1, where several proposed localisation schemes (that is, sets of definitions ofnlocalised states and position operators) are reviewed: none of these is deemednwholly satisfactory. Introductory remarks are also made in this chapter aboutnthe remarkable phenomenon of 'probability backflow', which is discussed further below, and which is relevant to our discussion of localisation.In Chapter 2 we show that while strict localisation of the free Dirac particle is not describable within the usual mathematical formalism, in terms of Dirac's position operator x it is possible to describe sequences of positive-energy states whose spread Dx = ((x -- x0)2) about a point X0 approaches zero.In Chapter 3 the component states of these sequences are used as the basis of a full causally well-behaved solution to the localisation problem for the free electron, with natural space-time transformation properties. It is shown that although x is not an observable in the usual sense, the four-vector density (p(x, t), j(x, t)/c) is observable, and can be localised arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalised at the same time.n n